From objects to diagrams for ranges of functors

1 1 Pierre Gillib ert and F riedric h W ehrung Address (Gilliber t): Charles Universit y in Prague F aculty of Mathematics and Physics Department of Algebra Sokolo vsk ´ a 83 186 0 0 P raha Czech Republic e-mail (Gillib ert): gillib er@ka rlin.mff.cuni.cz , pgillibert @yahoo.fr URL (Gilliber t): htt p://ww w.mat h.unicaen.fr/~giliberp/ Address (W ehrung): LMNO, CNRS UMR 6139 D ´ epartement de Math´ ematiques, BP 518 6 Univ er sit´ e de C a en, Ca mpus 2 14032 Ca en cedex F rance e-mail (W ehr ung): wehru ng@ma th.uni caen.fr , fwehru ng@ya hoo.fr URL (W ehrung): http ://ww w.math .unicaen.fr/~wehrung/ F rom ob jects to diagra ms for ranges of functors Septem b er 25, 2018 Springer ∗ The ﬁrst autho r was partially supported by the institutional grant MSM 0021 620839 3 2010 Mathematics Subje ct Classiﬁc ation : Primary 18 A30, 18A25, 18A20, 18A35; Secondary 03E05, 05D10, 06A07, 06A12, 06 B 20, 08B1 0 , 08A30, 08B25 , 08C15 , 20E10. Key w ords: Category; functor; larder; lifter; condensate; L¨ owenheim-Sk olem Theorem; w eakly present ed; Armature Lemma; Buttress Lemma; Condens a te Lifting Lemma; Kuratowski’s F ree Set Theorem; lifter; Erd˝ os cardinal; c r iti- cal point; product; colimit; epimorphism; monomorphism; section; retraction; retract; pr o jection; pro jectable; ps eudo join- semilattice; a lmost jo in-semilat- tice; pro jectabilit y witness; quasiv ariety; semilattice; lattice; congr uence; dis- tributive; mo dular Con tents 1 Bac kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1-1 Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1-1.1 The search for functorial solutions to certain representation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1-1.2 Partially functorial solutions to repr esentation problems 15 1-1.3 Conten ts of the b o ok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1-1.4 How not to read the b o ok . . . . . . . . . . . . . . . . . . . . . . . . . 22 1-2 Ba sic co ncepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1-2.1 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1-2.2 Stone duality for Bo o lean a lgebras . . . . . . . . . . . . . . . . . . 25 1-2.3 Partially or dered sets (p osets) a nd lattices . . . . . . . . . . . 25 1-2.4 Catego ry theor y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1-2.5 Directed co limits of ﬁr st-order structures . . . . . . . . . . . . 30 1-3 K appa-presented and weakly k appa-pr e sented ob jects . . . . . . . 33 1-4 E xtension of a functor b y directed colimits . . . . . . . . . . . . . . . . . 35 1-5 P ro jectability witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 Bo olean algebras scaled with respe ct to a p oset . . . . . . . . . . . 45 2-1 P seudo join-semila ttices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2-2 P -no r med spaces, P -scale d Bo o le an algebr as . . . . . . . . . . . . . . . 48 2-3 Direc ted colimits and ﬁnite pr o ducts in Bo ol P . . . . . . . . . . . . . 51 2-4 Finitely pr esented P -scaled Bo o lean a lgebras . . . . . . . . . . . . . . . 53 2-5 Nor mal mo rphisms of P -sca led Bo olea n alg ebras . . . . . . . . . . . 55 2-6 Nor m-cov erings of a p oset; the s tr uctures 2 [ p ] a nd F ( X ) . . . . . 57 3 The Condens ate Lifting Lemma (CLL) . . . . . . . . . . . . . . . . . . . . 61 3-1 The functor A 7→ A ⊗ → S ; condensates . . . . . . . . . . . . . . . . . . . . . 61 3-2 Lifters and the Arma ture Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 64 3-3 The L¨ owenheim-Sk olem Condition and the Buttress Lemma . 68 3-4 La rders a nd the Condens ate Lifting L e mma . . . . . . . . . . . . . . . . 70 3-5 Inﬁnite co m binato rics and lambda-lifters . . . . . . . . . . . . . . . . . . 73 5 6 Con tent s 3-6 Lifters , retr acts, and pse udo -retrac ts . . . . . . . . . . . . . . . . . . . . . . 80 3-7 Lifting dia grams without assuming lifter s . . . . . . . . . . . . . . . . . . 84 3-8 Left a nd rig h t larder s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4 Larders from ﬁrst-order structures . . . . . . . . . . . . . . . . . . . . . . . . 89 4-1 The category of all monoto ne-indexed structures . . . . . . . . . . . 90 4-2 Direc ted colimits of mo no tone-indexed str uctures . . . . . . . . . . . 93 4-3 Co ngruence lattices in gener alized q uasiv arieties . . . . . . . . . . . . 96 4-4 P reserv ation of directed colimits for co ngruence la ttices . . . . . 101 4-5 Idea l-induced mor phisms a nd pro jectabilit y witnesses . . . . . . . 103 4-6 An extens ion o f the L¨ owenheim-Sk olem Theor em . . . . . . . . . . . 106 4-7 A dia g ram version of the Gr¨ atzer-Schmidt Theorem . . . . . . . . . 10 8 4-8 Right ℵ 0 -larders from ﬁrst-or der structure s . . . . . . . . . . . . . . . . 112 4-9 Rela tive critical p o ints b etw een quas iv arieties . . . . . . . . . . . . . . 11 5 4-10 Finitely gener ated v arieties of algebr as . . . . . . . . . . . . . . . . . . . . 121 4-11 A p otential use of larder s on non- r egular car dinals . . . . . . . . . . 122 5 Congruence-preserving e xtensions . . . . . . . . . . . . . . . . . . . . . . . . 125 5-1 The category of semila ttice-metric s paces . . . . . . . . . . . . . . . . . . 12 6 5-2 The category of all semilattice- metric covers . . . . . . . . . . . . . . . 12 7 5-3 A family of unliftable squa r es of semila ttice-metric spaces . . . 128 5-4 A left la r der inv olving a lg ebras and se mila ttice-metric spaces . 132 5-5 CP CP-retra cts and CP C P -extensions . . . . . . . . . . . . . . . . . . . . . 13 4 6 Larders from v on Neumann regular rings . . . . . . . . . . . . . . . . . 13 9 6-1 Idea ls o f reg ular ring s a nd of la ttices . . . . . . . . . . . . . . . . . . . . . . 139 6-2 Right lar ders from regular rings . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 Sym b ol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Sub ject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Author inde x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 F orew ord The aim of the present work is to intro duce a g e neral metho d, applicable to v arious ﬁelds o f mathematics, that enables us to g ather infor ma tion on the range o f a functor Φ , thus making it p oss ible to s olve prev iously intractable representation problems with resp ect to Φ . This method is esp ecia lly eﬀec- tive in ca se the problems in question a re “cardina lit y-s ensitive”, that is , an analogue of the cardinality function turns out to pla y a crucial role in the description of the members of the rang e of Φ . Let us ﬁrst give a few exa mples of such problems. The ﬁrst three b elong to the ﬁeld of universal alg ebra, the fourth to the ﬁeld of ring theor y (nonsta ble K-theory of rings). Con text 1. The cla s sical Gr¨ atzer -Schmidt The o rem, in universal alg ebra, states that every ( ∨ , 0 )-semilattice is isomorphic to the compact congru- ence lattice of some a lgebra. Can this result b e extended to diagr ams of ( ∨ , 0)-s emilattices? Con text 2. F or a member A of a quasiv ariety A o f a lgebraic sys tems , we denote b y Con A c A the ( ∨ , 0)-semilattice of all compact elements of the lattice of all congruences of A with quotient in A ; further, we denote by Con c , r A the class of all isomorphic copies of Co n A c A where A ∈ A . F or qua s iv arieties A and B of a lg ebraic s y stems, we denote by crit r ( A ; B ) ( r elative critic al p oint b etw een A and B ) the least poss ible car dinality , if it exists, of a mem b er of (Con c , r A ) \ (Con c , r B ), a nd ∞ otherwise. What are the p ossible v alues o f cr it r ( A ; B ), say for A a nd B both with ﬁnite language? Con text 3. Let V be a nondistributive v ariety of lattices and let F be the free la ttice in V o n ℵ 1 generator s. Doe s F hav e a congrue nc e - pe rmutable, congruence- preserving extension? Con text 4. Let E b e an exchange ring. Is there a (von Neumann) reg ular ring, o r a C*-alge bra of r eal r ank zero, R with the same nonstable K-theor y as E ? 7 8 F oreword It turns o ut that each of these pro ble ms can b e r educed to a catego ry- theoretical problem of the following gener al kind. Let A , B , S be categories, let Φ : A → S and Ψ : B → S b e functors. W e are also given a s ub ca tegory S ⇒ of S , of which the ar rows will b e ca lled double arr ows and written f : X ⇒ Y . W e assume that for “many” obje cts A of A , there are an ob ject B of B and a double arrow χ : Ψ ( B ) ⇒ Φ ( A ). W e also need to a ssume that our ca tegorica l data forms a so -called lar der . In such a case, w e establish that under certain combinatorial assumptions o n a p oset P , for “ many” diagr ams − → A =  A p , α q p | p ≤ q in P  from A , a similar conclusion holds a t the diagra m Φ − → A , that is, there a re a P - indexed dia gram − → B fr o m B and a double arrow − → χ : Ψ − → B ⇒ Φ − → A from S P . The combinatorial assumptions on P imply tha t every principa l ideal of P is a join-semilattice and the set of all uppe r b ounds of any ﬁnite subset is a ﬁnitely gener ated upp er subset. W e argue by concentrating all the rele v an t proper ties of the diagra m − → A int o a c ondensate of − → A , which is a special kind of dir ected colimit of ﬁnite pro ducts of the A p for p ∈ P . Our ma in r esult, the Condensate Lifting L emma (CLL), reduces the liftabilit y o f a diagr am to the liftability of a condensate, mo dulo a list o f elementary v eriﬁca tions o f ca tegorical na ture. The impa ct of CLL o n the four problems ab ov e ca n b e summar ized a s follows: Con text 1. The Gr¨ atzer-Schmidt Theor em can b e extended to any di- agram o f ( ∨ , 0)-semila ttices a nd ( ∨ , 0)-homomor phisms indexed by a ﬁ- nite p oset (r esp., assuming a pro per cla ss of E rd˝ os car dinals, an arbitr ary po set), lifting with algebr as of v ariable simila rity t yp e. Con text 2. W e prove that in a host of s ituations, either cr it r ( A ; B ) < ℵ ω or crit r ( A ; B ) = ∞ . This holds, in particular, if A is a lo cally ﬁnite quasiv ariety with ﬁnitely man y relations while B is a ﬁnitely generated, congruence- mo dular v ariety with ﬁnite similarity type of algebr as o f ﬁnite t yp e (e.g., gr oups , lattic es , mo dules over a ﬁnite ring ). Con text 3. The free V - la ttice F has no co ngruence-p ermutable, congru- ence-preser ving extensio n in any similarity type con taining the lattice t yp e. Due to earlier work by Gr¨ atzer, Laks er, and W ehrung [3 0], if V is lo cally ﬁnite, then the cardinality ℵ 1 is o ptimal. Con text 4. By using the results of the present w ork, the second author prov es in W ehrung [71] that the answer is no (both for regular rings and for C*-algebras of real rank zer o), with a co un tere x ample of cardinalit y ℵ 3 . W e als o pave the wa y to solutions of further befor ehand intractable op en problems: — the determination o f all the p oss ible critica l points b etw een ﬁnitely gen- erated v a rieties of lattices (Gillibert [20]), then b etw een v a rieties A and B of algebr a s such A is lo cally ﬁnite while B is ﬁnitely generated with ﬁnite similarity type and omits tame congruence theory types 1 and 5 (Gilli- ber t [22 ]). F oreword 9 — the pr oblem whether every lattice o f c ardinality ℵ 1 , in the v ariet y g en- erated by M 3 , has a co ngruence m -p ermutable, co ngruence-pre s erving extension for some p ositive integer m (Gilliber t [21]); — a 1962 pr oblem by J´ onsson ab out co o rdinatizability of sectionally com- plement ed mo dula r la ttices without unit (W ehrung [69]). Chapter 1 Bac kground 1-1 Introduct ion The present work o riginates in a collection of attempts to solving v arious op en problems, in diﬀerent topics in mathematics (mainly , but not restr ic ted to, universal algebr a and lattice theory ), all rela ted by a co mmon feature : How lar ge c an the r ange of a functor b e? If it is lar ge on obje cts, then is it also lar ge on diagr ams? “Larg e ness” is mean t her e in the sense of con tainment (not car dinality): for example, lar g eness of a cla ss C of sets co uld mean that C co ntains, a s a n element, ev er y ﬁnite set; or ev ery countable set; or e very cartesian pro duct of 17 sets; a nd so on. Our primary in terests are in lattic e the ory , u niversal algebr a , a nd ring the ory ( nonstable K-t he ory ). O ur further aims include, but are not restricted to, gr oup t he ory and mo dule the ory . One o f our main to ols is inﬁnite c ombinatori cs , articulated around the new notio n of a λ -lifter . Our main g oal is to introduce a new metho d, of categorica l nature, orig - inally designed to solve functorial r epresentation problems in the ab ove- men tioned topics. Our main result, which we call the Condensate Lifting L emma , CLL in abbreviation (Lemma 3-4 .2), a s well as its main precursor the Armatur e L emma (Lemma 3-2 .2), are theorems o f c ate gory the ory . They enable us to solve several until now seemingly intractable op en problems , outside the ﬁeld of categ ory theory , by re ducing them to the veriﬁcation of a list of mo stly elementary categor ical prop er ties. The large st part of this work b e longs to the ﬁeld of categor y theory . Nev- ertheless, its prime in tent is strong ly oriented tow ar ds outside applications, including the afore men tioned topics. Hence, unlike most w ork s in category theory , these notes will sho w at some places mo re detail in the s ta temen ts and pr o ofs o f the purely categor ical results. 11 12 1 Background 1-1.1 The s e ar ch for functorial s olutions to c ertain r epr esentation pr oblems Roughly speaking , the kind of problem that the prese n t bo ok aims to help solving is the follo wing. W e are given categ ories B and S toge ther with a functor Ψ : B → S . W e a r e trying to desc rib e the mem b ers of the (categor ic al) range of Ψ , that is, the ob jects of S that ar e isomorphic to Ψ ( B ) for some ob ject B of B . Our metho ds are likely to s he d some light on this problem, even in some cases solving it completely , mainly in case the answer turns out to b e c ar dinality-sensitive . At this p oint, cardinality is meant so mewhat heuristically , inv olving an a ppropriate notion of “size” for o b jects of S . Y et we need to b e more s pe c iﬁc in the form ulation of our general proble m, th us inevitably adding some complexit y to our statemen ts. This requires a slight recasting of our problem. W e are now given categories A , B , S together with functors Φ : A → S and Ψ : B → S , as illustr ated on the left hand side of Figure 1.1. S S A Φ ? ?        B Ψ _ _ ? ? ? ? ? ? ? A Φ ? ?        Γ / / B Ψ _ _ ? ? ? ? ? ? ? Fig. 1. 1 A few catego ri es and functors W e assume that for “many” o b jects A of A , there exis ts an ob ject B of B such that Φ ( A ) ∼ = Ψ ( B ). W e ask to what extent the assig nmen t ( A 7→ B ) can b e made fun ctorial . Ideally , we would b e able to prov e the existence of a functor Γ : A → B such that Φ and Ψ ◦ Γ ar e iso morphic functors—then we say that Γ lifts Φ with r esp e ct to Ψ , as illustrated on the right hand side of Figure 1.1. (Tw o functors a re isomorph ic if there is a natural transformation from o ne to the other whose comp onents are all isomo rphisms). This highly des irable goal is se ldom rea ched. Here are a f ew instances where this is nevertheless the case (and then this never ha ppe ns for trivia l rea sons), and some other instances wher e it is still not known whether it is the case. In b oth E x amples 1-1.1 and 1-1.2, S is the ca tegory of all distributive ( ∨ , 0)- semilattices with ( ∨ , 0)-homomor phisms, A is the subcateg o ry of S consis ting of all distributive lattic es with zero and 0-preserving lattic e emb e ddings , a nd Φ is the inclusion functor from A into S . While Example 1-1.1 is in lattice theory , Examples 1-1.2 and 1 - 1.3 are in ring theor y , and E xample 1-1.4 is in universal alg ebra. Examples 1- 1.5 and 1- 1.6 inv olve S -v alued versions of a discrete set a nd a poset, resp ectively , where S is a given ( ∨ , 0)-semilattice. 1-1 Introduction 13 Example 1-1. 1. B is the ca tegory o f all lattices with lattice homomo r- phisms, a nd Ψ is the natural e x tension to a functor of the assignment that to ev ery lattice L ass igns the ( ∨ , 0)-semilattice Con c L o f all co mpact (i.e., ﬁnitely genera ted) congruences of L . The statement ( ∀ D ∈ A )( ∃ L ∈ B )( D ∼ = Con c L ) was ﬁr st established b y Schm idt [56]. The stronger statemen t that the assignment ( D 7→ L ) can b e made functor ial was established by Pudl´ ak [50]. That is, ther e exis ts a functor Γ , fro m distributiv e 0-lattices with 0-la ttice embeddings to lattices a nd la ttice embeddings, such that Con c Γ ( D ) ∼ = D naturally o n all distributive 0-la ttices D . F urther mo re, in Pudl´ ak [50], Γ ( D ) is ﬁnite atomistic in case D is ﬁnite. Pudl´ ak’s approa ch was mo tiv a ted by the sear ch fo r the solution o f the Congruenc e L attic e Pr oblem CLP, p osed by Dilw or th in the forties , which asked whether ev ery distr ibutive ( ∨ , 0)-semilattice is isomor phic to Con c L for some lattice L . Pudl´ ak asked in [50] the stronger question whether CLP could hav e a functorial answer. This approach prov ed extremely fruitful, although it ga ve r ise to unpredictable dev elopments. F or instance, Pudl´ a k’s question was ﬁnally answered, in the negativ e, in T ˚ uma a nd W ehrung [60], but this was not of m uch help for the full negative solution of CLP in W ehrung [6 7], which required even mu ch mo re work. Example 1-1. 2. An a lgebra R ov er a ﬁeld F is • matricial if it is isomorphic to a ﬁnite pro duct of full matrix rings ov er F ; • lo c al ly matricial if it is a directed colimit of matr icial alge bras. Denote b y Id c R the ( ∨ , 0)-semilattice of all compact (i.e., ﬁnitely generated) t wo-sided ideals o f a ring R . Then Id c extends naturally to a functor from rings with ring homomo rphisms to ( ∨ , 0)-semilattices with ( ∨ , 0)-homomor - phisms. Let B b e the catego ry of lo cally matricia l F -alge br as and let Ψ be the Id c functor from B to S . R ˚ u ˇ z iˇ ck a proved in [52] that for every distributive 0- lattice D , there exists a locally matr icial F -algebra R such that Id c R ∼ = D . Later o n, R ˚ uˇ ziˇ ck a extended his result in [53] b y proving that the assig n- men t ( D 7→ R ) can be ma de functor ial, from distributive 0 -lattices with 0-preser ving lattice embeddings to lo cally ma tricial ring s (ov er a g iven ﬁeld) and r ing embedding s . Example 1-1. 3. F o r a (not nece s sarily unital) ring R and a p ositive in te- ger n , we denote b y M n ( R ) the ring o f all n × n squar e matrices with entries from R , and w e iden tity M n ( R ) with a subring of M n +1 ( R ) via the embedding x 7→  x 0 0 0  . Setting M ∞ ( R ) := S n> 0 M n ( R ), we sa y that idempotent matri- ces a, b ∈ M ∞ ( R ) are e quivalent , in notation a ∼ b , if there are x, y ∈ M ∞ ( R ) such that a = xy and b = y x . The rela tio n ∼ is a n equiv alence r elation, and we deno te b y [ a ] the ∼ -eq uiv alence class o f an idempotent matrix a ∈ M ∞ ( R ). Equiv alence cla sses ca n b e added, via the rule 14 1 Background [ a ] + [ b ] :=  a 0 0 b  , for all idempo tent matrices a, b ∈ M ∞ ( R ). This wa y , w e get a commutativ e monoid, V ( R ) := { [ a ] | a ∈ M ∞ ( R ) idemp otent } , that enco de s the so-called nonstable K-the ory (or, sometimes, the nonstable K 0 -the ory ) of R . It is easy to see that V extends to a fun ct or , from rings with ring homomorphisms to commutativ e monoids with monoid ho momorphisms, via the formula V ( f )([ a ]) = [ f ( a )] for any homomor phism f : R → S of rings and an y idemp otent a ∈ M ∞ ( R ). The monoid V ( R ) is c onic al , that is, it satiﬁes the following statement: ( ∀ x , y )( x + y = 0 ⇒ x = y = 0) , F or example, if R is a division ring, then V ( R ) is isomorphic to the monoid Z + := { 0 , 1 , 2 , . . . } of all non-negative integers, while if R is the endomor- phism ring of a vector space of inﬁnite dimension ℵ α (ov er an arbitrary divi- sion ring), then V ( R ) ∼ = Z + ∪ {ℵ ξ | ξ ≤ α } . Every conical commutativ e monoid is isomorphic to V ( R ) for some hered- itary ring R : this is prov ed in Theorems 6.2 and 6.4 o f Bergman [5] for the ﬁnitely genera ted cas e, and in Bergman and Dicks [6, page 315 ] for the general case with order-unit. The genera l, non-unital case is prov ed in Ara and Go o dea rl [4, P rop osition 4.4]. In light of this result, it is natura l to a sk whether this solution ca n be made functorial , that is, whether there exist a functor Γ , from the category of conic a l commutativ e monoids with monoid homomorphisms to the categor y of rings and ring homomorphisms, such that V ◦ Γ is iso morphic to the identit y . This pr o blem is still o pen. V ar iants are discuss ed in W ehr ung [71]. Example 1-1. 4. Deﬁne b oth A and S as the category o f all ( ∨ , 0 , 1)-semi- lattices with ( ∨ , 0 , 1)-embe dding s, Φ a s the identit y functor on A , B as the category of all gr oup oids (a g r oup oid is a nonempty set endow ed with a binary op era tion), and Ψ a s the C o n c functor from B to S . Lampe prov es in [4 2] that for each ( ∨ , 0 , 1 )-semilattice S there e x ists a group o id G such that Con c G ∼ = S . Ho wev er, it is not kno wn whether the as signment ( S 7→ G ) can b e made functorial, from ( ∨ , 0 , 1 )-semilattices with ( ∨ , 0 , 1)-emb eddings to gro up oids and their embeddings. Example 1-1. 5. F o r a ( ∨ , 0)- s emilattice S , an S -value d distanc e on a set Ω is a map δ : Ω × Ω → S such that δ ( x, x ) = 0 , δ ( x, y ) = δ ( y , x ) , δ ( x, z ) ≤ δ ( x, y ) ∨ δ ( y , z ) ( triangular ine quality ) , 1-1 Introduction 15 for all x, y , z ∈ Ω . F ur thermore, for a p ositive integer n , we say that δ is a V-distanc e of t yp e n if for all x, y ∈ Ω and all α 0 , α 1 ∈ S , δ ( x, y ) ≤ α 0 ∨ α 1 implies the existence o f z 0 , . . . , z n +1 ∈ Ω such tha t z 0 = x , z n +1 = y , and δ ( z i , z i +1 ) ≤ α i m od 2 for each i ≤ n . It is not ha rd to modify the pro o f in J´ onsson [36] to obtain that every ( ∨ , 0)-s emilattice with mo dular ideal lattice is the r ange of a V-distance of t yp e 2 on some set. O n the o ther hand, it is prov ed in R ˚ uˇ ziˇ ck a, T ˚ uma, and W e hrung [55] that this res ult can be made functorial on distributive semilattices. It is also prov ed in that pap er that there exists a distributive ( ∨ , 0 , 1)-semila ttice, of car dina lit y ℵ 2 , that is not generated by the ra nge o f any V-distance of type 1 on any set. Example 1-1. 6. F o r a ( ∨ , 0)-semilattice S , an S -value d me asur e on a p oset P is a map µ : P × P → S s uch that µ ( x, y ) = 0 in case x ≤ y , and µ ( x, z ) ≤ µ ( x, y ) ∨ µ ( y , z ) , for all x, y , z ∈ Ω . It is pr ov ed in W ehrung [68] that every distributive ( ∨ , 0)-semilattice S is join-genera ted by the range o f an S -v alued meas ure on s ome p oset P , which is also a meet-semilattice with zero, such that for all x ≤ y in P and all α 0 , α 1 ∈ S , if µ ( y , x ) ≤ α 0 ∨ α 1 , then ther e are a pos itive integer n and a decomp osition x = z 0 ≤ z 1 ≤ · · · ≤ z n = y such that µ ( z i +1 , z i ) ≤ α i m od 2 for each i < n . How ever, although the assignment ( S 7→ ( P , µ )) ca n be made “ functo- rial on lattice-indexed diagrams” , we do not k now whether it can b e made functorial (starting with ( ∨ , 0)-semilattices with ( ∨ , 0) -emb e ddi ngs ). 1-1.2 Partial ly functorial s olutions to r epr esentation pr oblems Let us c onsider aga in the settings of Section 1-1.1, that is , categories A , B , and S together with functors Φ : A → S a nd Ψ : B → S . The mo st commo nly encountered situation is that for all “not to o large”, but not all, ob jects A of A there exists an ob ject B of B s uch that Φ ( A ) ∼ = Ψ ( B ). It turns out that the extent of “not to o large” is often related to comb inato rial prop er ties of the functor s Φ a nd Ψ . More pr ecisely , a large pa rt of the present b o ok aims at explaining a for merly mysterious rela tion b etw een the tw o following statements: • F or e ach ob ject A of A o f “ c ardinality” at most ℵ n , ther e exists an ob ject B of B such that Φ ( A ) ∼ = Ψ ( B ). • F or every diagra m − → A of “ﬁnite” ob jects in A , indexed by { 0 , 1 } n +1 , there are a { 0 , 1 } n +1 -indexed dia gram − → B in B suc h that Φ − → A ∼ = Ψ − → B . 16 1 Background Example 1-1. 7. I n R ˚ uˇ ziˇ ck a, T ˚ uma, a nd W ehr ung [55 ] the question whether every distributiv e ( ∨ , 0 )-semilattice is iso mo rphic to Con c A for some congru- ence-p ermutable alg ebra A is solved in the negative. F or instance , if F L ( X ) denotes the fr ee la ttice on a set X , then Co n c F L ( ℵ 2 ) is not isomor phic to Con c A , for any congr uence-p ermutable algebra A . In par ticular, the ( ∨ , 0)-s emilattice Con c F L ( ℵ 2 ) is neither isomor phic to the ﬁnitely generated normal subgroup ( ∨ , 0)-semilattice of any group, nor to the ﬁnitely generated submo dule lattice of a mo dule. In these res ults the cardinality ℵ 2 is optimal. On the other hand, every { 0 , 1 } 2 -indexed diag ram (we say squar e ) of ﬁnite distributive ( ∨ , 0)-semilattices can be lifted, with respect to the Con c functor, by a square of ﬁnite rela tively complemented (thus co ngruence-p ermutable) lattices, see T ˚ uma [58] and Gr¨ a tzer, Lakser, and W ehrung [3 0]. It is prov ed in R ˚ u ˇ ziˇ ck a, T ˚ uma, and W ehrung [55] that this result do es not extend to cub es , that is, dia grams indexed by { 0 , 1 } 3 . Example 1-1. 8. I t is prov ed in Plo ˇ sˇ cica, T ˚ uma , and W ehrung [49] that there is no (von Neumann) regular r ing R such that the ideal la ttice of R is iso- morphic to Con F L ( ℵ 2 ). O n the other hand, it is pr ov ed in W ehrung [63] that every square o f ﬁnite Bo olean semila ttices and ( ∨ , 0)-ho momorphisms ca n b e lifted, with r esp ect to the Id c functor, by a square of regular rings. As all rings hav e p er m utable co ngruences, it follo ws from [55, Corollar y 7.3] that this result cannot b e extended to the cub e deno ted there b y D ac . Example 1-1. 9. Deno te b y L ( R ) the la ttice of all principal right ideals of a regular ring R . The a ssignment ( R 7→ L ( R )) can be natur a lly extended to a functor , fro m regular rings with ring homomorphisms to sectionally complemented mo dular lattices with 0-lattice ho momorphisms (see Chapter 6 for details). A lattice is c o or dinatizable if it is iso morphic to L ( R ) for s ome regular ring R . Denote b y M ω the lattice of length t wo with ω atoms a n , with n < ω . The assignment ( a n 7→ a n +1 ) deﬁnes an endomorphism ϕ o f M ω . The seco nd author prov es in [66] that • one ca nnot hav e r ing s R and S , a unital r ing homomor phism f : R → S , and a natural eq uiv alence b etw een the dia grams L ( f ) : L ( R ) → L ( S ) and ϕ : M ω → M ω ; • there exists a non-co or dinatizable 2-distr ibutive complemented mo dular lattice, of cardinality ℵ 1 , c o ntaining a copy of M ω with the same zero and the same unit (we say sp anning ). The c a rdinality ℵ 1 is optimal in the seco nd p oint ab ove, as the sec o nd author prov ed, b y metho ds extending thos e o f W ehrung [66] (and ment ione d there without pro of ), that Every c ountable 2 - distributive c omplemente d mo d- ular lattic e with a sp anning M ω is c o or dinatizable . Example 1-1. 10. In nonstable K 0 -theory, three pa r ticular classes of rings enjoy a specia l impor tance: namely , the ( von Neumann ) r e gular rings , the 1-1 Introduction 17 C*-algebr as of r e al r ank zer o , and the exchange rings . Every r egular r ing is an exchange r ing, the conv er se fails; and a C*-a lgebra has rea l ra nk zero iff it is an exchange ring. The sec o nd author ﬁnds in W ehr ung [7 1] a diagra m, indexed by the p ow- erset { 0 , 1 } 3 of a three-element set, that ca n b e lifted both b y a comm utative diagram of C*- a lgebras of real rank one and b y a commutative diagram o f exchange rings, but that cannot be lifted b y any commutativ e diagr am of either regula r rings or C*-a lgebras of real ra nk zer o. T his lea ds, in the s ame pap er, to the construction o f a dimension group with order -unit whose p os - itive cone ca n b e represented as V ( A ) for a C*-algebr a A of real r ank one, and also as V ( R ) for an exchange ring R , but never as V ( B ) for either a C*-algebr a of real rank zer o or a regula r ring B . Due to the cub e { 0 , 1 } 3 having order- dimension thr ee, the cardinality of this counterexample jumps up to ℵ 3 . It is conce iv able, although y et unknown, that reﬁning the metho ds used co uld yield a co unterexample of cardinality ℵ 2 . On the o ther hand, it is well-kno wn that pos itive co nes of dimension groups of cardinality ℵ 1 do not separate the nonstable K 0 -theories of exchange ring s , C * -algebr a s of real rank zero, and regula r rings. Example 1-1. 11. The original solution to CLP (see W ehrung [67]) pro duces a distributive ( ∨ , 0 , 1)-semilattice of car dinality ℵ ω +1 that is no t isomorphic to the compact co ngruence semilattice of a ny lattice. The bound is improved to the o ptimal one, namely ℵ 2 , by R ˚ u ˇ ziˇ ck a in [54]. On the other hand, every square o f ﬁnite ( ∨ , 0)-semilattices with ( ∨ , 0)- homomorphisms can b e lifted with resp ect to the Con c functor o n lattices , see Gr¨ a tzer, Lakser, and W ehrung [30]. It is not known whether this result can b e e xtended to { 0 , 1 } 3 -indexed dia grams. Many of the combinatorial patterns encoun tered in Examples 1-1.7 – 1-1 .1 1 app ear in th e following notion, ﬁrs t form ulated within P roblem 5 in the surv ey pap er T ˚ uma and W ehrung [59] a nd then extensively s tudied by the ﬁrst author in [17, 18, 19, 20, 21]. Deﬁnition 1 - 1.12. F or v arieties A and B of algebras (not necessarily on the same similarity type), we deﬁne • Con c A is the clas s of all ( ∨ , 0)-s emilattices that a r e isomorphic to Con c A , for some A ∈ A ; • crit( A ; B ), the critic al p oint of A and B , is the least pos s ible cardinality of a ( ∨ , 0)-s emilattice in (Con c A ) \ (Co n c B ) if Con c A 6⊆ Con c B , ∞ otherwise. The following r esult is prov ed by the ﬁrst author in [18, Co r ollary 7.1 3]. Theorem 1-1. 13 (Gillib ert). L et A and B b e varieties of algebr as with A lo c al ly ﬁnite and B ﬁnitely gener ate d c ongruenc e-distributive. Then Con c A 6⊆ Con c B implie s that crit( A ; B ) < ℵ ω . 18 1 Background The res ults of the pre s ent bo ok mak e it possible to extend Theorem 1-1.13 considerably in Gillibert [22], yielding the following res ult (prov ed there in the more genera l context of quasivarieties and r elative c ongruenc e lattic es ). Theorem 1-1. 14 (Gillib ert). L et A and B b e lo c al ly ﬁ nite varieties of al- gebr as such that for e ach A ∈ A ther e ar e only ﬁnitely many ( up to isomor- phism ) B ∈ B such that Con c A ∼ = Con c B , and every such B is ﬁnit e. Then Con c A 6⊆ Con c B implie s that crit( A ; B ) ≤ ℵ 2 . Due to known examples with v arieties of lattices, the bo und ℵ 2 in Theo- rem 1- 1.14 is s ha rp. If B is, in addition, ﬁnitely g enerated with ﬁnite similarity t yp e, then, due to results from [33], the condition of Theorem 1-1.14 holds in case B o mits tame cong ruence theory types 1 and 5 , which in turns holds if B satisﬁes a nontrivial congruence lattice iden tit y . In particular, this holds in cas e B is a ﬁnitely genera ted v a riety of groups, lattices, lo o ps, or mo dules ov er a ﬁnite ring . As to the present writing, the only known p os sibilities for the critical p oint betw een tw o v arieties of a lg ebras, o n ﬁnite simila rity t yp es, are either ﬁnite, ℵ 0 , ℵ 1 , ℵ 2 , or ∞ (cf. Problem 3). The pro ofs of Theo rems 1- 1.13 a nd 1-1.14 inv olve a deep analysis o f the relationship betw een liftability of obje cts and liftabilit y o f diagr ams with resp ect to the Con c functor o n alg ebras. 1-1.3 Contents o f the b o ok The main res ult of this work, the Condensate Lifting Lemma (CLL, Lemma 3-4 .2), is a co mplex, unfriendly-lo o king categor ical statemen t, that most rea ders might, at ﬁr st sig ht , disc a rd as v ery unlikely to hav e a ny application to any previously formulated problem. Such is CLL’s primary pr e cursor, the Arma- tur e L emma (Lemma 3- 2.2). The for mu latio n of CLL ’s sec ondary pr ecursor , the But tr ess L emma (Lemma 3-3.2), is ev en more tec hnical, although its pro of is easy and its intuitiv e conten t ca n be very roughly descr ib e d as a dia gram version of the L¨ owenheim-Skolem Theorem of mo del theor y . Therefore, altho ugh most of the technical diﬃculty of o ur bo ok lies in thirty pa g es of a relatively easy pr eparation in category theory , plus a com- bined six-page pro of of CLL together with its precur sors (the Armature Lemma—Lemma 3-2.2, and the Buttress L e mma—Lemma 3-3.2), together with tw o sections on inﬁnite combinatorics, a la rge part of our bo ok will consist of deﬁning and using contexts of p ossible a pplications of CLL . Our larg est suc h “applications” chapter is Chapter 4, that deals with ﬁr st- order s tructures. W e are having in mind the Gr¨ a tzer-Schmidt Theorem [31], that says that every ( ∨ , 0)-semila ttice is isomo r phic to Con c A for so me al- gebra A , with no p ossibility of assigning a spec iﬁed similarity t yp e to the algebra A (this caveat b eing due to the ma in res ult of F r eese, Lampe, and T aylor [12]). Hence w e consider the categ ory o f a ll ﬁr s t-order structures as 1-1 Introduction 19 a whole, with the existence of a homomorphism from A to B re q uiring the language (i.e., similarity type) of A to be contained in the languag e of B . W e denote b y MInd the c ategory thus for med, and we call it the category of all monotone-indexe d structur es . As a pplications of CLL in that context, we point the following: • (Extending the Gr¨ atzer -Schmidt Theorem to po set-indexed diag rams of ( ∨ , 0)-s emilattices and ( ∨ , 0)-homomorphisms, The o rem 4-7 .2) Every dia- gr am of ( ∨ , 0 ) -semilattic es and ( ∨ , 0) -homomorphi sms, indexe d by a ﬁ n ite p oset P , c an b e lifte d, with r esp e ct to the Con c functor, by a diagr am of algebr as ( with variable similarity typ es ). In the pr e sence of large cardina ls (e.g., existence of a proper class of Erd˝ os cardinals ), the ﬁniteness as- sumption on P can be removed. W e emphasize that it sounds rea sonable that a clos e r scrutiny of the pro of of the Gr¨ atzer -Schmidt Theorem could conceiv ably lead to a direct functorial pro o f of that re s ult. How ever, our pro of that the result for ob jects implies the result for diag rams requires no knowledge about the deta ils of the proo f of the Gr ¨ atzer-Schmidt Theorem. • Theorem 1 -1.13 (ab out critical p oints b eing either ∞ or less than ℵ ω ) can be , in a host of situations, e xtended to quas iv arieties of algebr a ic systems and the relative cong ruence lattice functor (cf. Theorem 4-9.4). By using the results of c ommutator the ory for c ongruenc e-mo dular va- rieties [13], we prov e that this applies to A b eing a lo c al ly ﬁnite qua- sivariety with ﬁ nitely many r elatio n symb ols and B a ﬁnitely gener ate d, ( say ) c ongruenc e-mo dular variety of algebr as with ﬁnite typ e (cf. Theo- rem 4-10.2). In pa rticular, B can b e a ﬁnitely gener ated v ar iety of gr oups , mo dules (ov er a ﬁnite ring), lo ops , lattic es , and so on. Actually , by using the res ults of [33], we can ev en form ulate o ur r esult in the mo re gener al context o f v arieties avoiding the tame c ongruenc e the ory types 1 and 5 (instead o f just congruence-mo dular ); on the other hand it is not so easy to ﬁnd such examples which are no t c o ngruence-mo dula r (Polin’s v a riety is such an example). How ever, even for groups, mo dules, or lattices, the result of Theorem 4 -10.2 is hig hly non-trivia l. The main pa r t of Chapter 4 co ns ists o f c hecking, o ne after another, the many conditions that nee d to be veriﬁed for our v ar ious a pplications o f CLL , as well as for new potential ones. Most of these v er iﬁca tions ar e elementary , with the poss ible exception of a c o ndition, called the “L¨ owenheim-Sk olem Condition” a nd denoted by e ither (LS µ ( B )) (for lar ders) or (LS r µ ( B )) (for right larders). Another a pplication chapter of CLL is Chapter 5. Its main purp ose is to solve the problem, until now o p en, whether every lattice of cardinality ℵ 1 has a congruence-p ermutable, congruence-pr e serving extension. (By deﬁnition, an algebra B is a c ongru en c e-pr eserving extension of a subalgebr a A if every congruence of A extends to a unique congruence of B .) The solution turns out to be negative: 20 1 Background • L et V b e a nondistributive variety of lattic es. Then the fr e e lattic e ( r esp. fr e e b ounde d lattic e ) on ℵ 1 gener ators within V has no c ongruenc e-p ermut- able, c ongruenc e-pr eserving extension (cf. Corollar y 5-5.6) . Once ag a in, the la rgest part of Chapter 5 consists of c hecking, one after another, the v arious conditions that need to b e veriﬁed for our application of CLL. Mo st of these veriﬁcations ar e elementary . Once they ar e p erfor med, a diag ram counterexample, describ ed in Sectio n 5-3, can be turned to an ob ject co unt er e x ample. Our ﬁnal application chapter is Chapter 6, whic h deals with the context of (von Neumann) reg ular rings . The f unctor in question is the o ne, denoted here by L , that s ends every reg ular ring R to the la ttice L ( R ) of all its principal right ideals. Aside from pa ving the road for further w ork on co o r dinatization, one of the goals of Chapter 6 is to provide “blac k box”-like to o ls. One of these to ols enables the second author to prov e in [69] the following statement, thus solving a problem of J´ onsso n stated in his 196 2 pap er [37]: • Ther e exists a se ctional ly c omplemente d mo dular lattic e, with a lar ge 4 - fr ame, of c ar dinali ty ℵ 1 , t hat is not c o or dinatizable. Let us now go bac k to the categ orical framework underly ing CLL. The context in which this r esult lives is v er y m uch r elated to the one of the monogra ph b y Ad´ amek and Rosick´ y [2]. The star ting a ssumptions of CLL are ca tegories A , B , and S , to g ether with functors Φ : A → S and Ψ : B → S . W e are also given a p oset (=par tially order ed set) P and a dia gram − → A =  A p , α q p | p ≤ q in P  from A . The r aison d’ ˆ et re of CLL is to co nstruct a c ertain o b ject A of A suc h that any lifting, with resp ect to the functor Ψ , of the obje ct Φ ( A ) creates a lifting, with resp ect to Ψ , of the diagr am Φ − → A . Of co ur se, this can b e done only under additiona l conditions , on the cat- egories A , B , S , on the functors Φ and Ψ , but a lso on the p o set P . The ob ject A , denoted in the statemen t of CLL by the notation F ( X ) ⊗ − → A , is a s o-called c ondensate of the diagram − → A . It c o ncentrates, in one ob ject, enough prop erties of the dia gram − → A to imply our statement on liftability . It is the directed colimit of a s uitable diagra m of ﬁnite products of the A p , for p ∈ P (cf. Section 3-1). The notio n of lifting itself requires a mo diﬁca- tion, via the intro duction of a suitable s ub ca tegory , denoted by S ⇒ , of S , whose ar rows are calle d double arr ows . While, in applica tions such a s The- orems 4-7 .2 and 4-9 .4, double arrows ca n alwa ys b e re duce d to single ar - rows (isomorphisms) via a condition called pr oje ctability (cf. Section 1- 5), double arrows cannot alwa ys b e eliminated, the most pro minen t example being Cor ollary 5-5.6 (ab out lattices of cardinality ℵ 1 without congruence- per mut able cong ruence-pres erving extensions). 1-1 Introduction 21 The construction and bas ic prop erties of the c ondensate F ( X ) ⊗ − → A a r e explained in Chapter 2. This requir e s a slight expansion of the category of Bo olean algebra s. This expansion inv olves a p os et parameter P , and it is the dual of the notio n of what w e shall call a P -norme d Bo ole an sp ac e (cf. Deﬁnition 2-2.1). By deﬁnition, a P -normed space is a top o lo gical spa ce X , endow ed with a map ν (the “norm” ) from X to the set Id P of ide als (non- empt y , directed lower subsets) of P such that { x ∈ X | p ∈ ν ( x ) } is op en fo r each p ∈ P . How ever, the statement of C L L in volv es dir ected c olimits , not dir e cted lim- its . Hence we c hose to form ulate most of the res ults o f Chapter 2 in the co ntext of the dual ob jects of P -no rmed Boo lean spa ces, whic h we shall call P -s c ale d Bo ole an alg ebr as (Deﬁnition 2-2.3). The duality b etw een the catego ry BT op P of P -nor med Bo olea n spaces and the ca tegory Bo ol P of P -sca le d Bo olea n al- gebras is (easily ) established in Section 2- 2. All the other results of Chapter 2 are for m ulated “algebr a ically”, that is, within the categor y Bo ol P . F urther - more, the in tro duction of the “ free P -sc aled B o olean algebras” (o n certain generator s and rela tions) F ( X ) (cf. Section 2-6) is far more natural in the algebraic con text Bo ol P than in the topolo gical one BT o p P . Chapter 2 is still to be considered only as an int ro duction of the CL L framework. The statemen t of CLL inv olves three catego rical add-ons to the initial data A , B , S , Φ , a nd Ψ . One of them we alr eady discusse d is the subcateg ory S ⇒ of “ double arrows” in S . The o ther tw o a re the class A † (resp., B † ) o f all “small” ob jects of A (resp., B ). In case the cardinal pa r ameter λ is equal to ℵ 0 , then A † (resp., B † ) may be thought of as the “ﬁnite” ob jects of A (resp. B ). The statement o f CLL also in volves a condition on the p oset P . This condition is stated as P having a “ λ -lifter”. This condition, introduced in Deﬁnition 3-2.1, has combinatorial nature, inspired by Kur atowski’s F ree Set Theorem [41]. There are p osets, even ﬁnite, for whic h this is forbidden—that is, there a re no lifters. Actually , the ﬁnite p osets for whic h there are lifters are exa ctly the ﬁnite disjoint unions of ﬁnite pose ts with zer o in whic h every principal ideal is a join-semila ttice (cf. Coro llary 3 -5.10). An example of a ﬁnite p oset for which this do es not hold is re pr esented on the r ight hand side of Fig ure 2 .1, page 47. F or inﬁnite p osets, the situation is more complicated: even for the dual chain of ω + 1 = { 0 , 1 , 2 , . . . } ∪ { ω } , which is triv ia lly a p oset with zer o in whic h ev ery principal ideal is a join-semilattice, there is no (2 ℵ 0 ) + -lifter (cf. Co rollar y 3-6.4). Nevertheless our theory ca n be developed quite far, even for inﬁnite po s ets. Three classes of po sets emerge: the pseudo join-semilattic es , the supp orte d p osets , and the almo st join-semilattic es (Deﬁnition 2-1.2). Intriguingly, some of ou r deﬁnitions ar e closely r elate d to deﬁnitions use d in domain the ory, se e, in p articular, [1, C ha pter 4]. E very almo s t join-semilattice is supp orted, and every supp orted p oset is a pseudo join- s emilattice; none of the converse implications hold. Every p os e t with a λ -lifter (where λ is an inﬁnite cardi- 22 1 Background nal) is the disjoint union of ﬁnitely many a lmost join- s emilattices with zero (Prop osition 3-5.7). In the ﬁnite case, or even in the inﬁnite case provided P is lower ﬁnite (meaning tha t every pr incipal ideal of P is ﬁnite) and in the presence of a suitable large cardinal assumption, the co nv erse holds (Corol- lary 3-5.8). Some of the inﬁnite combinatorial as pec ts of lifters , in volving our ( κ, < λ ) ❀ P nota tion (cf. Deﬁnition 3-5.1), a re of indep endent interest and are developed further in our pap er [23]. Part o f the conclusion of CLL, namely the one ab o ut go ing from “ob ject representation” to “dia gram repr esentation”, can b e reached even fo r dia- grams indexed by p os ets without lifters (but still almost join-se mila ttices), at the exp ense of losing CLL’s condensa te F ( X ) ⊗ − → A . This result is presented in Cor ollary 3-7.2. This co rollar y inv olves the lar ge c a rdinal axiom deno ted, using the notation from the Erd˝ os, Ha jnal, M´ at´ e, and Rado monogra ph [1 0], by ( κ, <ω , λ ) → λ . It is, for example, inv olved in the pro of of Theo rem 4-7.2. 1-1.4 How not to r e ad the b o ok In the pr e s ent section we w ould like to give so me hints about how to use the present w ork as a to olb ox, ideally a list of “black box principles” that would enable the reade r who, although open-minded, is not necessa rily in the mo o d to g o thr o ugh a ll the details of our b o ok, to so lve his or her own problem. As our w ork inv olves a fair amount of topics as distan t as categor y theory , universal a lgebra, o r inﬁnite com binato r ics, such qualms are not unlikely to o ccur. In order for our work to hav e any r elev ance to our reader’s problem, it is quite likely that this proble m w ould still inv olve categor ies A , B , S tog ether with functors Φ : A → S a nd Ψ : B → S . In addition, the reader’s problem might also involv e suitable choices of a sub categor y S ⇒ of S (the “double arrows” of S ) to gether with classes of “small ob jects” A † ⊆ A , B † ⊆ B . Then, the r eader may need to relate liftability , with resp ect to the func- tor Ψ (and poss ibly the class of all double ar rows of S ), of obje cts o f the form Φ ( A ), and of diagr ams o f the form Φ − → A . A typical case is the following. The reader may hav e found, through pre- vious research, a diagram − → A from the categor y A , indexed by a po set P , such that there are no P -indexed diagram − → B fr om B and no double ar row Ψ − → B ⇒ Φ − → A . This is, for example, the case for the family of s quare diagra ms of Lemma 5-3.1 (ab out CP CP-retra cts) or a certain ω 1 -indexed diagram of sectionally complemented mo dular lattices in [6 9]. A preliminar y question that needs to b e a ddressed is the following. Is the p oset P an almost join-semilattic e (cf. Deﬁnition 2 - 1.2), or , at least, can the problem b e reduced to the case where P is an almost join-semilattice (like in Theorem 4-7.2)? 1-1 Introduction 23 If not, then, as to the pr esent wr iting , there is no form of CL L that can be used (cf. Problem 1 in Chapter 7). If yes, then this lo oks like a case where CLL (or its pair of a nc e s tors, the Armature L e mma a nd the Buttress Lemma) c o uld be applied. The hard core of the present b o ok lies in the pro of of CLL (Lemma 3-4.2), how ever the reader may not need to go throug h the details thereo f. The setting up of the ca rdinal parameters λ and κ := ca rd X dep ends of the nature of the problem. The inﬁnite combinatorial asp ects, which can be looked up in Section 3-5 and 3 -6, are articulated aro und the notio n of a λ -lifter . The exis tence of a lifter is a n esse n tial assumption in the statement of bo th CLL a nd the Armature Lemma. A more amenable v ariant of the existence of a lifter, the ( κ, <λ ) ❀ P relation, is introduced in o ur pap er Gilliber t and W ehrung [23] and reca lled in Deﬁnition 3 -5.1. As most of the diﬃculty underlying the formulation of CLL (Lemma 3-4.2) is related to the deﬁnition o f a larder, we s plit that deﬁnition in t wo par ts, namely left lar ders and right lar ders (cf. Section 3-8). Roughly sp eaking, the left larder co rresp onds to the left ha nd side arr ow Φ : A → S of the left part of the diagram of Figure 1.1, while the r ight larder corresp onds to the rig ht hand side arrow Ψ : B → S . In all cases encountered s o far, left larder ho o d nev er led to any diﬃculty in veriﬁcation. The situa tio n is diﬀer ent for rig ht la rders, that inv olve a deep er analysis of the structures in volv ed. In such applications as the one in W eh- rung [69 ], it is impo rtant to keep some control on how the P -scaled Bo olean algebra F ( X ) a nd the co ndensate F ( X ) ⊗ − → A are created. F or such no tio ns, the relev ant part o f o ur work to lo o k up is Chapter 2. In writing Chapter 4, we made the b et that many rig h t la rders would in- volv e ﬁrst-or der structur es and c ongruenc e lattic es (with relations possibly allow ed), a nd so we included in that chapter so me detail abo ut most of the basic tools required for chec king those la rders. In that c hapter, the functor Ψ has to b e thoug ht of as the relative c ompact co ngruence se milattice functor (cf. Deﬁnition 4-9.1) in a given quasiv ariety , or ev en in a gener alize d quasi- variety (cf. Deﬁnition 4 -3.4). While so me r esults in that chapter a r e alrea dy present in references such as Gorbunov’s mono graph [26], the refer ences fo r some other items can be quite hard to tra ce—such a s the de s cription of di- rected colimits in MInd (Propo sition 4-2.1) or th e r elative congr uence lattice functor on a gener alized quas iv ariet y (Section 4-3). F urther res ults of Cha p- ter 4 also appe ar in print here for the ﬁrst time (to our knowledge)—suc h as the preserv ation of all small dire c ted colimits by the Co n V c functor (Theo- rem 4-4 .1). Mos t results in Chapter 4 a re articula ted ar ound Theorem 4 -7.2 (diagram version of the Gr¨ a tzer-Schmidt Theo rem), Theorem 4- 9 .2 (estimates of relative critica l p oints via combinatorial prop erties of indexing p ose ts ), and Theorem 4-9 .4 (Dichotom y Theor em for r elative critical p oints b etw een quasiv arieties). Chapter 6 g ives another class of right larders , this time arising from the L functor on regula r r ings (cf. Ex ample 1-1.9). W e also wro te that c hapter in 24 1 Background our “to olb ox” spir it. F urthermore, although we b elieve that all of the basic results presen ted there already a ppea red in pr int somewhere in the case of unital regular rings, this do es not seem to b e the case for no n-unital rings, in particular Lemma 6-1 .8 and thus Pro po sition 6-1 .9 (Id R ∼ = NId L ( R )). It is certainly Chapter 5 that in volves the most unusual kind of larder, tai- lored to solve a sp eciﬁc problem ab o ut congr uence-p ermutable, co ngruence- preserving extensions of lattices, th us p erhaps providing the b es t illustra tion of the versatility of our larder to ols. Indeed, Ψ can b e des c rib ed there as the forgetful functor from a catego ry o f pairs ( B ∗ , B ) to the ﬁrs t comp onent B ∗ . W e end the present section with a list of all the pla ces in our w or k stating larderho o d of either a structure or a class of s tructures. • Prop ositio n 5-4.3 (left larder, from algebras and s emilattice-metric spaces). • Theorem 4-7.2 (Claim 1 for a left larder, Claim 2 for a rig ht λ -larder, for regular uncountable λ , fro m the co mpact congruence lattice functor on the category M Al g 1 of all unar y a lg ebras). • Theorem 4-8.2 ( ℵ 0 -larder, from the relative compa c t congr uence semilat- tice functor on a cong r uence-prop er and loca lly ﬁnite q ua siv ariety). • Theorem 4-1 1.1 ( λ -larder , for an y uncoun table ca rdinal λ , from the relative compact co ngruence semilattice functor on a quasiv ariety on a λ -small language). • Prop ositio n 5-2 .2 (right ℵ 0 -larder, from a forgetful functor from s emilat- tice-metric cov ers to se mila ttice-metric spaces). • Theorem 6 -2.2 (rig ht λ -larder , from the L functor on a class of re g ular rings with mild directed colimits-r elated assumptions and closure under homomorphic ima ges). The (tr iv ial) statemen t that br ings together left and right larders in to larders is Pro po sition 3-8 .3. 1-2 Basic c oncepts 1-2.1 Set the ory W e shall use basic set-theoretica l notation and terminology abo ut ordinals and cardina ls. W e denote by dom f (r esp., rng f ) the domain (resp., the range) of a function f , and by f “ ( X ), or f “ X (resp., f − 1 X ) the image (resp., inv erse image) of a set X under f . Cardinals ar e initial ordinals. W e denote by cf ( α ) the coﬁnality of an ordina l α . W e denote by ω := { 0 , 1 , 2 , . . . } the ﬁrst limit ordinal, mos tly denoted by ℵ 0 in case it is viewed a s a car dinal. W e denote by κ + the succes s or cardinal of a cardina l κ , and we deﬁne κ + α , for an ordina l α , b y κ +0 := κ , κ +( α +1) := ( κ + α ) + , and κ + λ := sup α<λ κ + α for every limit ordinal λ . W e set ℵ α := ( ℵ 0 ) + α , fo r each or dinal α . W e deno te by P ( X ) the p owerset of a s et X , and we put 1-2 Basic conc epts 25 [ X ] κ := { Y ∈ P ( X ) | card Y = κ } , [ X ] <κ := { Y ∈ P ( X ) | card Y < κ } , [ X ] 6 κ := { Y ∈ P ( X ) | card Y ≤ κ } , for ev ery car dina l κ . A set X is κ - smal l if card X < κ . W e shall often add an ex tra largest element ∞ to the class of all car dinals, and we shall say , by conv ention, “small” (res p., “ﬁnite”) instea d o f “ ∞ -small” (resp., “ ℵ 0 -small”). 1-2.2 Stone duality for Bo ole an algebr as A top ologic a l s pace is Bo ole an if it is compa ct Hausdorﬀ and every op en subset is a union o f clop en (i.e., c lo sed op en) subsets. W e denote by Clo p X the Bo olea n algebra of clop en subsets of a top olo gical space X . W e denote b y Ult B the Bo olean spa ce of all ultraﬁlters of a B o olean a lgebra B (a topolog i- cal spa ce is Bo ole an if it is compa c t Hausdor ﬀ and it has a basis consis ting of clop en sets). The pair (Ult , Clop) can b e extended to the well-kno wn Stone duality b etw een the catego ry Bo ol of Boo lean algebras with homomorphisms of Bo olea n algebra s and the category BT op of Bo olea n spaces with contin u- ous maps, in the following wa y . F or a homomo rphism ϕ : A → B of Bo ole a n algebras , we put Ult ϕ : Ult B → Ult A , b 7→ ϕ − 1 b . (1-2.1) F or a contin uous map f : X → Y b etw een Bo olean spac e s, we put Clop f : Clop Y → Clop X , V 7→ f − 1 V . (1-2.2) W e denote by At A the set o f atoms of a Bo o le a n algebr a A . 1-2.3 Partial ly or der e d s ets (p osets ) and lattic es All our p osets will b e no nempt y . A p oset P is the disjoint union of a family ( P i | i ∈ I ) of sub-p osets if P = S ( P i | i ∈ I ) and an y element of P i is incom- parable with any element of P j , for all distinct indices i , j ∈ I . F or p ose ts P and Q , a map f : P → Q is isotone (resp., ant itone ) if x ≤ y implies that f ( x ) ≤ f ( y ) (resp., f ( x ) ≥ f ( y )), for all x, y ∈ P . F or x, y ∈ P , let x ≺ y hold if x < y and there is no z ∈ P suc h that x < z < y . W e also s ay that y is an upp er c over of x and that x is a lower c over of y . W e denote by 0 P the leas t element o f P if it ex is ts, and by Min P (resp., Max P ) the set of all minimal (r esp., maximal) e lement s of P . An elemen t p in a p oset P is join-irr educible if p = W X implies that p ∈ X , for every (poss ibly 26 1 Background empt y) ﬁnite subset X of P ; we denote by J( P ) the set of all join-irreducible elements of P , endow ed with the induced partial ordering. Me et-irr educible elements are deﬁned dually . W e s e t Q ↓ X := { q ∈ Q | ( ∃ x ∈ X )( q ≤ x ) } , Q ↑ X := { q ∈ Q | ( ∃ x ∈ X )( q ≥ x ) } , Q  X := { q ∈ Q | ( ∃ x ∈ X )( q < x ) } , Q ⇈ X := { q ∈ Q | ( ∃ x ∈ X )( q > x ) } , Q ⇓ X := { q ∈ Q | ( ∀ x ∈ X )( q ≤ x ) } , Q ⇑ X := { q ∈ Q | ( ∀ x ∈ X )( q ≥ x ) } , for all s ubsets Q and X of P ; in case X = { a } is a singleto n, then w e shall write Q ↓ a instead of Q ↓ { a } , a nd so on. A subset Q of P is a lower su bset of P ( upp er subset of P , c oﬁnal in P , resp ectively) if P ↓ Q = Q ( P ↑ Q = Q , P ↓ Q = P , resp ectively). A low er subset Q of P is ﬁn itely gener ate d if Q = P ↓ X for so me ﬁnite subset X of P . The dual deﬁnition holds for ﬁnitely generated upp er subsets o f P . Joins (=suprema) and meets (=inﬁma) in po sets are denoted by ∨ and ∧ , resp ectively . W e say that the p oset P is • lower ﬁn it e if P ↓ a is ﬁnite for ea ch a ∈ P ; • wel l-founde d if every nonempty subset of P ha s a minimal elemen t (equiv- alently , P has no strictly decr easing ω -sequence); • monotone σ -c omplete if ev er y increasing sequence (indexed by the set ω of all natural num b ers) of elements of P has a jo in; • dir e cte d if every ﬁnite subset of P has an upper bo und in P . An ide al of P is a nonempt y , directed, low er subset of P ; we deno te b y Id P the set o f all ideals of P , o rdered by co ntainmen t. Observe that Id P need b e neither a meet- nor a join-semilattice. A princip al ide al of P is a subset of P of the form P ↓ x , for x ∈ P . F or a cardinal λ , we say that the p oset P is • lower λ -smal l if car d( P ↓ a ) < λ fo r ea ch a ∈ P ; • λ -dir e cte d if ev ery X ⊆ P such that card X < λ has an upp er b ound in P . Deﬁnition 1 - 2.1. F or a monotone σ -co mplete poset P , a subset X ⊆ P is σ -close d c oﬁnal if the following conditions hold: (i) X is co ﬁnal in P ; (ii) the lea st upp er b ound of any increa sing sequence of ele men ts of X belo ngs to X . W e will s o metimes abbreviate the s tatement (ii) ab ov e b y saying that X is σ -close d . (This notion is named after the co rresp onding classical notio n of closed coﬁnal subsets in uncountable regula r cardina ls.) Prop ositio n 1-2.2. L et P b e a m onotone σ -c omplete p oset. Then the inter- se ction of any at most c ountable c ol le ction of σ -close d c oﬁnal subsets of P is σ -close d c oﬁnal. 1-2 Basic conc epts 27 Pr o of . Let ( X n | n < ω ) be a sequence of σ -clo s ed coﬁnal subsets o f P . As the intersection X := T ( X n | n < ω ) is obviously σ -clo sed, it remains to prov e that it is coﬁnal. By r eindexing the X n s, w e ma y assume that the set { n < ω | X n = X m } is inﬁnite fo r each m < ω . Let p 0 ∈ P . If p n is already constructed, pic k p n +1 ∈ X n such that p n ≤ p n +1 . The supr e mum W ( p n | n < ω ) lies ab ov e p 0 and it b elo ng s to X . ⊓ ⊔ W e sha ll denote b y P op the dual of a p o s et P , that is, the p oset with the same underlying set as P and opp osite order. An element a in a lattice L is c omp act if for every subset X o f L such that W X exists and a ≤ W X , there exists a ﬁnite subset Y of X s uch that a ≤ W Y . W e say that L is algebr aic if it is complete a nd every element of L is a join of compact elements. A subset of a lattice L is alg ebr aic (see Gorbunov [26]) if it is c lo sed under a rbitrary meets a nd arbitrar y nonempty directed joins; in particula r, an y algebr aic subs et of an algebr aic lattice is an algebraic lattice under the induced or dering (see [26, Section 1.3]). A lattice L with zero is se ctional ly c omplemente d if for all x ≤ y in L there exists z ∈ L such that x ∨ z = y while x ∧ z = 0 (abbreviatio n: y = x ⊕ z ). Elements x and y in a lattice L are p ersp e ctive , in notation x ∼ y , if there exists z ∈ L suc h that x ∨ z = y ∨ z while x ∧ z = y ∧ z . In ca s e L is sectionally complemented and mo dular, w e ma y assume that x ∧ z = y ∧ z = 0 while x ∨ z = y ∨ z = x ∨ y . An ideal I in a lattice is neutr al if the sublattice of the ideal lattice of L g e nerated b y { I , X , Y } is distributiv e, fo r all ideals X and Y o f L . If this holds, then I is a distributive ide al of L , that is, the binary relation ≡ I on L deﬁned by x ≡ I y ⇔ ( ∃ u ∈ I )( x ∨ u = y ∨ u ) , for all x, y ∈ L , is a lattice cong ruence o f L . Then we denote by L/I the quotient la ttice L / ≡ I , and w e denote by x/ I := x/ ≡ I the ≡ I -equiv alence class o f x , for each x ∈ L . In ca se L is sectionally complemented and modula r, I is distributive iff it is neutral, iff x ∼ y and x ∈ I implies that y ∈ I , for all x, y ∈ L (cf. [7, Theorem I I I.13.20]). F or further unexplained no tio ns in la ttice theo r y we refer to the mono- graphs by Gr¨ atzer [28, 29]. 1-2.4 Cate gor y the or y Our categorica l background will be ma inly b or row ed fr o m Ma c L a ne [44], Ad´ amek and Rosick´ y [2], Johnstone [34], and from the second author’s earlier pap ers [64, 65]. W e sha ll often use sp ecia l sym b ols for sp ecia l sorts of arr ows in a given categor y C : • f : A ֌ B for m onomorphisms , or (for concrete categories) one-to-one maps ; 28 1 Background • f : A ։ B for epimorph isms , or (for concr e te categ ories) surje ctive maps ; • f : A ֒ → B for emb e ddings , that is, usually , sp ecia l classes of monomor- phisms pr e serving s ome a dditional s tructure (e.g., or der-emb e ddi ngs ); • f : A ⇒ B for double arr ows , that is , morphisms in a distinguished sub- category C ⇒ of C (see the end of the present section); • f : A . → B for natu r al tra nsformations b etw een functors; • f : A . ⇒ B for natural tra nsformations consisting of families of double arrows. W e denote by Ob A (resp., Mor A ) the class of ob jects (resp., mor phis ms ) of a categor y A . W e say that A has smal l hom-sets (resp., has κ -smal l hom- sets , where κ is a car dinal) if the class of all morphis ms from X to Y is a set (resp., a set with less than κ elements), for a ny ob jects X and Y of A . A morphism f : A → B in a categ ory C factors t hr ough g : B ′ → B (or , if g is understoo d, factors t hr ough B ′ ) if there exists h : A → B ′ such that f = g ◦ h . F or an ob ject C of C and a subca tegory C ′ of C , the c omma c ate gory C ′ ↓ C is the category whose ob jects are the mor phisms of the for m x : X → C in C wher e X is an ob ject o f C ′ , and where the morphisms from x : X → C to y : Y → C a re the morphisms f : X → Y in C ′ such that y ◦ f = x . W e write x E y the statement tha t there e x ists such a mor phis m f . W e identify , the us ua l wa y , preor dered sets with categorie s in which there exists a t most one morphism from any ob ject to any other ob ject. F or a po set P , a P -indexe d dia gr am in a category C is a functor from P (viewed as a ca tegory) to C . It can b e viewed as a system − → S =  S p , σ q p | p ≤ q in P  where σ q p : S p → S q in C , σ p p = id S p , and σ r p = σ r q ◦ σ q p , for all p ≤ q ≤ r in P . Hence a co cone above − → S consis ts o f a family ( T , τ p | p ∈ P ), fo r an ob ject T of C and morphisms τ p : S p → T such that τ p = τ q ◦ σ q p , for all p ≤ q in P . In case − → S has a colimit with under lying ob ject S , w e shall often denote by σ p : S p → S the corresp onding limiting mo rphism, for p ∈ P . A sub diagr am of − → S is a comp os ite of the form − → S ◦ ϕ , wher e ϕ is an o rder-embedding from some po set into P (o f c o urse ϕ is viewed as a functor ). F or a ca tegory A a nd a functor Φ : A → S , we say that a diagra m − → S in S is liftable with r esp e ct to Φ if there exists a diagram − → A in A such that Φ − → A ∼ = − → S . W e say then that − → S is lifte d b y A , or that A is a lifting of S . F or an inﬁnite cardinal κ , we say that − → S is • κ -dir e cte d if the p o set P is κ - directed; • c ontinuous if − → S preserves a ll sma ll directed colimits, that is, S p = lim − → q ∈ X S q (with the o bvious transition and limiting morphisms), for ev ery nonempty directed subset X of P with leas t upp er b ound p = W X . • σ -c ontinuous if − → S pres erves all directed colimits of incr e asing se quenc es in P . A colimit is monomorphic if all its limiting morphisms (thus also all its transition morphisms) are monic. 1-2 Basic conc epts 29 Deﬁnition 1 - 2.3. Let λ b e an inﬁnite cardinal and let C † be a full sub cate- gory of a categor y C . W e say that (i) C † has al l λ -smal l dir e cte d c olimits within C if every diag ram in C † indexed by a nonempty dir e cte d λ -s ma ll p oset P ha s a co limit in C whose underlying ob ject be longs to C † ; (ii) C has al l λ -smal l dir e ct e d c olimits if it ha s all λ -small directed colimits within itself. According to [2, Theorem 1.5] (see also [35, Theo rem B.2.6.13]), C † has all λ -small dir ected colimits within C iff every diagr am in C † , indexed by a nonempty λ -sma ll ﬁlter e d c ate gory , has a colimit whose underlying ob ject belo ngs to C † . (A categ ory is ﬁlt er e d if for all ob jets A and B there are an ob ject C with morphisms f : A → C and g : B → C , and for all o b jects A and B with morphisms u, v : A → B there are an ob ject C and a morphism f : B → C such that f ◦ u = f ◦ v .) F or a category C and a p os et P , we de no te by C P the ca tegory o f all P - indexed diagrams in C , with natura l tra nsformations as morphisms. In particular, in ca s e P = 2 = { 0 , 1 } , the tw o- element chain, we obtain the category C 2 of arrows of C . Deﬁnition 1 - 2.4. Let C b e a category . A pr oje ction in C is the canonical pro jection from a nonempty ﬁnite pro duct in C to one of its fac to rs. An arrow f : A → B in C is an ext en de d pr oje ction if there are diag rams − → A =  A i , α j i | i ≤ j in I  and − → B =  B i , β j i | i ≤ j in I  in C , b oth index ed by a directed p oset I , toge ther with co limits ( A, α i | i ∈ I ) = lim − → − → A , ( B , β i | i ∈ I ) = lim − → − → B , and a natural tr ansformation ( f i | i ∈ I ) from − → A to − → B with limit mo rphism f , such that each f i , for i ∈ I , is a pro jection. In par ticular, every isomo r phism is a pro jection, and every extended pr o- jection is a dir ected colimit, in the category C 2 of all a r rows of C , o f pro jections in C . Deﬁnition 1 - 2.5. Let λ b e an inﬁnite cardinal. A s ubca tegory C ′ of a cate- gory C is close d under λ -smal l dir e cte d c olimits , if for every λ -sma ll set I and all diagrams − → A =  A i , α j i | i ≤ j in I  and − → B =  B i , β j i | i ≤ j in I  in C ′ , together with colimits in C ( A, α i | i ∈ I ) = lim − → − → A , ( B , β i | i ∈ I ) = lim − → − → B , 30 1 Background and a natural transfor mation ( f i | i ∈ I ) from − → A to − → B with limit mor- phism f : A → B , if each f i , for i ∈ I , is an arrow of C ′ , then so is f . In cas e we formulate the a bove requirement with − → A (resp., − → B ) c onstant (i.e., all morphisms ar e the identit y), w e say that C ′ is left ( r esp., right ) close d under λ -smal l dir e cte d c olimits . F o r example, C ′ is righ t closed under λ -small directed colimits iff for every directed colimit dia g ram ( A, α i | i ∈ I ) = lim − →  A i , α j i | i ≤ j in I  in A with card I < λ , every ob ject B in B , and every f : A → B , if f ◦ α i is a mo rphism o f C ′ for each i ∈ I , then so is f . In case C ′ is a full sub catego ry of C , this is equiv a lent to verifying that an y colimit co c o ne in C of a ny directed p oset-indexed diagr am in C ′ is co nt ained in C ′ . Double arro ws W e shall often encounter the situation of a sub c ategory S ⇒ of a categ ory S , of whic h the arrows are denoted in the fo r m f : X ⇒ Y a nd called t he double arr ows of S . In suc h a context, for a po s et P and o b jects of S P (i.e., P - indexed diagr ams) − → X and − → Y , a double a rrow from − → X to − → Y will be deﬁned as an a r row − → f : − → X → − → Y in S P (i.e., a na tural transformation from − → X to − → Y ), say − → f = ( f p | p ∈ P ), suc h that f p : X p ⇒ Y p for each p ∈ P ; and then we shall write − → f : − → X . ⇒ − → Y . 1-2.5 Dir e cte d c ol imits o f ﬁrst-or der structur es W e shall use sta ndard deﬁnitions a nd facts ab out ﬁrst-o rder structur e s a s presented, for exa mple, in C ha ng and Keisler [8, Section 1 .3]. W e shall still denote by Lg( A ) the languag e of a ﬁrst-order structure A . This se t br eaks up as the set Cst( A ) of all constant symbo ls of A , the set Op( A ) of all oper ation symbols of A , and the set Rel( A ) of all relation symbols of A . W e denote by ar( s ) the ar it y of a symbol s ∈ Lg( A ); hence ar( s ) is a p ositive in teger , unless s ∈ Cst( A ) in which ca se ar( s ) = 0. W e sha ll also de no te the univ erse of a ﬁrst-order structure b y the corre- sp onding lightface c har a cter, so, for instance, A will b e the universe of A . W e denote by s A the interpretation of a symbol s o f Lg( A ) in the structure A , so that s A is an element of A if s is a constant symbol a nd an n -ary op era tio n (resp., relation) on A if s is an n -a r y op era tion (res p., rela tion) sy mbo l. A map f : A → B betw een ﬁrst-order structures of the same langua ge L is an elementary emb e dding if A | = F ( a 1 , . . . , a n )) iff B | = F ( f ( a 1 ) , . . . , f ( a n )) 1-2 Basic conc epts 31 for every ﬁrst-order formula F of L , of arity n , and all a 1 , . . . , a n ∈ A . Requiring this condition only for F atomic means that f is an emb e dding . Both co nditions trivially imply that f is one-to -one. W e shall need the following description of dir ected colimits in the cat- egory M o d L of a ll mo dels for a given ﬁrst-o rder language L , see for ex- ample [26, Se c tion 1 .2.5]. F o r a dir ected p o s et I a nd a p oset-index e d dia- gram  A i , ϕ j i | i ≤ j in I  in Mo d L , we deﬁne an equiv alence relation ≡ on A := S ( A i × { i } | i ∈ I ) by ( x, i ) ≡ ( y , j ) ⇐ ⇒ ( ∃ k ∈ I ⇑ { i, j } )  ϕ k i ( x ) = ϕ k j ( y )  , then we se t A := A / ≡ and we deﬁne ( ϕ i : A i → A , x 7→ ( x, i ) / ≡ ), for ea ch i ∈ I . Then the followin g r elation ho lds in the categor y Set of sets (with maps a s ho momorphisms): ( A, ϕ i | i ∈ I ) = lim − →  A i , ϕ j i | i ≤ j in I  (1-2.3) The directed co limit co cone ( A, ϕ i | i ∈ I ) is also uniquely characterized by the prop erties A = [ ( ϕ i “( A i ) | i ∈ I ) , (1-2.4) ϕ i ( x ) = ϕ i ( y ) ⇔ ( ∃ j ∈ I ↑ i )  ϕ j i ( x ) = ϕ j i ( y )  , fo r a ll i ∈ I and all x, y ∈ A i . (1-2.5) The set A can b e given a unique s tr ucture o f mo del of L such that (i) F or ea ch co nstant sym b ol c in L , c A = ϕ i ( c A i ) for each i ∈ I . (ii) F or ea ch op er ation s ymbol f in L , say of arity n , and for ea ch i ∈ I , the equation f A ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) = ϕ i  f A i ( x 1 , . . . , x n )  holds fo r a ll x 1 , . . . , x n ∈ A i . (iii) F or eac h r elation symbo l R in L , sa y of arit y n , and for each i ∈ I , the following equiv alence holds: ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) ∈ R A ⇐ ⇒ ( ∃ j ∈ I ↑ i )  ( ϕ j i ( x 1 ) , . . . , ϕ j i ( x n )) ∈ R A j  , for all x 1 , . . . , x n ∈ A i . (1-2.6) F urthermore, the following relatio n holds in the ca tegory Mo d L : ( A , ϕ i | i ∈ I ) = lim − →  A i , ϕ j i | i ≤ j in I  . (1-2.7) The conditions (i)–(iii) a bove, together with (1-2.4) and (1-2.5) , determine the colimit up to isomorphis m. In ca tegorical terms, the dis cussion ab ov e 32 1 Background means that t he for getful functor fr om Mo d L to Set cr e ates and pr eserves al l smal l dir e cte d c olimits . F or a theor y T in the languag e L , we denote b y Mo d ( T ) the full sub- category of M o d L consisting of the mo dels that satisfy all the axioms of T . It is well-known that if T consists only of axioms of the form ( ∀ − → x )  E ( − → x ) ⇒ ( ∃ − → y ) F ( − → x , − → y )  , (1-2.8) with ea ch of the formulas E and F either a ta utology , or an antilogy , or a conjunction o f atomic formulas, then Mo d ( T ) is closed under directed co l- imits (see, for e xample, the eas y direction of [8, Exer cise 5.2.24]). While there are more genera l ﬁrst-or der sentences pres e rving dire cted colimits, those of the form (1 - 2.8) have the additional adv an tage of pres erving direct pro ducts, which will b e o f impo rtance in the sequel. Example 1-2. 6. I f T consists of universal Horn sentenc es (cf. [26, Sec- tion 1.2.2]), then all the axioms in T ha ve the form (1-2.8), th us Mo d ( T ) is closed under directed co limits. This is the case for the following examples: (i) T is the theory of all gro ups, in the lang ua ge {· , 1 , − 1 } , where · is a bi- nary oper ation sym b ol, 1 is a constan t sym b ol, and − 1 is a unary o pe ration symbol. (ii) T is the theory of all partially ordered abelia n groups, in the language {− , 0 , ≤} , wher e − is a binary op eratio n symbol, 0 is a consta nt symbol, and ≤ is a binar y r e la tion symbol. (iii) T is the theory of all ( ∨ , 0)-semilattices, in the langua ge {∨ , 0 } , where ∨ is a binar y o pe r ation symbol and 0 is a constant symbol. Many other examples are given in Gorbunov [26]. Example 1-2. 7. L et L := {∨ , 0 } , where ∨ is a binary op eration symbol and 0 is a constant symbol, and let T cons ist of the a xioms for ( ∨ , 0)-semi- lattices (i.e., idemp otent commutativ e monoids) together with the axio m ( ∀ x , y , z )  z ∨ x ∨ y = x ∨ y ⇒ ( ∃ x ′ , y ′ )( x ′ ∨ x = x a nd y ′ ∨ y = y and z = x ′ ∨ y ′  . Then all a xioms of T have the form (1-2.8) and the mo dels of T are exactly the distributive ( ∨ , 0) -s emilattic es . Example 1-2. 8. L et L := { + , 0 } , wher e + is a binary opera tion sym b ol, 0 is a constant symbol, and let T consist of all the ax ioms of the theory of commutativ e mo no ids, tog ether with the r eﬁnement axiom , ( ∀ x 0 , x 1 , y 0 , y 1 )  x 0 + x 1 = y 0 + y 1 ⇒ ( ∃ i,j < 2 z i,j ) ^ ^ i< 2 ( x i = z i, 0 + z i, 1 and y i = z 0 ,i + z 1 ,i )  , 1-3 Kappa-presented and w eakly k appa -pr esen ted ob jects 33 where V V denotes conjunction of a list of formulas. Then all axioms of T hav e the form (1-2.8) and the mo dels of T are exa ctly the r eﬁnement monoids . If we add to T the c onic ality axiom (cf. Exa mple 1- 1.3) ( ∀ x , y )( x + y = 0 ⇒ x = y = 0) , then we obtain the theory of c onic al re ﬁn emen t monoids . Example 1-2. 9. L et L := {− , · , 0 } , where − and · are b oth binar y op era tion symbols and 0 is a constant s y m b ol, and T consis ts of all axioms of rings (they are identities) together with the a xiom ( ∀ x )( ∃ y )( x · y · x = x ) . Then all a xioms of T have the form (1-2.8) and the mo dels of T are exactly the (not necessar ily unital) r e gular rings . The given description of dir ected colimits gives a particula r ly s imple wa y to chec k whether a coco ne ( A , ϕ i | i ∈ I ), of mo dels of a theory whos e ax- ioms all hav e the for m (1-2.8), ab ov e a dir ected p oset-indexed diagra m  A i , ϕ j i | i ≤ j in I  , is a colimit of that dia g ram: namely , w e only need to chec k the statements (1-2.4), (1-2.5), and (1-2.6). In pa rticular, in all the examples describ ed abov e exc e pt the one o f pa rtially ordered ab elian groups, the directed colimit is c hara cterized by (1- 2.4) and (1-2.5); in the ca se of partially order ed ab elian gro ups , one also nee ds to chec k that the statement ϕ i ( x ) ≥ 0 ⇒ ( ∃ j ∈ I ↑ i )  ϕ j i ( x ) ≥ 0  holds fo r a ll i ∈ I a nd all x ∈ A i . 1-3 Kappa-presen ted and w eakly k appa-presen ted ob jects In the prese nt section we sha ll intro duce (cf. Deﬁnition 1-3.2) a weak ening of the usual deﬁnition o f a κ -present ed ob ject in a ca tegory . W e ﬁr st re c a ll that deﬁnition, as given in Gabriel and Ulmer [15, Deﬁnition 6 .1 ], see also Ad´ amek and Rosick´ y [2, Deﬁnitions 1.1 and 1.1 3]. Deﬁnition 1 - 3.1. F or an inﬁnite regular cardinal κ , an ob ject A in a cate- gory C is κ -pr esente d if for every κ -directed colimit ( B , b i | i ∈ I ) = lim − →  B i , b j i | i ≤ j in I  in C , the following statements hold: 34 1 Background (i) F or every f : A → B , there exists i ∈ I such that f factor s through B i (i.e., f = b i ◦ f ′ for some morphism f ′ : A → B i ). (ii) F or ea ch i ∈ I a nd morphisms f , g : A → B i such that b i ◦ f = b i ◦ g , there ex ists j ≥ i in I suc h that b j i ◦ f = b j i ◦ g . In case κ = ℵ 0 , we will say ﬁnitely pr esen t e d instead of κ -prese nted. In the following de ﬁnitio n we do not assume r e gularity of the ca rdinal κ . Deﬁnition 1 - 3.2. F or an inﬁnite c a rdinal κ , a n ob ject A in a category C is we akly κ - pr esente d if for e very set Ω and every c ontinuous dir e c ted colimit ( B , b X | X ∈ [ Ω ] <κ ) = lim − →  B X , b Y X | X ⊆ Y in [ Ω ] <κ  in C , every mor phism from A to B factor s through b X : B X → B for some X ∈ [ Ω ] <κ . T rivially , in ca se κ is regular , κ -presented implies weakly κ -presented. F or c o c omplete ca teg ories (i.e., catego ries in whic h every small—not necessar- ily directed—diagram has a co limit), w eakly κ -presented structures can be characterized as fo llows. Prop ositio n 1-3.3. L et κ b e an inﬁnite c ar dinal and let C b e a c o c omplete c ate gory. Then an obje ct A of C is we akly κ - pr esente d iff for every ( non ne c essarily dir e cte d ) p oset I and every c oli mit c o c one ( B , b i | i ∈ I ) = lim − →  B i , b j i | i ≤ j in I  in C , every morphism fr om A to B factors t hr ough lim − → i ∈ J B i for some κ -smal l subset J of I . Pr o of . Supp ose that A satisﬁes the given condition and consider a morphism f : A → B with a contin uous directed colimit  B , b X | X ∈ [ Ω ] <κ  = lim − →  B X , b Y X | X ⊆ Y in [ Ω ] <κ  in C . As this colimit is contin uous, B X = lim − → Y ∈ [ X ] <ω B Y for eac h X ∈ [ Ω ] <κ , th us  B , b X | X ∈ [ Ω ] <ω  = lim − →  B X , b Y X | X ⊆ Y in [ Ω ] <ω  in C . By a ssumption, there exists a κ -small subset I of [ Ω ] <ω such that f factor s through lim − → X ∈ I B X . It follows that the union X of all the elements of I is a κ -small subset of Ω ( as every memb er of I is ﬁn it e, t his do es not r e quir e any r e gularity assumption on κ ) and f factors through B X . Conv ersely , supp ose that A is w eakly κ -prese nted. Let ( B , b i | i ∈ I ) = lim − →  B i , b j i | i ≤ j in I  in C . As C is co co mplete, we can deﬁne C X := lim − → i ∈ X B i for each X ∈ [ I ] <κ , with the ob vious transition and limiting mo rphisms. As B = lim − → X ∈ [ I ] <κ C X is a 1-4 Extension of a functor by directed colimi ts 35 contin uous directed co limit and A is weakly κ -presented, every morphism from A to B factor s through C J = lim − → i ∈ J B i , for so me J ∈ [ I ] <κ . ⊓ ⊔ Corollary 1 - 3.4. L et λ and κ b e inﬁn ite c ar dinals with λ ≤ κ and let C b e a c o c omplete c ate gory. Then every we akly λ -pr esente d obje ct of C is also we akly κ -pr esente d. Example 1-3. 5. F o r an inﬁnite ca rdinal κ , we ﬁnd a co mplete lattice (thus a coco mplete categ ory) L κ in whic h the unit is w eakly κ -presented in L κ but not weakly α -presented for any α < κ . Denote b y A κ the set of all inﬁnite cardinals smaller than κ and endo w the set L ∗ κ := { ( α, x ) | α ∈ A κ and x ∈ [ α ] <α } with the partial order ing deﬁned by ( α, x ) ≤ ( β , y ) ⇐ ⇒ ( α = β and x ⊆ y ) . W e set L κ := L ∗ κ ∪ { 0 , 1 } , fo r a new smallest (resp., larg est) ele men t 0 (res p., 1). Then L κ is a complete lattice, thus, viewed a s a ca tegory , L κ is co co mplete. Nontrivial joins in L κ are given by ( α, x ) ∨ ( β , y ) = 1 if α 6 = β , (1-3.1) _ (( α, x i ) | i ∈ I ) = (  α, S ( x i | i ∈ I )  , if ca r d S ( x i | i ∈ I ) < α , 1 , if car d S ( x i | i ∈ I ) = α . (1-3.2) Suppo se that the unit o f L κ is the colimit (i.e., join) o f a family (( α i , x i ) | i ∈ I ) of elements of L ∗ κ , we m ust ﬁnd a κ -small subset J if I such that 1 = W (( α i , x i ) | i ∈ J ). If α i 6 = α j for some i, j then, using (1 - 3.1), 1 = ( α i , x i ) ∨ ( α j , x j ) and we ar e done. Now supp ose that α i = α for all i ∈ I . It follows from (1-3.2) that the union x of all x i has cardinality α . Pick i ξ ∈ I such that ξ ∈ x i ξ , for each ξ ∈ x . Then the set J := { i ξ | ξ ∈ x } has ca rdinality at most α (th us it is κ - small), and 1 = W (( α i , x i ) | i ∈ J ). Therefore, by Prop ositio n 1-3.3, 1 is weakly κ -presented in L κ . Now let α ∈ A κ . Then 1 = W (( α, { ξ } ) | ξ < α ) in L κ , but (using (1-3 .2)) there is no α -small subset u of α suc h that 1 = W (( α, { ξ } ) | ξ ∈ u ). Ther efore, by Prop osition 1-3.3, 1 is not weakly α -prese nted in L κ . 1-4 Extension of a functor b y directed colimits The main result of the present section, Prop ositio n 1-4.2, sta tes a very in tu- itive and probably mostly well-known fact, pa rts of which are already prese n t in the literature such a s Pudl´ ak [5 0], although w e could not trace a reference 36 1 Background where it is s ta ted explicitly in the strong for m that w e shall r equire. W e are given categories A a nd S , tog ether with a full s ub ca tegory A † of A a nd a functor Φ : A † → S . W e wish to e x tend Φ to a functor, whic h preserves all small dir ected co limits, fro m all small dir ected colimits of diagrams from A † to S . Prop osition 1 - 4.2 states suﬃcien t conditions under which this can b e done. Lemma 1 - 4.1. L et A † b e a ful l sub c ate gory of a c ate gory A , let S b e a c at- e gory with al l smal l dir e ct e d c olimits, and let Φ : A † → S b e a functor. L et f : A → B b e a morphism in A , to gether with c olimits ( A, α i | i ∈ I ) = lim − → − → A , with − → A =  A i , α i ′ i | i ≤ i ′ in I  , (1-4.1) ( B , β j | j ∈ J ) = lim − → − → B , with − → B =  B j , β j ′ j | j ≤ j ′ in J  , (1-4.2) with b oth p osets I and J dir e ct e d, al l A i ﬁnitely pr esente d, and al l A i and B j b elonging t o A † . S et K := { ( i, j, x ) | ( i, j ) ∈ I × J, x : A i → B j , and β j ◦ x = f ◦ α i } ,  A, α i | i ∈ I  := lim − → Φ − → A ,  B , β j | j ∈ J  := lim − → Φ − → B . Then ther e exists a unique morphism f : A → B such t hat β j ◦ Φ ( x ) = f ◦ α i for e ach ( i, j, x ) ∈ K . Note. While I a nd J are a ssumed to be s e ts, K may b e a pro p er class . Pr o of . W e deﬁne a par tial o rdering ≤ o n the cla s s K by the rule ( i, j, x ) ≤ ( i ′ , j ′ , x ′ ) ⇐ ⇒ ( i ≤ i ′ , j ≤ j ′ , and β j ′ j ◦ x = x ′ ◦ α i ′ i ) . W e claim that K is directed. Let ( i 0 , j 0 , x 0 ) , ( i 1 , j 1 , x 1 ) ∈ K . W e pick i ≥ i 0 , i 1 in I . As A i is ﬁnitely presented, there exists j ∈ J such that j ≥ j 0 , j 1 and the morphism f ◦ α i : A i → B factors thro ugh B j ; the la tter means that ther e exists x : A i → B j such that β j ◦ x = f ◦ α i . It follows tha t β j ◦ x ◦ α i i 0 = f ◦ α i ◦ α i i 0 = f ◦ α i 0 = β j 0 ◦ x 0 = β j ◦ β j j 0 ◦ x 0 , and thus, as A i 0 is ﬁnitely presented, there exists j ′ ∈ J ↑ j such that β j ′ j ◦ x ◦ α i i 0 = β j ′ j ◦ β j j 0 ◦ x 0 , that is, β j ′ j ◦ x ◦ α i i 0 = β j ′ j 0 ◦ x 0 . This implies that ( i, j ′ , β j ′ j ◦ x ) b elongs to K ↑ ( i 0 , j 0 , x 0 ). A similar ar g ument , using the ﬁnite presentabilit y of A i 1 , yields an elemen t j ′′ ∈ J ↑ j ′ such 1-4 Extension of a functor by directed colimi ts 37 that ( i, j ′′ , β j ′′ j ◦ x ) b elongs to K ↑ ( i 1 , j 1 , x 1 ). Therefore, ( i, j ′′ , β j ′′ j ◦ x ) is an element of K ab ov e b oth elements ( i 0 , j 0 , x 0 ) and ( i 1 , j 1 , x 1 ), whic h completes the pro of of our claim. Both pro jections λ : K → I , ( i, j, x ) 7→ i and µ : K → J , ( i, j, x ) 7→ j ar e isotone. Let i ∈ I . As A i is ﬁnitely presen ted, there exists j ∈ J such that the morphism f ◦ α i : A i → B factors thro ugh B j ; furthermore, j can b e chosen ab ov e any given element o f J . By deﬁnition, there exists x : A i → B j such that ( i, j, x ) ∈ K . This proves that λ is surjective (thus, a fortiori , it ha s coﬁnal range) a nd µ has coﬁnal ra ng e. As bo th λ and µ are isotone with coﬁnal range and K is directed, it follo ws from [2 , Section 0.11] that  A, α i | ( i, j, x ) ∈ K  = lim − → Φ − → A λ ,  B , β j | ( i, j, x ) ∈ K  = lim − → Φ − → B µ . By the deﬁnition of the order ing of K , the family ( Φ ( x ) | ( i, j, x ) ∈ K ) deﬁnes a na tural transfo rmation from Φ − → A λ to Φ − → B µ . The universal pr op erty of the colimit gives the desired conclusion. ⊓ ⊔ Prop ositio n 1-4.2. L et A † b e a ful l su b c ate gory of ﬁnitely pr esente d obje cts in a c ate gory A , let S b e a c ate gory with al l smal l dir e ct e d c olimits. We assume that every obje ct in A is a smal l dir e cte d c olimit of obje cts in A † . Then every functor Φ : A † → S ex tends to a functor Φ : A → S which pr eserves al l c olimits of dir e cte d p oset-indexe d diagr ams in A † . F urthermor e, if A † has smal l hom- sets, then Φ pr eserves al l smal l dir e cte d c olimits. Pr o of . The ﬁrs t part of the pro of, up to the preser v ation of directed colimits from A † , is established in a similar fashion as in the proo f of the Cor o llary in Pudl´ ak [50, pag e 101]. F or every ob ject A o f A , we pick a r epresentation of the form ( A, α i | i ∈ I ) = lim − →  A i , α i ′ i | i ≤ i ′ in I  in A , (1-4.3) which we shall call the c anonic al r epr esentation of A , in such a wa y that I is directed, A i ∈ A † for all i ∈ I , and I = {⊥ } with A ⊥ = A in ca se A ∈ A † . W e deﬁne Φ on o b jects by picking a co cone  Φ ( A ) , α i | i ∈ I  such that  Φ ( A ) , α i | i ∈ I  = lim − →  Φ ( A i ) , Φ ( α i ′ i ) | i ≤ i ′ in I  in S . By Lemma 1-4.1, for ea ch mor phism f : A → B in A with canonical repre- sentations (1-4.1) and (1-4 .2), there exists a unique morphism Φ ( f ) : Φ ( A ) → Φ ( B ) in S suc h that β j ◦ Φ ( x ) = Φ ( f ) ◦ α i for all ( i, j ) ∈ I × J and all x : A i → B j such that β j ◦ x = f ◦ α i . W e ﬁrst prov e that Φ is a functor. It is clear that Φ sends identities to ident ities. Now let f : A → B and g : B → C in A , put h := g ◦ f , and let the 38 1 Background canonical re pr esentations o f A , B , and C b e resp ectively giv en by (1-4.1), (1-4.2), and ( C, γ k | k ∈ K ) = lim − →  C k , γ k ′ k | k ≤ k ′ in K  . Let ( i, k ) ∈ I × K and z : A i → C k such that γ k ◦ z = h ◦ α i . As A i is ﬁnitely pres ent ed, ther e a r e j ∈ J and x : A i → B j such that β j ◦ x = f ◦ α i . As B j is ﬁnitely presented, ther e are k ′ ≥ k in K and y : B j → C k ′ such that γ k ′ ◦ y = g ◦ β j . As γ k ′ ◦ y ◦ x = g ◦ β j ◦ x = g ◦ f ◦ α i = h ◦ α i = γ k ◦ z and A i is ﬁnitely presented, ther e exists k ′′ ≥ k ′ in K such that γ k ′′ k ′ ◦ y ◦ x = γ k ′′ k ◦ z . Hence, by repla c ing k ′ by k ′′ and y by γ k ′′ k ′ ◦ y , we may assume that k ′ = k ′′ , and he nc e y ◦ x = γ k ′ k ◦ z (see Fig ure 1.2). C k γ k ′ k   γ k   Φ ( C k ) Φ ( γ k ′ k )   γ k   A i x / / α i   z 6 6 n n n n n n n n n n n n n n n B j y / / β j   C k ′ γ k ′   Φ ( A i ) Φ ( x ) / / α i   Φ ( z ) 3 3 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Φ ( B j ) Φ ( y ) / / β j   Φ ( C k ′ ) γ k ′   A f / / B g / / C Φ ( A ) Φ ( f ) / / Φ ( B ) Φ ( g ) / / Φ ( C ) Fig. 1. 2 Proving that Φ i s a functor It follows (see again Fig ure 1.2) that Φ ( g ) ◦ Φ ( f ) ◦ α i = Φ ( g ) ◦ β j ◦ Φ ( x ) = γ k ′ ◦ Φ ( y ) ◦ Φ ( x ) = γ k ′ ◦ Φ ( γ k ′ k ) ◦ Φ ( z ) = γ k ◦ Φ ( z ) . Therefore, b y deﬁnitio n, Φ ( g ) ◦ Φ ( f ) = Φ ( h ) = Φ ( g ◦ f ), and so Φ is a functor. Now let A be an ob ject of A , with canonical re pr esentation (1-4.3) a nd for which another representation ( A, β j | j ∈ J ) = lim − →  B j , β j ′ j | j ≤ j ′ in J  , is given, where J is dire c ted and B j ∈ A † for all j ∈ J . W e shall prov e tha t 1-4 Extension of a functor by directed colimi ts 39  Φ ( A ) , Φ ( β j ) | j ∈ J  = lim − →  Φ ( B j ) , Φ ( β j ′ j ) | j ≤ j ′ in J  . (1-4.4) W e put  B , β j | j ∈ J  := lim − →  Φ ( B j ) , Φ ( β j ′ j ) | j ≤ j ′ in J  , (1-4.5) U := { ( i, j, x ) | ( i, j ) ∈ I × J, x : A i → B j , and β j ◦ x = α i } , V := { ( i, j, y ) | ( i, j ) ∈ I × J, y : B j → A i , and β j = α i ◦ y } . By Lemma 1- 4 .1, there are unique morphisms u : Φ ( A ) → B and v : B → Φ ( A ) such that β j ◦ Φ ( x ) = u ◦ α i for all ( i, j, x ) ∈ U and α i ◦ Φ ( y ) = v ◦ β j for a ll ( i, j, y ) ∈ V . As in the par agra ph ab ove, it follows that v ◦ u ◦ α i = α i for each i ∈ I ; whence v ◦ u = id Φ ( A ) . Similarly , u ◦ v = id B , and thus u a nd v ar e mutually inv ers e iso morphisms. F or all j ∈ J , Φ ( β j ) is the unique morphism from Φ ( B j ) to Φ ( A ) such that Φ ( β j ) = α i ◦ Φ ( y ) for each i ∈ I a nd all y : B j → A i with β j = α i ◦ y . It follows that Φ ( β j ) = v ◦ β j , and hence, b y (1-4.5) and as v is an isomo rphism, (1 -4.4) follows. So far we hav e pro ved that Φ pres e rves all colimits of directed p oset- indexed diagrams in A † . No w w e assume that A † has small ho m-sets. W e a re given a n arbitra r y small directed colimit in A as in (1 -4.3), where the A i are no lo nger as s umed to be in A † , we must prove that the following statement holds:  Φ ( A ) , Φ ( α i ) | i ∈ I  = lim − →  Φ ( A i ) , Φ ( α i ′ i ) | i ≤ i ′ in I  in S . (1-4.6) F or each i ∈ I , we pick a representation  A i , α i i,j | j ∈ J i  = lim − →  A i,j , α i,j ′ i,j | j ≤ j ′ in J i  in A , (1-4.7) where the p oset J i is dir e cted and a ll A i,j belo ng to A † . W e put P := [ ( { i } × J i | i ∈ I ) . F or ( i, j ) , ( i ′ , j ′ ) ∈ P , w e deﬁne a morphism from ( i, j ) to ( i ′ , j ′ ) as a mor- phism x : A i,j → A i ′ ,j ′ in A † such that α i ′ i ′ ,j ′ ◦ x = α i ′ i ◦ α i i,j (this r e- quires i ≤ i ′ ). This deﬁnes a ca teg ory P with underlying set P . As A † has small ho m-sets, P is a small categ o ry . W e put α i ′ i,j := α i ′ i ◦ α i i,j , for all i ≤ i ′ in I and all j ∈ J i . Claim. The categ ory P is ﬁlter ed. Pr o of . First let ( i 0 , j 0 ) , ( i 1 , j 1 ) ∈ P . There exists i ≥ i 0 , i 1 in I . As A i l ,j l is ﬁnitely presented, there exists j ′ l ∈ J i such that α i i l ,j l factors through A i,j ′ l , for all l < 2. Hence, ta king j ≥ j ′ 0 , j ′ 1 in J i , b oth mor phisms α i i 0 ,j 0 and α i i 1 ,j 1 40 1 Background factor through A i,j , which yields x l : ( i l , j l ) → ( i, j ), for all l < 2. Part of the argument can b e follow ed on Figure 1 .3. A i l ,j l α i l i l ,j l / / x l   α i i l ,j l L L L L & & L L L L A i l α i i l   A i,j α i i,j / / A i Fig. 1. 3 A commutat ive square in A Next, let x, y : ( i 0 , j 0 ) → ( i 1 , j 1 ) in P , so α i 1 i 0 ,j 0 = α i 1 i 1 ,j 1 ◦ x = α i 1 i 1 ,j 1 ◦ y . As A i 0 ,j 0 is ﬁnitely presented, there exists j ≥ j 1 in J i 1 such that α i 1 ,j i 1 ,j 1 ◦ x = α i 1 ,j i 1 ,j 1 ◦ y . Therefore, α i 1 ,j i 1 ,j 1 : ( i 1 , j 1 ) → ( i 1 , j ) co equalizes x a nd y in P . ⊓ ⊔ Claim Now w e put α i,j := α i ◦ α i i,j , for all ( i , j ) ∈ P , and we deﬁne a functor A : P → A by A ( i, j ) := A i,j for ( i, j ) ∈ P and A ( x ) := x for every mo rphism x in P . It is str a ightforw ard to verify that ( A i,j , α i,j | ( i, j ) ∈ P ) is a co cone ab ov e A . W e shall now establish the statement ( A, α i,j | ( i , j ) ∈ P ) = lim − → A . (1-4.8) Let ( B , β i,j | ( i , j ) ∈ P ) b e a co cone ab ov e A . In pa rticular, for all i ∈ I , the family ( B , β i,j | j ∈ J i ) is a co cone a b ov e  A i,j , α i,j ′ i,j | j ≤ j ′ in J i  , thus, by (1-4.7), there exists a unique morphism β i : A i → B suc h that β i,j = β i ◦ α i i,j for all j ∈ J i . Let i ≤ i ′ in I and le t j ∈ J i . As α i ′ i ◦ α i i,j : A i,j → A i ′ = lim − → j ′ ∈ J i ′ A i ′ ,j ′ and A i,j is ﬁnitely presented, there a re j ′ ∈ J i ′ and x : ( i, j ) → ( i ′ , j ′ ) in P . Hence, β i ′ ◦ α i ′ i ◦ α i i,j = β i ′ ◦ α i ′ i ′ ,j ′ ◦ x = β i ′ ,j ′ ◦ x = β i,j = β i ◦ α i i,j . As this holds for all j ∈ J i , it follows that β i ′ ◦ α i ′ i = β i . Hence, a s A = lim − → i ∈ I A i , there exis ts a unique morphism ϕ : A → B such that β i = ϕ ◦ α i for each i ∈ I . It follows that 1-4 Extension of a functor by directed colimi ts 41 ϕ ◦ α i,j = ϕ ◦ α i ◦ α i i,j = β i ◦ α i i,j = β i,j , for all ( i, j ) ∈ P . Now let ψ : A → B suc h that ψ ◦ α i,j = β i,j , for all ( i, j ) ∈ P . F or i ∈ I and j ∈ J i , we compute ψ ◦ α i ◦ α i i,j = ψ ◦ α i,j = β i,j = β i ◦ α i i,j . As this holds for all j ∈ J i , it follows that ψ ◦ α i = β i . As this holds for all i ∈ I and by the uniqueness statement deﬁning ϕ , it follows that ϕ = ψ , th us co mpleting the pr o of o f (1-4.8). As, by the claim above, P is a small ﬁltered c a tegory, it follows fr o m [2, Theorem 1.5 ] that there are a small directed poset P and a coﬁnal functor from P to P . As we ha ve seen that Φ preser ves all co limits of directed p oset- indexed diagra ms in A † , it follows from [2, Section 0.1 1] and (1-4.8) that  Φ ( A ) , Φ ( α i,j ) | ( i, j ) ∈ P  = lim − → Φ A . (1-4.9) Now we can conclude the pr o of of (1-4.6). Let ( S, σ i | i ∈ I ) b e a co cone in S ab ov e  Φ ( A i ) , Φ ( α i ′ i ) | i ≤ i ′ in I  . Put σ i,j := σ i ◦ Φ ( α i i,j ), for all ( i, j ) ∈ P . F or every morphism x : ( i, j ) → ( i ′ , j ′ ) in P , σ i ′ ,j ′ ◦ Φ ( x ) = σ i ′ ◦ Φ ( α i ′ i ′ ,j ′ ) ◦ Φ ( x ) = σ i ′ ◦ Φ ( α i ′ i ) ◦ Φ ( α i i,j ) = σ i ◦ Φ ( α i i,j ) = σ i,j . This prov es that ( S, σ i,j | ( i, j ) ∈ P ) is a co cone ab ove Φ A , thus, by (1-4.9), there exists a unique ϕ : Φ ( A ) → S s uch that σ i,j = ϕ ◦ Φ ( α i,j ) for all ( i, j ) ∈ P . F or a ll ( i, j ) ∈ P , σ i ◦ Φ ( α i i,j ) = σ i,j = ϕ ◦ Φ ( α i,j ) = ϕ ◦ Φ ( α i ) ◦ Φ ( α i i,j ) . Fix i ∈ I . As the equation above is satisﬁed for all j ∈ J i , it follows that σ i = ϕ ◦ Φ ( α i ). Finally , let ψ : Φ ( A ) → S s atisfy σ i = ψ ◦ Φ ( α i ) for all i ∈ I . Then σ i,j = ψ ◦ Φ ( α i,j ) for all ( i, j ) ∈ P , and th us, b y the uniqueness statement deﬁning ϕ , we get ϕ = ψ , which concludes the pro of. ⊓ ⊔ Remark 1- 4 .3. Under mild se t- theoretical as s umptions, it is easy to remov e from the statement of the ﬁnal sentence of Pro po sition 1-4.2 the hypothesis that A † has sma ll hom-sets. In the pro o f of P rop osition 1- 4.2, we need to replace P by a smal l sub ca tegory P ∗ satisfying the following conditions : 42 1 Background (i) Ob P ∗ = P . (ii) The mo r phism α i,j 1 i,j 0 belo ngs to Mor P ∗ (( i, j 0 ) , ( i, j 1 )), for all i ∈ I a nd all j 0 ≤ j 1 in J i . (iii) Mor P (( i, j ) , ( i ′ , j ′ )) 6 = ∅ iﬀ Mor P ∗ (( i, j ) , ( i ′ , j ′ )) 6 = ∅ , for all ele- men ts ( i, j ) and ( i ′ , j ′ ) in P . F or ex ample, such a P ∗ can b e constructed in case the ambien t set- theoretical universe satisﬁes the Ber nays-G¨ odel class theo ry with axiom of foundation. How ever, as we will ne e d Prop os ition 1-4 .2 only in cas e A = Bo o l P (cf. Section 2-2), which has small hom-sets, we sha ll not ex- pand on this further here. Remark 1- 4 .4. Let κ b e a n inﬁnite regular cardinal. If we assume only , in the statement o f Prop os itio n 1- 4 .2, tha t S has a ll κ -small directed colimits, that e very ob ject in A is a co limit o f a κ -small direc ted po set-indexed dia gram in A † , and tha t A † has κ -small hom-sets, then the following a nalogue o f the conclusion of Pr op osition 1 -4.2 remains v alid: The funct or Φ extends to a functor Φ : A → S that pr eserves al l κ - smal l dir e cte d c olimi ts . W e shall use this result only in cas e A = Bo ol P (cf. Section 2-3), where thes e assumptions will b e automatically satisﬁed. Remark 1- 4 .5. It is not hard to verify that any t wo extensions of Φ to A pre- serving all co limits of directed p os e t-indexed diagra ms in A † are isomor phic ab ov e Φ . Hence w e shall call the functor Φ c onstructed in Pr op osition 1-4.2 the natur al ext ension of Φ to A . 1-5 Pro jectabilit y wit nesses Given c ategories A and B together with a functor Ψ : A → B , it will o f- ten be the case (though not a lwa ys, a notable exception being developed in Chapter 5) tha t certain a r rows o f the form ϕ : Ψ ( A ) → B c an b e tur ne d into isomorphi sms of the form ε : Ψ ( A ) → B , for a “ca no nical quotient” A of A . Lemma 1-5 .2 will be our key idea to turn statements involving the “double arrows”, mentioned in Se c tio n 1- 1 .3 and intro duced for mally in Se c tion 3- 4, to isomor phisms. Roughly sp eaking, the existence of pro jectability witnesses is a categor ical combination of the First and Second Is omorphism Theo rems for algebra ic systems (cf. Lemmas 4-1.5 and 4- 1.6). W e recall a deﬁnition from W ehrung [65]. Deﬁnition 1 - 5.1. Let Ψ b e a functor from a categor y A to a categ ory B , let A ∈ Ob A , B ∈ Ob B , and ϕ : Ψ ( A ) → B . A pr oje ctability witness for ( ϕ, A, B ) (or, abusing no tation, “for ϕ : Ψ ( A ) → B ”) with res pec t to Ψ is a pair ( a, ε ) sa tisfying the fo llowing conditions: (i) a : A ։ A is a n epimor phism in A . 1-5 Pro jectability witnesses 43 (ii) ε : Ψ ( A ) → B is an isomor phism in B . (iii) ϕ = ε ◦ Ψ ( a ). (iv) F or e very f : A → X in A and ev ery η : Ψ ( A ) → Ψ ( X ) suc h that Ψ ( f ) = η ◦ Ψ ( a ), there exists g : A → X in A such that f = g ◦ a a nd η = Ψ ( g ). W e observe that the mor phism g in (iii) ab ov e is necessarily unique (be - cause a is an epimorphis m). F urthermo r e, the pr o jectability witness ( a, ε ) is unique up to isomorphism. Deﬁnition 1-5.1 is illus trated on Figure 1.4. Ψ ( A ) ϕ / / Ψ ( a )   B Ψ ( A ) Ψ ( a )   Ψ ( f ) ' ' P P P P P P P P P P P P A a     f   > > > > > > > > Ψ ( A ) ε ∼ = | | | = = | | | Ψ ( A ) η = Ψ ( g ) / / Ψ ( X ) A g / / X Fig. 1. 4 ( a, ε ) is a pro j ectabilit y witness for ϕ : Ψ ( A ) → B Lo osely sp eak ing, the following proper t y says that the existence of enough pro jectability witnesses entails the existence of liftings. Lemma 1 - 5.2. L et I , A , and S b e c ate gories, let D : I → A , Ψ : A → S , and S : I → S b e functors. L et τ : Ψ D . → S b e a natur al tr ansformation such that τ i : Ψ D ( i ) → S ( i ) has a pr oje ctability witn ess ( a i , η i ) , with a i : D ( i ) ։ E ( i ) , for e ach i ∈ Ob I . Then E c an b e uniquely extende d t o a fun ctor fr om I to A such that a : D . → E is a natu r al tr ansformation and η : Ψ E . → S is a natu ra l e quivalenc e with τ i = η i ◦ Ψ ( a i ) for e ach i ∈ O b I . Pr o of . Note that η i : Ψ E ( i ) ∼ = → S ( i ) and τ i = η i ◦ Ψ ( a i ), for each i ∈ Ob I . W e need to extend E to a functor. Let f : i → j in I . As ( η − 1 j ◦ S ( f ) ◦ η i ) ◦ Ψ ( a i ) = η − 1 j ◦ S ( f ) ◦ τ i = η − 1 j ◦ τ j ◦ Ψ D ( f ) = Ψ ( a j ◦ D ( f )) , there ex ists a unique E ( f ) : E ( i ) → E ( j ) such that Ψ E ( f ) = η − 1 j ◦ S ( f ) ◦ η i and E ( f ) ◦ a i = a j ◦ D ( f ), s ee Figure 1.5. The unique ne s s statement on E ( f ), together with routine ca lculations, show easily that E is a functor. ⊓ ⊔ 44 1 Background D ( i ) D ( f ) / / a i     D ( j ) a j     Ψ E ( i ) Ψ E ( f ) / / η i ∼ =   Ψ E ( j ) η j ∼ =   E ( i ) E ( f ) / / E ( j ) S ( i ) S ( f ) / / S ( j ) Fig. 1. 5 The functor E created from pro j ectabilit y witnesses Chapter 2 Bo olean alg ebras t hat are scaled with resp ect to a p ose t Abstract. Our main result, CLL (Lemma 3 -4.2), in volv es a construction that turns a diagr am − → A , indexed b y a p os e t P , from a category A , to an obje ct of A , called a c ondensate of − → A (cf. Deﬁnition 3 -1.5). A condensate of − → A will be wr itten in the from B ⊗ − → A , where B is a Bo olea n algebr a with addi- tional structure—we shall say a P - sc ale d Bo ole an algebr a (Deﬁnition 2-2.3). It will turn o ut (cf. P rop osition 2-2.9) that P -scale d Bo olean a lg ebras are the dual ob jects of top ologica l ob jects ca lled P -norme d Bo ole an sp ac es (cf. Deﬁnition 2- 2 .1). By deﬁnitio n, a P -no rmed topo logical space is a top o lo g- ical space X endow ed with a map (the “nor m function”) fro m X to Id P which is c ontinuous with resp e c t to the given top o lo gy of X and the Scott top ology o n Id P . In case X is a o ne - po int s pace w ith no rm an ideal H of P and B is the c o rresp onding P -scaled Bo olea n algebra , B ⊗ − → A = lim − → p ∈ H A p . In cas e X is ﬁnite and ν ( x ) = P ↓ f ( x ) (where f ( x ) ∈ P ) for each x ∈ X , then B ⊗ − → A = Q  A f ( x ) | x ∈ X  . The latter situation describ es the case where B is a ﬁ nitely pr esente d P -s caled Bo olean a lg ebra (cf. Deﬁnition 2 - 4.1 and Corollary 2-4.7). In the general c a se, there is a dir ected colimit repre- sentation B = lim − → i ∈ I B i where all the B i are ﬁnitely pr esented (cf. Prop o s i- tion 2-4.6) and then B ⊗ − → A is deﬁned as the corresp onding directed colimit of the B i ⊗ − → A . That this can b e done, and that the resulting functor B 7→ B ⊗ − → A preserves all small directed colimits, will follow fro m Pr op osition 1 - 4.2. 2-1 Pseudo join-semilattices, supported p osets, and almost join-semilattices The statement of CLL (Lemma 3-4 .2) in volves p osets P for which ther e exists a “ λ -lifter” ( X , X ); here X is a p oset, endo wed with an isoto ne map ∂ : X → P , and X is a certain set of ideals of X . Deﬁning the condensate 45 46 2 Bo olean algeb ras scaled wi th resp ect to a poset F ( X ) ⊗ − → A , inv olved in the statement of CLL, req uires the construction of a certain P -s caled Bo olean a lgebra, deﬁned by generator s and relations, F ( X ). And the deﬁnition of F ( X ) will require X be a pseudo join-semilattic e (cf. Deﬁnition 2 -1.2). T og ether with pseudo join-semilattices, we will also need to int ro duce supp orte d p osets and almost join-semilattic es . Notation 2-1. 1. Le t X b e a subset in a p oset P . W e denote by ▽ X , o r ▽ P X in case P needs to be sp eciﬁed, the s et of all minimal elements of P ⇑ X . W e shall write a 0 ▽ · · · ▽ a n − 1 , or ▽ i ℵ 0 , then the ▽ -closure Z ▽ of a λ -small subset Z of X is also λ -small; this a lso holds for λ = ℵ 0 , beca us e in that case X is suppo rted. In particular , in any case, X is the dir ected union of the set [ X ] <λ ▽ of all its λ -small ▽ -closed subsets. By using (CONT( Φ )), Lemma 2 -6.6, and Prop osition 3-1 .4, w e obtain that for each x ∈ X = , χ ◦ ϕ x : S x → Φ  F ( X ) ⊗ − → A  = lim − →  Φ  F ( Z ▽ ) ⊗ − → A  | Z ∈ [ X ] <λ  (the trans ition and limiting mor phisms are all of the form Φ  f Z 1 Z 0 ⊗ − → A  , where the f Z 1 Z 0 are given by Lemma 2-6 .6). As S x is, by ass umption, weakly λ -presented, there exists V ( x ) ∈ [ X ] <λ such that χ ◦ ϕ x factors through Φ  F ( V ( x ) ▽ ) ⊗ − → A  . F urthermore , for each x ∈ X = , the set V ( x ) := [ ( V ( x ′ ) | x ′ ∈ X ↓ x ) 66 3 The Condensat e Lif ting Lemma (CLL) is λ -small (beca us e X ↓ x is cf ( λ )-small and all the sets V ( x ′ ) ar e λ -small). Hence V ( x ) ▽ is also λ -small. Therefore, repla c ing V ( x ) b y V ( x ) ▽ , we may assume that the map V is isoto ne, with v alues in [ X ] <λ ▽ . W e shall ﬁx the map V un til the end o f the pro of of Lemma 3-2 .2. W rite χ ◦ ϕ x = Φ  f X V ( x ) ⊗ − → A  ◦ ψ x , (3-2.1) for some morphism ψ x : S x → Φ  F ( V ( x )) ⊗ − → A  . The equation (3-2.1) can also b e visualized on Figure 3.2. S x ϕ x / / ψ x   S χ   Φ  F ( V ( x )) ⊗ − → A  Φ  f X V ( x ) ⊗ − → A  / / Φ  F ( X ) ⊗ − → A  Fig. 3. 2 The morphis ms ψ x As ( X , X ) is a λ -lifter of P , its no rm has a free isotone section σ : P ֒ → X with resp ect to the set mapping V , that is, V ( σ ( p )) ∩ σ ( q ) ⊆ σ ( p ) for all p < q in P . (3-2.2) Claim. The eq uation π X σ ( q ) ◦ f X V σ ( p ) = ε q p ◦ π X σ ( p ) ◦ f X V σ ( p ) is satisﬁed for all p < q in P . Pr o of . W e need to v erify that the giv en morphisms agree on the canonical generator s of F ( V σ ( p )), that is, the elemen ts ˜ u V σ ( p ) , for u ∈ V σ ( p ). W e pro ceed: ( π X σ ( q ) ◦ f X V σ ( p ) )( ˜ u V σ ( p ) ) = π X σ ( q ) ( ˜ u X ) = ( 1 , if u ∈ σ ( q ) 0 , otherwise , while ( ε q p ◦ π X σ ( p ) ◦ f X V σ ( p ) )( ˜ u V σ ( p ) ) = ( ε q p ◦ π X σ ( p ) )( ˜ u X ) = ( 1 , if u ∈ σ ( p ) 0 , otherwise . By (3-2.2), the tw o expres sions ag ree. ⊓ ⊔ Claim By applying the functor − ⊗ − → A (cf. P rop osition 3-1 .2) to the r e sult of the Claim ab ov e a nd as ε q p ⊗ − → A = α q p (cf. Lemma 3-1.3(ii)), we th us obtain the following equation, for all p < q in P : 3-2 Lif ters and the Armature Lemma 67 ( π X σ ( q ) ⊗ − → A ) ◦ ( f X V σ ( p ) ⊗ − → A ) = α q p ◦ ( π X σ ( p ) ⊗ − → A ) ◦ ( f X V σ ( p ) ⊗ − → A ) . (3-2.3) F urthermore, from p < q it follows that σ ( p ) ∈ X = , th us, substituting σ ( p ) to x in the dia gram of Figure 3.2, we obtain that the dia gram of Figure 3.3 is commutativ e. S σ ( p ) ϕ σ ( p ) / / ψ σ ( p )   S χ   Φ  F ( V σ ( p )) ⊗ − → A  Φ  f X V σ ( p ) ⊗ − → A  / / Φ  F ( X ) ⊗ − → A  Fig. 3. 3 The morphis ms ψ σ ( p ) Now w e can pro ceed: Φ ( α q p ) ◦ ρ σ ( p ) ◦ ϕ σ ( p ) = Φ ( α q p ) ◦ Φ  π X σ ( p ) ⊗ − → A  ◦ χ ◦ ϕ σ ( p ) (b y the deﬁnition o f ρ σ ( p ) ) = Φ ( α q p ) ◦ Φ  π X σ ( p ) ⊗ − → A  ◦ Φ  f X V σ ( p ) ⊗ − → A  ◦ ψ σ ( p ) (cf. Fig ure 3.3) = Φ  π X σ ( q ) ⊗ − → A  ◦ Φ  f X V σ ( p ) ⊗ − → A  ◦ ψ σ ( p ) (use (3-2.3)) = Φ  π X σ ( q ) ⊗ − → A  ◦ χ ◦ ϕ σ ( p ) (cf. Figure 3.3) = ρ σ ( q ) ◦ ϕ σ ( p ) (b y the deﬁnition of ρ σ ( q ) ) = ρ σ ( q ) ◦ ϕ σ ( q ) ◦ ϕ σ ( q ) σ ( p ) . S σ ( p ) ϕ σ ( q ) σ ( p ) / / ρ σ ( p ) ◦ ϕ σ ( p )   S σ ( q ) ρ σ ( q ) ◦ ϕ σ ( q )   Φ ( A p ) Φ ( α q p ) / / Φ ( A q ) Fig. 3. 4 Getting the desired natural transformation Therefore, the diagram of Fig ure 3.4 commutes, as desired. ⊓ ⊔ 68 3 The Condensat e Lif ting Lemma (CLL) 3-3 The L¨ ow enheim-Sk olem Condition and t he Buttress Lemma The main result of Section 3-3 is a technical result with no obvious mea ning, Lemma 3-3 .2. Ro ug hly spe a king, it sta tes the following. W e are g iven cate- gories B and S , tog e ther with a sub categ ory S ⇒ of S (the “double a rrows”), a full s ub ca tegory B † of B (the “ small o b jects” of B ), a functor Ψ : B → S , and ob ject B of B , and a family of double arrows ρ u : Ψ ( B ) ⇒ S u , where u ranges o ver a p oset U . The latter sa ys that all ob jects S u are “small”. A but - tr ess of the family ( ρ u | u ∈ U ) (cf. Deﬁnition 3-3 .1) is a n U -indexed diagr am in B † ↓ B that witnesses the c ol le ctive sma llness of ( S u | u ∈ U ). The main assumption required in Lemma 3- 3 .2 is a L¨ owenheim-Sk olem type prop erty , denoted ther e by (LS b λ ( B )). In this sense, Lemma 3 -3.2 may be cons ide r ed a diagr am version of the L¨ owenheim-Sk olem pro p e rty . All the uses that we have b een able to ﬁnd so far for Lemma 3-3.2 hav e the po s et U wel l- founde d . Howev er, the future may also bring uses o f that lemma in the non w ell-founded case, hence w e record it also for tha t case. There are other p oss ible v ariants of L e mma 3- 3.2, that w e shall not r ecord here; we tried to inc lude the o ne with the larg e st application rang e. Deﬁnition 3 - 3.1. Let B a nd S be catego ries together with a full sub cate- gory B † of B , a sub categor y S ⇒ of S , and a functor Ψ : B → S , let U b e a po set, let B b e an ob ject of B . A but tr ess of a family ( ρ u : Ψ ( B ) → S u | u ∈ U ) of morphisms in S is an U -indexed diagr am ( β u : B u → B , β v u : β u → β v | u ≤ v in U ) in B † ↓ B such that ρ u ◦ Ψ ( β u ) is a morphism of S ⇒ for each u ∈ U . The buttress is monic if all mo rphisms β u are monic (th us a ll morphisms β v u are monic). Observe that the deﬁnition of a buttress is form ulated relatively to the ob ject B o f B , the full subcateg ory B † of B , the subcatego ry S ⇒ of S , and the functor Ψ . In the following result, we sha ll refer to (LS b λ ( B )) as the L¨ owenheim-Skolem Condition with index λ at B . It is labe le d after the classica l L¨ owenheim- Skolem Theorem in mo del theory . Lemma 3 - 3.2 (The Buttress Lem ma). L et λ b e an inﬁnite c ar dinal, let B and S b e c ate gories to gether with a ful l sub c ate gory B † of B , a sub c ate gory S ⇒ of S , and a functor Ψ : B → S , let U b e a lower λ -smal l p oset, let B b e an obje ct of B , and let − → ρ = ( ρ u : Ψ ( B ) ⇒ S u | u ∈ U ) b e an U -indexe d family of morphisms in S ⇒ . We make the fol lowing assum ption: (LS b λ ( B )) F or e ach u ∈ U , e ach double arr ow ψ : Ψ ( B ) ⇒ S u , e ach λ -smal l set I , and e ach family ( γ i : C i ֌ B | i ∈ I ) of monic obje cts in B † ↓ B , 3-3 The L¨ owenheim-Sk olem Condition and the Buttress Lemma 69 ther e exists a monic obje ct γ : C ֌ B in B † ↓ B such that γ i E γ for e ach i ∈ I while ψ ◦ Ψ ( γ ) is a morphism of S ⇒ . F urthermor e, we assu m e that either U is wel l-founde d or the fol lowing addi- tional assumptions ar e satisﬁe d: (CLOS λ ( B † , B )) The ful l sub c ate gory B † has al l λ -smal l dir e cte d c olimits within B ( cf. Deﬁnition 1-2.3) . (CLOS r λ ( S ⇒ )) The sub c ate gory S ⇒ is right close d under all λ -smal l dir e cte d c olimits ( cf. Deﬁnition 1-2.5) . (CONT λ ( Ψ )) The funct or Ψ pr eserves al l λ -smal l dir e cte d c olimi ts. Then − → ρ has a bu ttr ess. Pr o of . W e ﬁrst assume that U is w ell-founded. Let w ∈ U and suppo se ha ving constructed an ( U ↓ w )-indexed diag r am in B † ↓ B , ( β u : B u ֌ B , β v u : β u ֌ β v | u ≤ v in U  w ) , with all β u : B u ֌ B mo nic. By (LS b λ ( B )), ther e ex ists a monic ob ject β w : B w ֌ B of B † ↓ B such that β u E β w for each u ∈ U  w while ρ w ◦ Ψ ( β w ) is a double arrow of S . Denote b y β w u : B u ֌ B w the unique morphism such that β u = β w ◦ β w u , for each u ∈ U  w . F or all u ≤ v < w , β w ◦ β w u = β u = β v ◦ β v u = β w ◦ β w v ◦ β v u , th us, as β w is monic , β w u = β w v ◦ β v u . This completes the pro of in case U is well-founded. Now remov e the well-foundedness assumption on U but as sume the con- ditions (CLOS λ ( B † , B )), (CLOS r λ ( S ⇒ )), and (CONT λ ( Ψ )). W e endo w the set b U := { ( u, a ) ∈ U × [ U ] <ω | a ⊆ U ↓ u } with the partial order ing deﬁned by ( u , a ) ≤ ( v , b ) iff u ≤ v and a ⊆ b . As b U is lo wer ﬁnite, it is w ell-founded, th us, by the paragr a ph ab ov e, the family  ρ u | ( u, a ) ∈ b U  has a buttress, say  β u,a : B u,a → B , β v, b u,a : β u,a → β v, b | ( u, a ) ≤ ( v , b ) in b U  . By a ssumption, fo r each u ∈ U , there exists a dir ected co limit c o cone in B  B u , δ u,a | a ∈ [ U ↓ u ] <ω  = lim − →  B u,a , β u,b u,a | a ⊆ b in [ U ↓ u ] <ω  (3-3.1) with B u ∈ B † . By the universal pr op erty of the colimit, ther e exists a unique morphism β u : B u → B suc h that β u ◦ δ u,a = β u,a for ea ch a ∈ [ U ↓ u ] <ω . Mo re- ov er, it follo ws from (CONT λ ( Ψ )) and (3-3.1) that the follo wing statemen t holds in S : 70 3 The Condensat e Lif ting Lemma (CLL)  Ψ ( B u ) , Ψ ( δ u,a ) | a ∈ [ U ↓ u ] <ω  = lim − →  Ψ ( B u,a ) , Ψ ( β u,b u,a ) | a ⊆ b in [ U ↓ u ] <ω  . (3-3.2) F or each a ∈ [ U ↓ u ] <ω , ( ρ u ◦ Ψ ( β u )) ◦ Ψ ( δ u,a ) = ρ u ◦ Ψ ( β u,a ) is a double arrow, th us, by the assumption (CLOS λ ( B † , B )) tog ether with (3-3.2), ρ u ◦ Ψ ( β u ) is a do uble arrow of S . F or all u ≤ v in U and a ll a ⊆ b in [ U ↓ u ] <ω , we obtain, by using (3-3 .1), that ( δ v, b ◦ β v, b u,b ) ◦ β u,b u,a = δ v, b ◦ β v, b u,a = δ v, b ◦ β v, b v, a ◦ β v, a u,a = δ v, a ◦ β v, a u,a , th us, by the univ er s al proper ty of the colimit, there exists a unique morphism β v u : B u → B v such that β v u ◦ δ u,a = δ v, a ◦ β v, a u,a for each a ∈ [ U ↓ u ] <ω . Hence β v ◦ β v u ◦ δ u,a = β v ◦ δ v, a ◦ β v, a u,a = β v, a ◦ β v, a u,a = β u,a = β u ◦ δ u,a . As this ho lds for each a ∈ [ U ↓ u ] <ω and by the universal pr op erty of the colimit, it follows that β v ◦ β v u = β u . A s imilar pro o f, with β w v in place of β v , for u ≤ v ≤ w , yields that β w u = β w v ◦ β v u . This completes the pro o f. ⊓ ⊔ Remark 3- 3 .3. Observe from the pr o of of Lemma 3 - 3.2 that if U is well- founded, then the buttress ca n b e taken monic. In fact, in most applications of L e mma 3-3 .2 through this work, U will b e X = for a λ -lifter ( X , X ) of a lower ﬁnite a lmost jo in- semilattice P , and in such a situatio n X may be taken low er ﬁnite (cf. Lemma 3-5.5). Therefo r e, in most o f (but no t all) our applications, U can be taken lower ﬁnite, and in that case , in order to get the conclusion of Lemma 3-3 .2, it is suﬃcient to replace (LS b λ ( B )) by (LS b ω ( B )). Situations corresp onding to the case where U is not lo wer ﬁnite ar e en- countered in Gilliber t [18]. It seems to us quite lik ely that future extensions of our w or k may req uire the emer gence of new v aria nt s of the Buttress L emma, while, quite to the contrary , the Arma tur e Lemma lo ok s mor e sta ble. 3-4 Larders and the C ondensate Lifting Lemma In this sectio n we shall complete, with the lar ders , the in tro duction of a ll the concepts needed to formulate and prove CLL (Lemma 3-4 .2). Recall (cf. Section 1-1.1) that the basic categ o rical cont ext of CLL consis ts of categor ies A , B , S , functor s Φ a nd Ψ , and a few a dd-ons. Deﬁnition 3-4.1 states what these add-ons sho uld be and what they should b e exp ected to satisfy . Deﬁnition 3 - 4.1. Let λ and µ b e inﬁnite ca rdinals. W e s ay that an o ctuple Λ = ( A , B , S , A † , B † , S ⇒ , Φ, Ψ ) is a ( λ, µ ) - lar der at an obje ct B if A , B , S are categorie s, B is an ob ject of B , Φ : A → S and Ψ : B → S are functors, A † 3-4 Larders and the Condensa te Lifting Lemma 71 (resp., B † ) is a full s ubca tegory of A (resp., B ), a nd S ⇒ is a sub categ o ry o f S satisfying the following conditions: (CLOS( A )) A ha s a ll small directed colimits. (PROD( A )) An y tw o o b jects in A ha ve a pro duct in A . (CONT( Φ )) The functor Φ pres erves all sma ll directed colimits. (PROJ( Φ, S ⇒ )) Φ ( f ) is a morphism in S ⇒ , for each extended pro jection f of A (cf. Deﬁnition 1 -2.4). (PRES λ ( B † , Ψ )) The ob ject Ψ ( B ) is weakly λ -pr esented in S , for each ob- ject B ∈ B † . (LS µ ( B )) F or ea ch S ∈ Φ “( A † ), each double a rrow ψ : Ψ ( B ) ⇒ S , each µ - small set I , and each family ( γ i : C i ֌ B | i ∈ I ) of monic ob jects in B † ↓ B , there exists a monic ob ject γ : C ֌ B in B † ↓ B such that γ i E γ for each i ∈ I while ψ ◦ Ψ ( γ ) is a mor phism in S ⇒ . W e say that Λ is a ( λ, µ ) -lar der if it is a ( λ, µ )-la rder at every ob ject of B . W e say that Λ is str ong if the following conditions are satisﬁed: (CLOS µ ( B † , B )) The full sub category B † has all µ -small directed colimits within B (cf. Deﬁnition 1-2.3). (CLOS r µ ( S ⇒ )) The sub catego ry S ⇒ is righ t closed under all µ -small di- rected c olimits (cf. Deﬁnition 1 -2.5). (CONT µ ( Ψ )) The functor Ψ pr e s erves all µ - s mall dir ected co limits. W e say that Λ is pr oje ctable if every double ar row ϕ : Ψ ( C ) ⇒ S , for ob jects C ∈ B and S ∈ S , has a pro jecta bilit y witness (cf. Deﬁnition 1 -5.1). W e shall usually say λ -larder instead of ( λ, cf ( λ ))-larder . The conditions (CLOS µ ( B † , B )), (CLOS r µ ( S ⇒ )), and (CONT µ ( Ψ )) were formulated within the statement of the Buttress Lemma (Lemma 3-3.2). The condition (LS µ ( B )) is a mo diﬁcation of the condition (LS b λ ( B )) formulated within the statement of the Buttress Lemma . Now at lo ng last we hav e reached the statement o f CLL. W e r efer to Deﬁnition 2-6.4 for nor m-cov erings , Deﬁnition 3-2 .1 for λ -lifters, and Deﬁni- tion 3-4.1 for λ -larder s. Lemma 3 - 4.2 (Co ndensate Lifting Lemm a). L et λ and µ b e inﬁnite c ar- dinals, let Λ := ( A , B , S , A † , B † , S ⇒ , Φ, Ψ ) b e a ( λ, µ ) -lar der at an obje ct B of B , and let P b e a p oset with a λ -lifter ( X, X ) . Supp ose t hat we ar e also given t he fol lowing addi tional da ta: • a P -indexe d diagr am − → A =  A p , α q p | p ≤ q in P  in A such that A p b elongs to A † for e ach non-maximal p ∈ P ; • a double arr ow χ : Ψ ( B ) ⇒ Φ  F ( X ) ⊗ − → A  in S . Then in e ach of the fol lowing c ases, (i) µ = ℵ 0 and X = is lower ﬁn ite, (ii) µ = cf ( λ ) and X = is wel l-founde d, 72 3 The Condensat e Lif ting Lemma (CLL) (iii) µ = cf ( λ ) and Λ is st r ong, ther e ar e an obje ct − → B of B P and a double arr ow − → χ : Ψ − → B ⇒ Φ − → A in S P such that B p ∈ B † for e ach non-maximal p ∈ P while B p = B for e ach maximal p ∈ P . F urthermor e, if Λ is pr oje ctable, then ther e ar e an obje ct − → B ′ of B P and a natur al e qu ivalenc e − → χ ′ : Ψ − → B ′ . → Φ − → A , while ther e exists a natura l tr ansformation fr om − → B to − → B ′ al l whose c omp onents ar e epimorphisms. Pr o of . It follows fr om (CLO S( A )) and (P ROD( A ) ) that the results of Sec- tion 3-1 apply to the ca tegory A . As, by Lemma 2- 6 .7, π X x is a normal mor- phism from F ( X ) to 2 , for ea ch x ∈ X , it follows fr om Pr op osition 3-1.6 that π X x ⊗ − → A is an extended pro jection of A . F rom (PROJ( Φ, S ⇒ )) it follows that Φ  π X x ⊗ − → A  is a double arrow of S , thus so is the morphism ρ x := Φ  π X x ⊗ − → A  ◦ χ : Ψ ( B ) ⇒ Φ ( A ∂ x ) . Now we observe that each of the assumptions (i)–(iii) allows an application of the Buttre s s Lemma (Lemma 3-3.2) to the family ( ρ x | x ∈ X = ), with S x := Φ ( A ∂ x ). W e obtain an X = -indexed diagr am in B † ↓ B , ( γ x : C x → B , γ y x : γ x → γ y | x ⊆ y in X = ) , such that ρ x ◦ Ψ ( γ x ) is a double ar row for each x ∈ X = . W e extend this diagram to an X -indexed diagram in B ↓ B (no longer necessarily in B † ↓ B ), by setting C x := B a nd γ x := γ y x = id B , fo r a ll x ⊆ y in X \ X = , (3-4.1) γ y x := γ x , fo r a ll ( x , y ) ∈ X = × ( X \ X = ) with x ⊆ y (it is straig h tforward to v erify that the diagr a m th us extended remains com- m utative). Obs erve that ρ x ◦ Ψ ( γ x ) = ρ x is also a double a rrow, for each x ∈ X \ X = ; thus ρ x ◦ Ψ ( γ x ) is a double ar row for each x ∈ X . Now we apply the Ar ma ture Lemma (Lemma 3-2.2) with S x := Ψ ( C x ) (whic h is not iden tical to the S x used ex c lusively in the para graph ab ove) , ϕ x := Ψ ( γ x ), and ϕ y x := Ψ ( γ y x ). If x ∈ X = , then C x ∈ B † , thus, by (PRES λ ( B † , Ψ )), S x is weakly λ -presen ted. W e thus obtain an isotone sec- tion σ of ∂ such that  ρ σ ( p ) ◦ ϕ σ ( p ) | p ∈ P  is a natural trans formation from  Ψ ( C σ ( p ) ) , Ψ ( γ σ ( q ) σ ( p ) ) | p ≤ q in P  to Φ − → A . W e set χ p := ρ σ ( p ) ◦ ϕ σ ( p ) , B p := C σ ( p ) , β q p := γ σ ( q ) σ ( p ) . All the χ p are double ar rows while − → χ := ( χ p | p ∈ P ) is a natural transforma tion from Ψ − → B to Φ − → A , a nd p maximal implies that B p = B (cf. (3 -4.1)). If Λ is pro jectable, then the morphism χ p : Ψ ( B p ) ⇒ Φ ( A p ) has a pro jectability witness ( a p , η p ), say a p : B p ։ B ′ p , for each p ∈ P . By Lemma 1-5 .2, we can ﬁnd a system o f morphisms β ′ q p : B ′ p → B ′ q , for p ≤ q 3-5 Inﬁnite com binatorics and lambda-lifters 73 in P , such that − → B ′ :=  B ′ p , β ′ q p | p ≤ q in P  is a P -indexed diagram and Ψ − → B ′ ∼ = Φ − → A . ⊓ ⊔ Remark 3- 4 .3. It may very well be the cas e that the part of CLL dealing with str ong lar ders is v acuous. This would mean that whenev er a p oset P has a λ -lifter, then it has a λ -lifter ( X , X ) with X = well-founded; this w ould imply , in particula r, that P \ Max P is well-founded. W e do not know whether this holds, see Problems 8 and 9 in Chapter 7. 3-5 Inﬁnite c ombina toric s and lam bda-lifters F or a given p oset P and an inﬁnite cardinal λ , the complexity of the deﬁnit ion of a λ -lifter mak es it quite unpractical to v erify whether suc h an ob ject exists. The main goal of the pre s ent section is to relate the existence of a lifter to the deﬁnition of an almost join-semilattice (Deﬁnition 2-1.2) on the one hand, and to known inﬁnite com binator ial issues on the other hand. One of the main results from this section (Corollary 3-5.8) relates the existence of a λ -lifter of P and an inﬁnite co mbinatorial statement that we in tro duced in Gillib ert and W ehrung [23], denoted ( κ, <λ ) ❀ P , that we shall recall here. F urther results ab out the ( κ, <λ ) ❀ P sta tement, in par ticular for sp eciﬁc choices of the pose t P , can b e found in [23]. A s ur vey pap er on the interaction b etw een those ma tters a nd the present work can also b e found in W ehrung [7 0]. Deﬁnition 3 - 5.1. F or cardinals κ , λ and a p ose t P , let ( κ, <λ ) ❀ P hold if for ev ery mapping F : P ( κ ) → [ κ ] <λ , there exists a one-to-one map f : P ֌ κ such that F ( f “( P ↓ x )) ∩ f “( P ↓ y ) ⊆ f “ ( P ↓ x ) , fo r a ll x ≤ y in P . (3-5.1) In ma ny cases, P is lower ﬁnite, and then it is of course s uﬃcie nt to v erify the conclusion ab ov e for F : [ κ ] <ω → [ κ ] <λ isotone . Deﬁnition 3 - 5.2. The Ku ra towski index of a ﬁnite po set P is deﬁned a s 0 if P is an an tichain, and the leas t p ositive integer n such that the relatio n ( κ +( n − 1) , < κ ) ❀ P holds for ea ch inﬁnite ca rdinal κ , otherwis e. W e denote this integer by kur( P ). W e prove in [23, Pr op osition 4.7] that the or der-dimension dim( P ) of a ﬁnite p os et P lies above the Kuratowski index of P ; that is, kur( P ) ≤ dim( P ). Deﬁnition 3 - 5.3. The r estricte d Kura towski index of a ﬁnite po set P is deﬁned as zero if P is an a nt ichain, a nd the least po sitive integer n suc h that the r e lation ( ℵ n − 1 , < ℵ 0 ) ❀ P holds, otherwise . W e denote this int ege r by kur 0 ( P ). 74 3 The Condensat e Lif ting Lemma (CLL) In par ticular, by deﬁnition, kur 0 ( P ) ≤ kur( P ). Example 3-5. 4. T he following example s hows that the inequa lity ab ove may be strict, that is, kur 0 ( P ) < kur( P ) may hold, for a certain ﬁnite la ttice P — at least in some gener ic extension of the universe. Start with a univ ers e V of ZF C satisfying the Generalized Contin uum Hypo thesis GCH . Ther e ar e a ﬁnite la ttice P and a generic ex tens ion o f the universe in which kur 0 ( P ) < kur( P ). In or der to see this, we set t 0 := 5, t 1 := 7, and for each po sitive integer n , t n +1 is the least p os itive in teger such that t n +1 → ( t n , 7 ) 5 (the latter notation meaning tha t for ea ch mapping f : [ t n +1 ] 5 → { 0 , 1 } , either there exists a t n -element subset X of t n +1 such that f “[ X ] 5 = { 0 } or there ex ists a 7-element subs et X of t n +1 such that f “ [ X ] 5 = { 1 } ). The existence of the s equence ( t n | n < ω ) is ensur ed b y Ramsey’s Theorem. Now w e order the set B m ( 6 r ) := { X ⊆ m | either card X ≤ r or X = m } by containmen t, for all integers m , r such that 1 ≤ r < m . The cardinal τ := ℵ ω +1 is regular, th us, by Ko mj´ ath and Shela h [40, The- orem 1], there exists a generic extension V [ G ] of V , with the s ame cardina ls as V , in which GCH holds b e low τ , 2 τ = τ +4 , and ( τ +4 , 4 , τ ) 6→ t 4 . Hence, it follows from [23, Pro p o sition 4.11 ] that kur B t 4 ( 6 4) ≥ 6 in V [ G ]. On the other hand, as GCH holds b elow τ in V [ G ], it follows fro m [10, The- orem 45.5] that the r e lation ( ℵ n , n , ℵ 0 ) → ℵ 1 holds in V [ G ], for each p o sitive int ege r n . In particular , ( ℵ 4 , 4 , ℵ 0 ) → t 4 , and thus, by [23, Prop os ition 4.11], ( ℵ 4 , < ℵ 0 ) ❀ B t 4 ( 6 4). The r efore, kur 0 B t 4 ( 6 4) ≤ 5. In fact, as B t 4 ( 6 4) has breadth 5, it follows fro m [2 3, Prop osition 4.8] that kur 0 B t 4 ( 6 4) = 5. In particular, kur 0 B t 4 ( 6 4) < kur B t 4 ( 6 4) in V [ G ]. Lemma 3 - 5.5. L et P b e a lower ﬁnite almo st join-semilattic e with zer o, let λ and κ b e inﬁnite c ar dinals such that every element of P has less than cf ( λ ) upp er c overs and ( κ, <λ ) ❀ P . Then ther e exist s a norm-c overing X of P such that (i) X is a lower ﬁnite almost join-semilattic e with zer o; (ii) car d X = κ ; (iii) X , to gether with the c ol le ction of al l it s princip al ide als, is a λ -lifter of P . Pr o of . If P is a singleton then the statemen t is trivial. Supp ose that P is not a singleton. F ro m ( κ, <λ ) ❀ P it follows that κ ≥ λ (otherwise, consider the constant mapping on P ( κ ) with v alue κ , get a c o ntradiction). F urther more, from the assumption o n P it follows, using an eas y induction pr o of, tha t { x ∈ P | card( P ↓ x ) ≤ n } is cf ( λ )-small, for ea ch n < ω . (3-5.2) 3-5 Inﬁnite com binatorics and lambda-lifters 75 In particula r , as P is low er ﬁnite, car d P ≤ c f ( λ ) (and ca rd P < cf ( λ ) if cf ( λ ) > ω ). F or any set K , denote by P h K i the s et of a ll pair s ( a, x ), where: (i) a ∈ P ; (ii) x is a function fr om a subset X of P ↓ a to K such that a ∈ ▽ X . ( O f c ourse, as P is lower ﬁnite, X is ne c essarily ﬁ nite .) W e order P h K i comp onent wise, that is, ( a, x ) ≤ ( b, y ) ⇐ ⇒ ( a ≤ b and y extends x ) , fo r a ll ( a, x ) , ( b, y ) ∈ P h K i . F urthermore, for each ( b, y ) ∈ P h K i , the function y extends only ﬁnitely many functions x ; as ▽ (dom x ) is ﬁnite for eac h of those x , it follows that P h K i ↓ ( b, y ) is ﬁnite. Hence P h K i is low er ﬁnite. F urthermo re, P h K i has a smallest element, namely (0 P , ∅ ). Let ( a 0 , x 0 ) , ( a 1 , x 1 ) ∈ P h K i b oth below so me element of P h K i . In par- ticular, x := x 0 ∪ x 1 is a function. Put X i := dom x i , for each i < 2, and X := dom x . W e cla im that ( a 0 , x 0 ) ▽ ( a 1 , x 1 ) = ( a 0 ▽ a 1 ) × { x } . (3-5.3) Let a ∈ a 0 ▽ a 1 . As a i ∈ ▽ X i for each i < 2, it follows from Lemma 2-1 .4(ii) that a ∈ ▽ X , and so ( a, x ) ∈ P h K i . Then it is obvious that ( a i , x i ) ≤ ( a, x ) for each i < 2, and tha t ( a, x ) is minimal such. This establishes the contain- men t from the r ight hand side int o the left hand side in (3-5.3). Conv er s ely , let ( b, y ) be lo ng to the left hand side of (3-5.3). As a i ≤ b for eac h i < 2, there exists a ∈ a 0 ▽ a 1 such that a ≤ b . Moreover, y obviously extends x , s o ( a i , x i ) ≤ ( a, x ) ≤ ( b, y ) for each i < 2, and so, by the minimality of ( b, y ), ( b, y ) = ( a, x ) b e longs to the r ight hand side of (3- 5 .3). This completes the pro of of (3-5.3). As P h K i has a zer o, it follows that P h K i is a pseudo join-semilattice. F ur- thermore, it follo ws from (3- 5.3) that an y pair { ( a 0 , x 0 ) , ( a 1 , x 1 ) } of ele ments of P h K i b elow some ( b, y ) ∈ P h K i has a join in P h K i ↓ ( b, y ), which is ( a, x ) where a is the unique elemen t of a 0 ▽ a 1 below b ( we use her e the assumption that P ↓ b is a join-semilattic e ) and x = x 0 ∪ x 1 . In particula r , P h K i is an almost join-semilattic e . W e put ∂ ( a, x ) := a , for all ( a, x ) ∈ P h K i . As we just veriﬁed that P h K i is a pseudo join-semilattice, this deﬁnes a nor m-cov ering of P . W e shall consider th e p os e t P h κ i . F ro m κ ≥ λ it follo ws that car d P h κ i = κ . W e must prove that P h κ i , together with the collection of all principal ideals o f P h κ i , is a λ -lifter of P . Let S : P h κ i → [ P h κ i ] <λ be an is otone mapping. F or each U ∈ [ κ ] <ω , denote by Φ ( U ) the set of all ( c, z ) ∈ P h κ i such that there are ( a, x ) ∈ P h U i and b ∈ P such that (F1) a ≺ b and c ≤ b ; (F2) ( c, z ) ∈ S ( a, x ); 76 3 The Condensat e Lif ting Lemma (CLL) (F3) ca rd( P ↓ a ) = car d U . F urthermore, set F ( U ) := S (rng z | ( c, z ) ∈ Φ ( U )). It fo llows from (3-5.2) that there a re les s than cf ( λ ) elemen ts a ∈ P such that card( P ↓ a ) = card U , and each o f tho se a has less than cf ( λ ) upp er cov ers. As P is lower ﬁnite, it follows tha t there are less than cf ( λ ) triples ( a, b, c ) ∈ P 3 satisfying b o th (F1) and (F3). As each S ( a, x ) is λ -small, it follows that Φ ( U ) is λ -sma ll, a nd th us F ( U ) is λ -small. As ( κ, <λ ) ❀ P , there exis ts a one- to -one map f : P ֌ κ such that F ( f “( P ↓ a )) ∩ f “( P ↓ b ) ⊆ f “( P ↓ a ) , for all a < b in P . (3-5.4) As P is low er ﬁnite, the elemen t σ ( a ) := ( a, f ↾ P ↓ a ) b e longs to P h κ i , for each a ∈ P . O f course, σ is isotone. It remains to prov e the containmen t S ( σ ( a )) ↓ σ ( b ) ⊆ P h κ i ↓ σ ( a ) , (3-5.5) for all a < b in P . As S is isotone and a ny closed interv a l of P has a ﬁnite maximal chain, it suﬃces to establish (3-5.5) in ca se a ≺ b . Let ( c, z ) b e an element of the le ft ha nd side of (3- 5.5). The r elation ( c, z ) ∈ σ ( b ) implies that c ≤ b , f e xtends z , and rng z ⊆ f “ ( P ↓ b ). As f is o ne- to-one, the s e t U := f “( P ↓ a ) has the sa me car dinality as P ↓ a , and ( c, z ) ∈ Φ ( U ); whence rng z ⊆ F ( U ). Using (3-5.4), it follows that rng z ⊆ F ( f “( P ↓ a )) ∩ f “( P ↓ b ) ⊆ f “( P ↓ a ) , th us, a s f is one-to- one and extends z , dom z is contained in P ↓ a . As P ↓ b is a join-semilattice co ntaining { c } ∪ dom z with c ∈ ▽ (dom z ), c is the jo in of dom z in P ↓ b ; whence c ≤ a . It follows that ( c, z ) ≤ σ ( a ), th us completing the pro of of (3-5.5). ⊓ ⊔ Recall that a p oset T with zero is a tr e e if T has a smalles t element, that we shall denote by ⊥ , and T ↓ a is a chain for each a ∈ T . The follo wing r esult is prov ed in the ﬁrs t author’s pap er [18 , Cor ollary 4.7 ]. Prop ositio n 3-5.6. L et λ b e an inﬁn ite c ar dinal and let T b e a nontrivial lower cf ( λ ) -smal l wel l-founde d tre e such that c a rd T ≤ λ . Then ther e exists a λ -lifter ( X , X ) of T such that X is a lower ﬁnite almost join-semilattic e with zer o and card X = λ . The λ -lifter in Prop os ition 3-5.6 is constructed as fo llows. The p oset X is the set o f all functions from a ﬁnite sub chain of T \ { ⊥} to λ , endow ed with the extensio n order. Althoug h it is not ex plic itly stated in [18] that X is an almost join-semilattice, the veriﬁcation o f that fact is straightforward. The norm on X is deﬁned by ∂ x := W dom( x ) for each x ∈ X . The set X consists of the so-called ex tr eme ide als of X : by deﬁnition, a sharp ideal x of X is extr eme if there is no sha rp ideal y of X s uch that x < y and ∂ x = ∂ y . 3-5 Inﬁnite com binatorics and lambda-lifters 77 In the context of L e mma 3-5.5, the λ -lifter ( X , X ) is here chosen in suc h a w ay that X is the c ollection o f a ll pr incipal idea ls of X . In addition, within the pro o f of Lemma 3-5.5, the ideals in the ra nge of σ are extre me ideals , so we could have restricted X further to the collec tio n of all extr eme principal ideals. It is in triguing that in all known cases, including P rop osition 3-5.6, every λ -liftable poset with zer o admits a λ -lifter of the for m ( X , X ) where X is a low er ﬁnite almos t jo in-semilattice with zero and X is the collection of all ex treme ideals of X . The following result yields neces s ary conditions for λ -liftability (cf. Deﬁ- nition 2-1.2 for almost join-semilattices ). It is r elated to [17, Le mme 3.3.1]. Prop ositio n 3-5.7. L et λ b e an inﬁn ite c ar dinal and let ( X , X ) b e a λ -lifter of a p oset P . Put κ := card X . Then the fol lowing statement s hold: (1) P \ Max P is lower cf ( λ ) -smal l. (2) P is an almost join-semilattic e. (3) P is the disjoint union of ﬁ nitely many p osets with z er o. (4) F or every isotone map f : [ X = ] < cf ( λ ) → [ X ] <λ , t her e exists an isotone se ction σ : P ֒ → X of ∂ such that ( ∀ a < b in P )  f ( σ “ ( P ↓ a )) ∩ σ “( P ↓ b ) ⊆ σ “( P ↓ a )  . (5) F or every isotone map f : [ κ ] < cf ( λ ) → [ κ ] <λ , ther e exists a one-to-one map σ : P ֌ κ such t hat ( ∀ a < b in P )  f ( σ “ ( P ↓ a )) ∩ σ “( P ↓ b ) ⊆ σ “( P ↓ a )  . Pr o of . F or each x ∈ X , it follows from the shar pness o f x that there exists  ( x ) ∈ x such that ∂  ( x ) = ∂ x . F ur thermore, if ∂ x is minimal in P , then for each x ∈ x , it follows from the minimalit y of ∂ x and the inequa lit y ∂ x ≤ ∂ x that ∂ x = ∂ x . Ho wev er , as X is a pseudo join-semilattice and x is an ideal of X , the set x ∩ Min X is nonempty . Therefore, we may assume that  ( x ) is minimal in X whenever ∂ x is minimal in P . Now we set S 0 ( x ) := {  ( y ) | y ∈ X ↓ x } , S ( x ) := (Min X ) ∪ [ ( u ▽ v | u, v ∈ S 0 ( x )) , for each x ∈ X = . It follows from the assumptions deﬁning λ -lifters (Def- inition 3 -2.1) that S ( x ) is a cf ( λ )-small subset of X , for every x ∈ X = . As ( X , X ) is a λ -lifter of P , there exists a n isoto ne section σ : P ֒ → X of ∂ such that S ( σ ( a )) ∩ σ ( b ) ⊆ σ ( a ) for all a < b in P . As σ res tricts to an order - embedding fro m P \ Ma x P into X = and the latter is low er cf ( λ )-sma ll, (1) follows. (2). Let A be a ﬁnite subset of P , and put ˙ a := σ ( a ), for each a ∈ A . In particular, ∂ ˙ a = a . Set ˙ A := { ˙ a | a ∈ A } . As X is a pseudo join-semilattice, 78 3 The Condensat e Lif ting Lemma (CLL) ▽ ˙ A is a ﬁnite subset of X . Hence, U := { ∂ ˙ u | ˙ u ∈ ▽ ˙ A } is a ﬁnite subset of P . W e claim tha t P ⇑ A = P ↑ U . F or the con tainment from the right int o the left, we observe that each ˙ u ∈ ▽ ˙ A lie s ab ove all elements of ˙ A , thus ∂ ˙ u lies ab ov e all elements of A (use the equatio n ∂ ˙ a = a that holds for each a ∈ A ), and thus U ⊆ P ⇑ A . Co nversely , let x ∈ P ⇑ A . F o r each a ∈ A , ˙ a ∈ σ ( a ) ⊆ σ ( x ), th us, as σ ( x ) is an ideal of the pseudo join-se milattice X and ˙ A is ﬁnite, there exists ˙ u ∈ ▽ ˙ A such that ˙ u ∈ σ ( x ). Hence, ∂ ˙ u ≤ ∂ σ ( x ) = x . This completes the pro of of our claim. In particular, P ⇑ A is a ﬁnitely genera ted upper subset of P . This completes the pro of of the statement that P is a pseudo join-semilattice. Now let a, b, e ∈ P such that a, b ≤ e , w e m ust pr ov e that { a, b } has a join in P ↓ e . P ut again ˙ a := σ ( a ) and ˙ b := σ ( b ). As σ ( e ) is a n ideal of X , { ˙ a, ˙ b } ⊆ σ ( e ), and X is a pseudo join-semilattice, ther e exists a n element ˙ x in ( ˙ a ▽ ˙ b ) ∩ σ ( e ). F rom ∂ ˙ a = a , ∂ ˙ b = b , and ∂ σ ( e ) = e it follows that a, b ≤ ∂ ˙ x ≤ e . Let c ∈ P ↓ e suc h tha t a, b ≤ c . F rom σ ( a ) ∈ X ↓ σ ( c ) it follows that ˙ a = σ ( a ) ∈ S 0 ( σ ( c )). Similarly , ˙ b ∈ S 0 ( σ ( c )). As ˙ x b elongs to ˙ a ▽ ˙ b , it belong s to S ( σ ( c )). On the other hand, ˙ x ∈ σ ( e ), so ˙ x b elo ngs to S ( σ ( c )) ∩ σ ( e ), th us to σ ( c ), and so ∂ ˙ x ≤ c . W e hav e prov ed that ∂ ˙ x is the join of { a, b } in P ↓ e . (3). F or all a ≤ b in P , (Min X ) ∩ σ ( b ) ⊆ S ( σ ( a )) ∩ σ ( b ) ⊆ σ ( a ), th us, as σ ( a ) ⊆ σ ( b ), w e obtain (Min X ) ∩ σ ( a ) = (Min X ) ∩ σ ( b ). No w let a, b ∈ P with a minimal in P and P ⇑ { a, b } 6 = ∅ ; pick c ∈ P ⇑ { a, b } . By the previous remark, we obtain that (Min X ) ∩ σ ( a ) = (Min X ) ∩ σ ( c ) = (Min X ) ∩ σ ( b ). As ∂ σ ( a ) = a is minimal in P and by the choice of the map  , the element ˙ a := σ ( a ) is minimal in X . Hence ˙ a b elong s to (Min X ) ∩ σ ( a ) = (Min X ) ∩ σ ( b ) ⊆ σ ( b ) , and thus a = ∂ ˙ a ≤ ∂ σ ( b ) = b . As, by (1) ab ov e, Min P is ﬁnite and P is the union o f a ll its subsets P ↑ a , for a ∈ Min P , it follows that this union is disjoint. (4). As f is iso tone, the assignment F ( x ) := f ( X ↓ x ) deﬁnes an isotone map F : X = → [ X ] <λ . Now set S ( x ) :=  “ F ( x ), for each x ∈ X = . Hence S : X = → [ X ] <λ is isoto ne. As ( X , X ) is a λ -lifter of P , its norm has a free isotone sec tio n σ : P ֒ → X with resp ect to S . Let a < b in P , w e shall prov e that f ( σ “( P ↓ a )) ∩ σ “ ( P ↓ b ) ⊆ σ “( P ↓ a ). Let c ∈ P ↓ b suc h that σ ( c ) ∈ f ( σ “( P ↓ a )), we must prov e tha t c ≤ a . As σ “ ( P ↓ a ) is contained in X ↓ σ ( a ) and f is isotone, we obtain f ( σ “ ( P ↓ a )) ⊆ f ( X ↓ σ ( a )) = F ( σ ( a )) , hence σ ( c ) ∈ F ( σ ( a )), and hence σ ( c ) ∈ S ( σ ( a )). As σ ( c ) ∈ σ ( c ) ⊆ σ ( b ), it follows that σ ( c ) b elongs to S ( σ ( a )) ∩ σ ( b ), th us to σ ( a ), and so c = ∂ σ ( c ) = ∂ σ ( c ) ≤ ∂ σ ( a ) = a . In or der to obtain (5) from (4), it suﬃce s to prov e that for ev er y isotone map f : [ X ] < cf ( λ ) → [ X ] <λ , there exists a one-to- o ne map σ : P ֌ X such 3-5 Inﬁnite com binatorics and lambda-lifters 79 that ( ∀ a < b in P )  f ( σ “ ( P ↓ a )) ∩ σ “( P ↓ b ) ⊆ σ “( P ↓ a )  . (3-5.6) By applying (4) to the restrictio n g of f to [ X = ] < cf ( λ ) , we obtain an isotone section σ : P ֒ → X of ∂ such that ( ∀ a < b in P )  g ( σ “( P ↓ a )) ∩ σ “ ( P ↓ b ) ⊆ σ “( P ↓ a )  . How ever, fo r a < b in P , σ ( a ) ∈ X = , thus σ “ ( P ↓ a ) ⊆ X = , and thus we get f ( σ “ ( P ↓ a )) = g ( σ “( P ↓ a )), so (3-5.6) is satisﬁed as well. ⊓ ⊔ In par ticular, we obtain the following characterization of λ -liftabilit y for low er ﬁnite and suﬃciently small p osets. This character ization is related to [17, Th´ eo r` e me 3.3 .2]. Corollary 3 - 5.8. L et λ and κ b e inﬁn it e c ar dinals and let P b e a lower ﬁnite p oset in which every element has less than cf ( λ ) upp er c overs. Then the fol lowing ar e e quivalent: (i) P has a λ -lifter ( X , X ) such that X c onsists of al l princip al ide als of X and ca rd X = κ , while X is a lower ﬁnite almost join-semilattic e. (ii) P has a λ -lifter ( X , X ) such that ca rd X = κ . (iii) P is t he disjo int union of ﬁnitely many almost join-semilattic es with zer o, and ( κ, < λ ) ❀ P . Pr o of . (i) ⇒ (ii) is tr ivial. (ii) ⇒ (iii). The statemen t that P is the disjoin t union of ﬁnitely man y almost join-semilattices with zero follows from P rop osition 3-5 .7(2,3). Now let F : [ κ ] <ω → [ κ ] <λ . W e ca n a sso ciate to F an isotone mapping G : [ κ ] < cf ( λ ) → [ κ ] <λ by setting G ( X ) := [  F ( Y ) | Y ∈ [ X ] <ω  for each X ∈ [ κ ] < cf ( λ ) . (3-5.7) Observe that F ( X ) ⊆ G ( X ) for every X ∈ [ κ ] <ω . Now apply Pr op osi- tion 3-5.7(5) to G . (iii) ⇒ (i). By ass umption, we can wr ite P as a disjoint union P = S ( P i | i < n ), where n is a p ositive in teger and each P i is an almost join- semilattice with zero. F or each i < n , as P i is a subset of P and ( κ, <λ ) ❀ P , we obtain that ( κ, <λ ) ❀ P i , a nd thus, by Lemma 3- 5.5, P i has a λ - lifter ( X i , X i ) such that X i is the set of a ll principal ideals o f X i and card X i = κ , while each X i is a lower ﬁnite almos t join-semilattice. W e may assume that the X i s are pairwise disjoin t. Then form the disjoint union (as po sets) X := S ( X i | i < n ) and set X := S ( X i | i < n ), which is also the set o f a ll pr incipal ideals of X . F urthermor e, let ∂ X extend e a ch ∂ X i . Le t S : X = → [ X ] <λ and deﬁne S i ( x ) := S ( x ) ∩ X i , fo r a ll i < n and all x ∈ X = i . 80 3 The Condensat e Lif ting Lemma (CLL) As ( X i , X i ) is a λ -lifter o f P i , it has an isoto ne sectio n σ i which is free with resp ect to S i . It follows ea sily that the union of the σ i is an iso tone sec tion of the lifter ( X, X ) which is free with resp ect to S . Therefore, ( X , X ) is a λ -lifter of P . ⊓ ⊔ Corollary 3 - 5.9. L et P and Q b e lower ﬁnite, disjoint unions of ﬁnitely many almost join-semilattic es with z er o and let λ and κ b e inﬁnite c ar dinals such that every element of P has less than cf ( λ ) upp er c overs. If P emb e ds into Q as a p oset and Q has a λ -lifter ( X, X ) such that card X = κ , then so do es P . Pr o of . Obser ve that the pro of of (ii) ⇒ (iii) in Corollar y 3 -5.8 do es not use the assumption that card P < cf ( λ ). It follows that the relation ( κ, <λ ) ❀ Q holds. As P em b eds in to Q , th e relatio n ( κ, <λ ) ❀ P holds as well. Ther efore, by applying a gain Corolla ry 3-5.8, the desired conclusio n follows. ⊓ ⊔ Observe also the fo llowing characterization of λ -liftable ﬁnite p osets. Corollary 3 - 5.10. L et P b e a ﬁnite p oset and let λ b e an inﬁnite c ar dinal. Then P has a λ -lif ter iff P is a ﬁn ite di sjoint union of almost joi n- semilat- tic es with zer o. F urthermor e, if P is not an ant ichain, then the c ar dinality of such a lifter c an b e taken e qual to λ +( n − 1) , wher e n := kur( P ) . Pr o of . If P has a λ -lifter, then it fo llows from Corollary 3-5.8 that P is a disjoint union of almost join-semilattices with zer o. Conv ersely , suppos e that P is not an antic hain. Set n := kur( P ) and κ := λ +( n − 1) . The relation ( κ, <λ ) ❀ P holds b y deﬁnition. The conclusio n follows from Corolla ry 3-5.8. ⊓ ⊔ 3-6 Lifters, r etracts, and pseudo-retracts It is ob vious th at if P is a subp ose t of a lo wer ﬁnite p o s et Q , then ( κ, <λ ) ❀ Q implies that ( κ, <λ ) ❀ P (cf. [23, Lemma 3.2]). W e do not k now whether a similar statement ca n be proved for lifters, the inﬁnite case included: that is, whether if P is a n almost join-semila ttice a nd Q has a λ -lifter, then so does P . Corollar y 3 -5.9 s hows additiona l assumptions on P and Q under which this holds. The following result shows a nother t yp e of ass umption under whic h this can b e done. Lemma 3 - 6.1. L et λ b e an inﬁn ite c ar dinal and let P b e a r etr act of a p oset Q . If Q has a λ -lifter ( Y , Y ) , then P has a λ -lifter ( X , X ) such that X and Y have the same un derlying set and X = is a subset of Y = . Pr o of . Denote by ρ a retr action of Q onto P and set R := { q ∈ Max Q | ρ ( q ) / ∈ Max P } . 3-6 Lif ters, retracts, and pseudo-retracts 81 W e endow the underlying s et of Y with the no rm ρ∂ ; we o btain a norm- cov ering X of P . W e s et X := { x ∈ Y | ∂ x / ∈ R } . Every element of X is a sharp ideal of Y with resp ect to ∂ , thus a fortio ri with resp ect to ρ∂ ; that is, X ⊆ Id s X . W e claim that X = (deﬁned with r e sp e ct to the norm ρ∂ ) is a subset of Y = (deﬁned w ith resp ect to the norm ∂ ). Let x ∈ X = and supp os e that x / ∈ Y = , that is, ∂ x ∈ Max Q . F rom x ∈ X it follo ws that ∂ x / ∈ R , thus ρ∂ x ∈ Max P , whic h contradicts the assumption x ∈ X = and th us prov es our claim. In particular , as Y = is low er cf ( λ )-small, so is X = . W e can extend a n y mapping S : X = → [ X ] <λ to a mapping S : Y = → [ Y ] <λ by setting S ( y ) := ∅ for each y ∈ Y = \ X = . As ( Y , Y ) is a λ -lifter of Q , it has an isotone section σ which is free with resp ect to S . It follows eas ily that σ ↾ P is a n isotone section of ρ∂ whic h is free with resp ect to S . ⊓ ⊔ W e do not know whether every liftable p oset is well-founded (cf. P roblem 8 in Chapter 7). How ever, the following sequence of results sheds some light on the gray zone where this co uld o ccur. W e use sta nda rd notation for the addition of ordinals, so, in particular , ω + 1 = { 0 , 1 , 2 , . . . } ∪ { ω } . Lemma 3 - 6.2. L et λ b e an inﬁnite c ar dinal and let P a non wel l-founde d p oset. If P is λ -liftable, then so is ( ω + 1 ) op . Pr o of . It follows from Pr op osition 3-5 .7(3) tha t we can wr ite P a s a disjoint union P = S ( P i | i < n ), where n is a p ositive in teger and each P i is an almost join-semilattice with zero. As each P i is a retra ct of P (send all the elements of P \ P i to the zero ele ment of P i ) and one of the P i is not well- founded, it follows from Lemma 3 -6.1 that we may repla ce P by that P i and th us a ssume tha t P is a n almost join-semilattice with zero . As P is not w ell-founded, there exists a str ictly decreasing ( ω + 1)-sequence ( p n | 0 ≤ n ≤ ω ) in P such that p ω is the ze r o element of P . Denote by ρ ( x ) the least n ≤ ω suc h that p n ≤ x , for each x ∈ P . As th e a ssignment ( n 7→ p n ) deﬁnes an or der-embedding from ( ω + 1) op int o P , with retraction ρ , the conclusion follows from Lemma 3-6.1. ⊓ ⊔ Prop ositio n 3-6.3. L et λ b e an inﬁnite c ar dinal su ch t hat µ ℵ 0 < λ for e ach µ < cf ( λ ) . (3-6.1) If a p oset P is λ -liftable, then P is wel l-founde d. Pr o of . Supp ose that P is not well-founded. By Lemma 3-6.2, we may assume that P = ( ω + 1) op . In particular, P \ Max P is not lower ﬁnite, th us, b y Prop ositio n 3-5.7(1), ℵ 1 ≤ cf ( λ ). 82 3 The Condensat e Lif ting Lemma (CLL) Claim. F or every ( not ne c essarily isotone ) map f : [ X = ] 6 ℵ 0 → [ X ] 6 ℵ 0 , there exists an isotone section σ : P ֒ → X of ∂ suc h that ( ∀ a < b in P )  f ( σ “ ( P ↓ a )) ∩ σ “( P ↓ b ) ⊆ σ “( P ↓ a )  . Pr o of . A slight mo diﬁcation o f the pro of of P rop osition 3 -5.7(4). W e set F ( x ) := [  f ( Z ) | Z ∈ [ X ↓ x ] 6 ℵ 0  , for each x ∈ X = . ( 3- 6.2) Let x ∈ X = . The cardinal µ := car d( X ↓ x ) is smaller than cf ( λ ). As f ( Z ) is at most countable for each Z ∈ [ X ↓ x ] 6 ℵ 0 , it follows tha t card F ( x ) ≤ µ ℵ 0 , th us, by the a s sumption (3- 6.1), F ( x ) is λ -sma ll. Hence F : X = → [ X ] <λ . As in the pro o f of P rop osition 3 -5.7, for each x ∈ X , there exists  ( x ) ∈ x such that ∂  ( x ) = ∂ x . W e set S ( x ) :=  “ F ( x ), for each x ∈ X = ; so S : X = → [ X ] <λ . As ( X, X ) is a λ -lifter of P , its norm has a free isotone section σ : P ֒ → X with r esp ect to S . Let a < b in P , we shall prov e the containmen t f ( σ “( P ↓ a )) ∩ σ “( P ↓ b ) ⊆ σ “( P ↓ a ). Let c ∈ P ↓ b s uch that σ ( c ) ∈ f ( σ “( P ↓ a )), we must pr ov e that c ≤ a . As σ “( P ↓ a ) is an a t most countable subset of X ↓ σ ( a ), it follows fro m (3-6 .2) that f ( σ “ ( P ↓ a )) ⊆ F ( σ ( a )) , hence σ ( c ) ∈ F ( σ ( a )), and hence σ ( c ) ∈ S ( σ ( a )). As σ ( c ) ∈ σ ( c ) ⊆ σ ( b ), it follows that σ ( c ) b elongs to S ( σ ( a )) ∩ σ ( b ), th us to σ ( a ), and so c = ∂ σ ( c ) = ∂ σ ( c ) ≤ ∂ σ ( a ) = a . ⊓ ⊔ Cla im W e shall apply the Cla im ab ov e to a sp eciﬁc f , as in the pro of o f [23, Prop ositio n 3.5]. Let U ≡ ﬁn V hold if the symmetric diﬀerence U △ V is ﬁnite, for all subsets U and V of X = . Let ∆ be a set that mee ts the ≡ ﬁn -equiv alence class [ U ] ≡ fin of U in exactly one elemen t, fo r ea ch at most coun table subs e t U of X = , a nd de no te b y f ( U ) the unique element of [ U ] ≡ fin ∩ ∆ . By our Cla im, there is a one - to-one map σ : P ֌ X = such that f ( σ “ ( P ↓ p )) ∩ σ “( P ↓ q ) ⊆ σ “( P ↓ p ) for a ll p < q in P . ( 3- 6.3) W e set U n := σ “( P ↓ n ) = { σ ( n ) , σ ( n + 1) , . . . } ∪ { σ ( ω ) } , for ea ch n < ω . By the deﬁnition of f , the diﬀerence U 0 \ f ( U 0 ) is ﬁnite, thus there exists a natural num ber m suc h that U 0 \ f ( U 0 ) ⊆ { σ (0) , σ (1) , . . . , σ ( m − 1 ) } ∪ { σ ( ω ) } . In particular , σ ( m ) / ∈ U 0 \ f ( U 0 ), but σ ( m ) ∈ U 0 , thus σ ( m ) ∈ f ( U 0 ). Now from U 0 ≡ ﬁn U m +1 it follows that f ( U 0 ) = f ( U m +1 ), thus σ ( m ) ∈ f ( U m +1 ). 3-6 Lif ters, retracts, and pseudo-retracts 83 As σ ( m ) ∈ U 0 , it follows from (3-6.3) that σ ( m ) ∈ U m +1 , a con tra diction. ⊓ ⊔ Corollary 3 - 6.4. If a p oset P is (2 ℵ 0 ) + -liftable, t hen it is wel l-founde d. Nevertheless, we shall see in Section 3-7 that a v ar iant of CLL can be formulated for po sets, suc h as ( ω + 1) op , that ar e not λ -liftable for certain inﬁnite cardinals λ . The key concept is the one of a pseudo-r etr act . Deﬁnition 3 - 6.5. A pseudo-r etr action p air for a pair ( P, Q ) of p osets is a pair ( e, f ) of isoto ne maps e : P → Id Q and f : Q → P s uch that P ↓ f “( e ( p )) = P ↓ p for each p ∈ P . W e say that P is a pseudo-r etr act of Q . Lemma 3 - 6.6. Every almost join-semilattic e P is a pseudo-r etr act of some lower ﬁnite almost join-semilattic e Q of c ar dinality card P . Pr o of . If P is ﬁnite ta ke Q := P . Now supp o se that P is inﬁnite, and deﬁne I := { X ⊆ P | X is a ﬁnite ▽ -clos ed subset of P } , ordered by co n tainment, a nd then Q := S ( { X } × X | X ∈ I ), or dered com- po nent wise. It is obvious that Q is low er ﬁnit e with the same cardinality a s P . Now let U := { ( X 0 , a 0 ) , . . . , ( X n − 1 , a n − 1 ) } (with n < ω ) b e a ﬁnite subset of Q . As P is a pseudo join- semilattice, A := ▽ i λ . 3-8 Left and right larders 87 Now supp os e that P is a n almost join- semilattice. W e ﬁrst deal with the case wher e P is low er ﬁnite. It follows from [23, P rop osition 3.4 ] that the relation ( κ, < λ ) ❀ ([ λ ] <ω , ⊆ ) holds. As the assignment ( x 7→ P ↓ x ) deﬁnes an order-embedding from ( P, ≤ ) in to ([ P ] <ω , ⊆ ) and card P ≤ λ , the relatio n ( κ, < λ ) ❀ P holds a s well. By Corollary 3 -5.8, P has a λ -lifter ( X , X ) with X low er ﬁnite. The des ired co nclusion follows from CLL (Lemma 3-4.2). In the genera l case, it follows from Lemma 3-6 .6 that P is a pseudo-retract of a low er ﬁnite almos t join- semilattice Q with the sa me c a rdinality as P . By the low er ﬁnite case, Lift( Φ − → A f ) ho lds . By Lemma 3 -7.1, Lift( Φ − → A ) holds as well. ⊓ ⊔ Remark 3- 7 .3. Corollary 3-7.2 makes it p ossible to lift diagrams indexed b y almost join-semilattices much mo re general than thos e taken c are of by CLL (Lemma 3-4.2). The easies t example is where P := ( ω + 1) op is the dual of the chain ω + 1 = { 0 , 1 , 2 , . . . } ∪ { ω } . Cor o llary 3 -7.2 says that under a suitable large cardinal ass umption, for every ob ject − → A of ( A † ) P , there ar e an ob ject − → B of B P and a do uble arrow − → χ : Ψ − → B ⇒ Φ − → A . This is not a trivia l consequence of Lemma 3-4.2, b ecause P is no t low er ﬁnite so Lemma 3-5 .5 do es not apply . In fact, by Corollary 3-6.4, P is no t (2 ℵ 0 ) + -liftable (cf. Pr o blem 8 in Chapter 7). F or an application of Corolla r y 3-7 .2, see the pro of of Theorem 4-7 .2. 3-8 Left and r igh t larders The deﬁnition of a larder (Deﬁnition 3-4.1) inv olves catego ries A , B , and S , with a few sub ca tegories, toge ther with functors Φ : A → S and Ψ : B → S . One half of the deﬁnition descr ib e s the interaction b etw een A and S , while the other half inv olves the interaction betw een B and S ; furthermor e, it is only the second pa r t that involv es the car dinal para meter λ . In most applications, the desir ed in teraction b et ween B a nd S is noticeably harder to establish than the one betw een A and S . Therefore , in orde r to ma ke our w or k more user -friendly in view of further applications, we sha ll s plit the deﬁnition of a larder in tw o parts. Deﬁnition 3 - 8.1. A lef t lar der is a quadruple Λ = ( A , S , S ⇒ , Φ ), where A and S a r e ca tegories , S ⇒ is a s ubca tegory of S (whose arrows w e shall call the double arr ows of S ), and Φ : A → S is a functor satisfying the following prop erties: (CLOS( A )) A ha s a ll small directed colimits. (PROD( A )) An y tw o o b jects in A ha ve a pro duct in A . (CONT( Φ )) The functor Φ pres erves all sma ll directed colimits. (PROJ( Φ, S ⇒ )) Φ ( f ) is a morphism in S ⇒ , for each extended pro jection f of A (cf. Deﬁnition 1 -2.4). 88 3 The Condensat e Lif ting Lemma (CLL) W e already formulated the conditions (CLOS( A )), (PR OD( A )), (CONT( Φ )), and (PROJ( Φ, S ⇒ )) within the statemen t of the Armature L e mma (Lemma 3- 2 .2). Deﬁnition 3 - 8.2. Let λ and µ be inﬁnite car dinals. A 6-uple Λ = ( B , B † , S , S † , S ⇒ , Ψ ) is a right ( λ, µ ) -lar der at an obje ct B if B and S are ca teg ories, B is an ob ject of B , B † (resp., S † ) is a full sub ca tegory of B (resp., S ), S ⇒ is a s ubca tegory of S (whose arrows w e shall call the double arr ows of S ), and Ψ : B → S is a functor sa tisfying the following prop erties: (PRES λ ( B † , Ψ )) The ob ject Ψ ( B ) is weakly λ -pr esented in S , for each ob- ject B ∈ B † . (LS r µ ( B )) F or ea ch S ∈ S † , each double arrow ψ : Ψ ( B ) ⇒ S , each µ -sma ll set I , and e ach family ( γ i : C i ֌ B | i ∈ I ) of monic ob jects in B † ↓ B , there exists a monic ob ject γ : C ֌ B in B † ↓ B such that γ i E γ for each i ∈ I while ψ ◦ Ψ ( γ ) is a mor phism in S ⇒ . W e say that Λ is a right ( λ, µ ) -lar der if it is a rig h t ( λ, µ )- larder a t every ob ject o f B . W e say that Λ is str ong if the following conditions are satisﬁed: (CLOS µ ( B † , B )) The full sub category B † has all µ -small directed colimits within B (cf. Deﬁnition 1-2.3). (CLOS r µ ( S ⇒ )) The sub catego ry S ⇒ is righ t closed under all µ -small di- rected c olimits (cf. Deﬁnition 1 -2.5). (CONT µ ( Ψ )) The functor Ψ pr e s erves all µ - s mall dir ected co limits. W e say that Λ is pr oje ctable if every double arr ow ϕ : Ψ ( C ) ⇒ S , for C ∈ Ob B and S ∈ Ob S , has a pro jectabilit y witness (cf. Deﬁnition 1 -5.1). W e already for m ulated the condition (PRES λ ( B † , Ψ )) in Deﬁnition 3 -4.1. The conditions (CLOS µ ( B † , B )), (CLOS r µ ( S ⇒ )), a nd (CONT µ ( Ψ )) w ere for- m ulated within the statement of the Buttress Lemma (Lemma 3-3.2). The condition (LS r µ ( B )) is a mo diﬁcation of the condition (LS b λ ( B )) formulated within the statement of the Buttress Lemma . Observe that in Deﬁnitions 3-8.1 a nd 3-8.2 ab ove, w e just split the items (CLOS( A ))–(PROJ( Φ, S ⇒ )) from Deﬁnition 3-4.1 in to the par ts dev oted to A and B , r esp ectively , while (LS r µ ( B )) is obtained from (LS µ ( B )) by r eplac- ing Φ “( A † ) by S † . The pr o of o f the following observ ation is trivial. Prop ositio n 3-8.3. L et λ and µ b e inﬁnite c ar dinals with λ r e gular, let A , B , S , A † , B † , S † , S ⇒ b e c ate gories, and let Φ , Ψ b e functors. If ( A , S , S ⇒ , Φ ) is a left lar der, ( B , B † , S , S † , S ⇒ , Ψ ) is a right ( λ, µ ) -lar der at an obje ct B of B , A † is a ful l sub c ate gory of A , and Φ “( A † ) is c ontaine d in S † , then ( A , B , S , A † , B † , S ⇒ , Φ, Ψ ) is a ( λ, µ ) -lar der at B . W e shall usually say right λ -larder instead of rig ht ( λ, cf ( λ ))-larder. Chapter 4 Getting larders from congruen c e lattices of ﬁrst-o rder str u ctures Abstract. One of the main origins of our w ork is the ﬁrst author’s pap er Gil- liber t [18], where it is pr ov ed, in par ticula r, that the critical po int cr it( A ; B ) betw een a lo cally ﬁnite v ariety A and a ﬁnitely generated congruence- distributive v ariety B s uch that Co n c A 6⊆ Con c B is a lwa ys less than ℵ ω . One of the goals of the present chapter is to show ho w ro utine categorical veriﬁcations abo ut alge braic sys tems make it p os s ible, using CLL, to extend this result to r elative compact congruence s emilattices of quasiv a rieties o f al- gebraic systems (i.e., the la nguages now ha ve relations as well as op erations, and w e are dealing with quasiv arieties rather than v arie ties ). Tha t pa rticular extension is stated and prov ed in Theorem 4-9 .4. W e a lso obtain a version o f Gr¨ atzer- Schm idt’s Theorem for p os et-indexed diagrams of ( ∨ , 0)-semilattices and ( ∨ , 0)-homomorphisms in Theorem 4-7.2. With further p otential appli- cations in v iew, most of C ha pter 4 is desig ned to build up a framework for being a ble to eas ily verify larderho o d o f man y structures arising from (ge n- eralized) quasiv arieties of algebra ic systems. Although we included in this chapter, fo r c o nv enience sake, a num b er of a lr eady known or folklo re results, it also con tains re sults which, although they could b e in principle obta ined from a lready published results, co uld not b e s o in a stra ightf or ward fashio n. Such res ults are Prop osition 4-2.3 (description of some w eakly κ -pres e nted structures in MInd ) o r Theorem 4-4.1 (preser v ation of all small directed col- imits b y the r elative compact congruence semila ttice functor within a given generalized quasiv ariety). The structures studied in Chapter 4 will be called monotone-indexe d struc- tur es . They form a ca teg ory , that we shall deno te b y MInd . The ob jects of MInd are just the ﬁrst-o rder structures. F or ﬁrst-order structur es A and B , a morphism from A to B in MInd can exist only if the language of A is contained in the language of B , and then it is deﬁned as a homomorphism (in the usual sense) from A to the reduct o f B to the langua g e of A . 89 90 4 Larders from ﬁrst-order structures 4-1 The categor y of all monotone-indexed str uct ures In the present section we shall in tro duce basic deﬁnitions and facts ab out ﬁrst-order structures, congruences, ho momorphisms, with the following t wist: we will allow homomor phisms b e t ween ﬁrst-or der structures with diﬀerent languages . Our deﬁnition o f a morphism will extend the standard deﬁnitio n of a ho - momorphism b etw een ﬁrst-order s tructures. A morphi sm from a ﬁrs t-order structure A to a ﬁrst-o rder structure B , deﬁned only in ca se Lg( A ) ⊆ Lg( B ) ( and not only, as usual, Lg( A ) = Lg ( B )), is a ma p ϕ : A → B whose re- striction from A to the Lg ( A )-reduct of B is a homomorphism of L g ( A )- structures: that is, ϕ ( c A ) = c B for each constant symbol c ∈ Lg( A ), and ϕ  f A ( x 1 , . . . , x n )  = f B ( ϕ ( x 1 ) , . . . , ϕ ( x n )) , (4-1.1) ( x 1 , . . . , x n ) ∈ R A ⇒ ( ϕ ( x 1 ) , . . . , ϕ ( x n )) ∈ R B , (4-1.2) for all n ∈ ω \ { 0 } , all x 1 , . . . , x n ∈ A , and each n -ary f ∈ Op( A ) (r esp., each n -ar y R ∈ Rel( A )). If, in addition, ϕ is one-to-one and the implica - tion in (4-1.2) is a logical e q uiv alence, we say that ϕ is an emb e dding . In the stronger ca se where ϕ is an inclusion map and the implica tion in (4-1.2) is still an equiv alence, we s ay that A is a substructur e of B . W e shall denote by MInd the categor y of all ﬁrst- order structures with this extended deﬁni- tion of a morphism, a nd w e shall call it the category of all monotone-indexe d structur es . The present deﬁnition is reminiscent of the few pro p o s ed for nonindexe d algebr as . A ma jor diﬀerence is that the category MInd ha s all small dir ected colimits (cf. Deﬁnition 1-2 .3). The following deﬁnition o f a c ongruence is equiv a lent to the one given in [26, Section 1.4]. Deﬁnition 4 - 1.1. A c ongruenc e of a ﬁrs t-order str ucture A is a pair θ =  θ, ( R θ | R ∈ Rel( A ))  , where (i) θ is an equiv alence relation on A ; (ii) θ is co mpatible with ea ch function in A , that is, for ea ch f ∈ Op( A ), say of arity n , and all x 1 , . . . , x n , y 1 , . . . , y n ∈ A , if ( x s , y s ) ∈ θ for each s ∈ { 1 , . . . , n } , then ( f ( − → x ) , f ( − → y )) ∈ θ ( her e and elsewher e, we u s e t he abbr evia tion − → x for the n - uple ( x 1 , . . . , x n )). (iii) R A ⊆ R θ ⊆ A ar( R ) , for each R ∈ Rel( A ); (iv) F or each R ∈ Rel( A ), say o f ar it y n , and a ll x 1 , . . . , x n , y 1 , . . . , y n ∈ A , if − → x ∈ R θ and ( x s , y s ) ∈ θ for each s ∈ { 1 , . . . , n } , then − → y ∈ R θ . Notation 4-1. 2. Le t α be a congruence o f a ﬁrst-order s tructure A . F or elements x, y ∈ A , let x ≡ y (mo d α ) hold if ( x, y ) ∈ α . Lik ewise, for R ∈ Rel( A ), say of arity n , and x 1 , . . . , x n ∈ A , let R ( x 1 , . . . , x n ) (mod α ) (often abbreviated R − → x (mo d α )) hold if ( x 1 , . . . , x n ) ∈ R α . 4-1 The categ ory of all monotone-indexed structures 91 The set Con A of all congr ue nc e s of a ﬁrst-or der structure A is par tially ordered co mp one nt wise, that is, α ≤ β iff α ⊆ β and R α ⊆ R β for ea ch R ∈ Rel( A ). It follows easily that Con A is an alg ebraic subset of the lattice P ( A × A ) × Y  P ( A ar( R ) ) | R ∈ Rel( A )  ; in par ticular, it is an a lgebraic lattice, see the c o mment s at the b eg inning o f [26, Section 1.4.2]. The smalles t cong ruence of A is 0 A :=  id A ,  R A | R ∈ Rel( A )  , while the larg e st congruence of A is 1 A :=  A × A,  A ar( R ) | R ∈ Rel( A )  . F or an equiv a lence rela tion θ on a se t A and an element x ∈ A , we shall denote by x/θ the blo ck (= equiv alence class) of x modulo θ . F or a n -uple ( x 1 , . . . , x n ), we abbreviate the n -uple ( x 1 /θ , . . . , x n /θ ) as − → x /θ . Deﬁnition 4 - 1.3. F or a congruence θ of a ﬁrst-order structure A , we shall denote b y A / θ the ﬁrst-order structure with univ erse A/θ , the same language as A , c A / θ = c A /θ for ea ch c ∈ Cst( A ), and f A / θ ( − → x /θ ) = f ( − → x ) /θ , − → x /θ ∈ R A / θ ⇔ − → x ∈ R θ , for all n ∈ ω \ { 0 } , a ll x 1 , . . . , x n ∈ A , and each n -ary f ∈ Op( A ) (res p., each n -ary R ∈ Rel( A )). The c anonic al homomorphi sm (or c anonic al pr oje ction ) from A on to A / θ is the ma p ( A ։ A/θ , x 7→ x/θ ). More generally , for congruences α and β o f a ﬁrst-order structure A , if α ≤ β , then the cano nical map ( A/α ։ A/β , x/α 7→ x/β ) is a morphism in MInd , that we shall also call the canonical pro jection fr om A / α o nto A / β . Deﬁnition 4 - 1.4. The kernel o f a morphism ϕ : A → B in MInd is the pair Ker ϕ :=  θ, ( R θ | R ∈ Rel( A ))  , where θ := { ( x, y ) ∈ A × A | ϕ ( x ) = ϕ ( y ) } , R θ := { − → x ∈ A ar( R ) | ϕ ( − → x ) ∈ R B } (where we set ϕ ( − → x ) := ( ϕ ( x 1 ) , . . . , ϕ ( x ar( R ) ))), for ea ch R ∈ Rel( A ). It is straightforw ar d to verify that the k ernel of a morphism fro m A to B is a congr uence of A . F urther more, the kernel of the canonica l pro jection from A on to A / θ is θ , for any cong ruence θ o f A . The following result extends [2 6, Pro po sition 1 .4.1]. As o ur notation diﬀers from the one used in that reference, we include a pro of for conv enience. 92 4 Larders from ﬁrst-order structures Lemma 4 - 1.5 (Fi rst Isomorphism Theorem). L et ϕ : A → B b e a mor- phism in MInd and let α b e a c ongruenc e of A . Denote by π : A ։ A / α the c anonic al pr oje ction. Then α ≤ Ker ϕ iff ther e exists a morphism ψ : A / α → B su ch that ϕ = ψ ◦ π , and if t his o c curs then ψ is unique. F urthermor e, ψ is an emb e ddi ng iff α = Ker ϕ . Pr o of . Set θ := K e r ϕ . Supp ose ﬁrst that ther e exis ts a morphism ψ : A / α → B such that ϕ = ψ ◦ π . F or all x, y ∈ A , ( x, y ) ∈ α means that π ( x ) = π ( y ), thus ϕ ( x ) = ϕ ( y ), that is, ( x, y ) ∈ θ ; whence α ⊆ θ . Now let R ∈ Rel( A ) and let − → x ∈ R α . This means tha t − → x /α ∈ R A / α , thus, as ψ is a morphism in M Ind , ψ ( − → x /α ) ∈ R B , that is, ϕ ( − → x ) ∈ R B , so − → x ∈ R θ . W e hav e proved the inequality α ≤ θ . Conv ersely , supp os e tha t α ≤ θ . As α ⊆ θ , ther e exists a unique map ψ : A/α → B such that ϕ = ψ ◦ π . W e m ust pr ov e that ψ is a mor phism in MInd . F or each f ∈ Op( A ), say with a rity n , a nd all x 1 , . . . , x n ∈ A , ψ  f A / α ( − → x /α )  = ψ  f A ( − → x ) /α  = ϕ  f A ( − → x )  = f B ( ϕ ( − → x )) = f B  ψ ( − → x /α )  . F urthermore, for any R ∈ Rel( A ) and any − → x ∈ A ar( R ) , − → x /α ∈ R A / α ⇔ − → x ∈ R α (b y the deﬁnition o f R A / α ) ⇒ − → x ∈ R θ (b y the assumption that α ≤ θ ) (4-1 .3) ⇔ ϕ ( − → x ) ∈ R B (b y the deﬁnition o f θ := Ker ϕ ) ⇔ ψ  − → x /α  ∈ R B (b y the deﬁnition o f ψ ) . This completes the pro of that ψ is a morphism in MInd . The uniquenes s statement on ψ follows trivially from the surjectivity o f the map π . If α = θ , then α = θ so ψ is one-to- one, and fur ther , th e implication (4-1.3) ab ov e is an equiv alence, s o ψ is an embedding. Co n versely , if ψ is an embed- ding, then similar arguments to those ab ov e show easily that α = Ker ϕ . ⊓ ⊔ F or congruences α , β of a ﬁrst-o rder structure A such that α ≤ β , we denote by β / α the kernel of the canonica l pro jection A / α ։ A / β . Obs e rve that this congr uence of A / α can b e describ ed by β / α := { ( x/α, y /α ) | ( x, y ) ∈ β } , R β / α := { ( x 1 /α, . . . , x n /α ) ∈ ( A/α ) n | ( x 1 , . . . , x n ) ∈ R β } , for each R ∈ Rel( A ) of a r ity , say , n . The following r e s ult is established in [26, Prop ositio n 1.4.3]. As our no- tation diﬀers from the one used in tha t reference, w e include a pro of for conv enience. Lemma 4 - 1.6 (Se cond Isomorphism Theo rem ). L et α b e a c ongruenc e of a ﬁrst-or der stru ctur e A . Then the assignment ( β 7→ β / α ) deﬁnes a lattic e 4-2 Dir ected colimits of monotone-indexe d structures 93 isomorphi sm fr om (Con A ) ↑ α onto Co n( A / α ) , and the assignment  x/β 7→ ( x/α ) / ( β / α )  deﬁnes an isomorphism fr om A / β onto ( A / α ) / ( β / α ) , for e ach β ∈ (Con A ) ↑ α . Pr o of . Again, similar to the case without relations (cf. [27, Sec tio n 1 .11]). W e set σ ( β ) := β / α , for each β ∈ (Con A ) ↑ α ; s o σ deﬁnes an iso tone ma p from (Con A ) ↑ α to Con( A / α ). W e need to deﬁne the conv erse map. T o ea ch congruence ˜ β ∈ Con( A / α ), we a sso ciate the kernel of the co mpo s ite of the canonical pro jections A ։ A / α ։ ( A / α ) / ˜ β . Hence τ ( ˜ β ) is a congruence of A co nt aining α , a nd τ ( ˜ β ) :=  β , ( R β | R ∈ Rel( A ))  with β := { ( x, y ) ∈ A × A | ( x/α, y /α ) ∈ ˜ β } , R β := { − → x ∈ A n | − → x /α ∈ R ˜ β } . Denote this cong ruence by β . Then β /α = ˜ β b y the deﬁnitio n of β , while, for each R ∈ Rel( A ), R β / α = { − → x /α | − → x ∈ R β } = { − → x /α | − → x /α ∈ R ˜ β } = R ˜ β , so ˜ β = β / α . This prov es that σ ◦ τ is the identit y on Con( A / α ). Conv ersely , let β ∈ (Con A ) ↑ α , set ˜ β := β / α a nd β ′ := τ ( ˜ β ). W e m ust prov e that β = β ′ . First, β ′ = { ( x, y ) ∈ A × A | ( x/α, y /α ) ∈ ˜ β } = { ( x, y ) ∈ A × A | ( x/α, y /α ) ∈ β /α } = β , and for all R ∈ Rel( A ) and all − → x ∈ A ar( R ) , − → x ∈ R β ′ iff − → x /α ∈ R ˜ β iff − → x /α ∈ R β / α iff − → x ∈ R β . Hence β = β ′ , which completes the pro o f that τ ◦ σ is the iden tity on (Con A ) ↑ α . As bo th σ and τ a re isoto ne , it follows that they are mut ually in verse isomor phisms. Finally , as β / α is the k ernel of the canonical pro jection from A / α on to A / β , it follows from Lemma 4-1 .5 that it induces a n iso morphism fro m ( A / α ) / ( β / α ) onto A / β . ⊓ ⊔ 4-2 Directed colimits of monotone-indexed structures The following result shows that the category M Ind has all small directed colimits (cf. Deﬁnition 1-2.3), and gives a description of those colimits. Of course, this res ult extends the clas sical description of dir ected c o limits for mo dels of a given languag e , g iven for example in [26, Section 1.2.5] and recalled in Section 1-2.5. 94 4 Larders from ﬁrst-order structures Prop ositio n 4-2.1. L et  A i , ϕ j i | i ≤ j in I  b e a dir e cte d p oset-indexe d di- agr am in MInd . We form t he c olimit ( A, ϕ i | i ∈ I ) = lim − →  A i , ϕ j i | i ≤ j in I  (4-2.1) in the c ate gory Set of al l sets, and we set L := S (Lg( A i ) | i ∈ I ) . Then A c an b e extende d to a unique ﬁrst-or der structu r e A such t hat Lg( A ) = L and t he fol lowing statements hold: (i) F or e ach c onstant symb ol c in L , c A = ϕ i ( c A i ) for al l lar ge enough i ∈ I . (ii) F or e ach op er ation symb ol f in L , say of arity n , and for e ach i ∈ I such that f ∈ Op( A i ) , the e quation f A ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) = ϕ i  f A i ( x 1 , . . . , x n )  holds for al l x 1 , . . . , x n ∈ A i . (iii) F or e ach r elation symb ol R in L , say of arity n , and for e ach i ∈ I such that R ∈ Rel( A i ) , t he e quivalenc e ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) ∈ R A ⇐ ⇒ ( ∃ j ∈ I ↑ i )  ( ϕ j i ( x 1 ) , . . . , ϕ j i ( x n )) ∈ R A j  , holds for al l x 1 , . . . , x n ∈ A i . F urthermor e, the fol lowing r elation holds in MInd . ( A , ϕ i | i ∈ I ) = lim − →  A i , ϕ j i | i ≤ j in I  . (4-2.2) In categorica l terms, Pr op osition 4-2.1 means that the for getful functor fr om MInd to Set cr e ates and pr eserves al l smal l dir e cte d c olimits . Pr o of . The construction o f ( A, ϕ i | i ∈ I ) is g iven just b efore (1-2.3). The existence of the structure A is then a straig ht for ward e x ercise. F or a co cone ( B , ψ i | i ∈ I ) in MInd a bove  A i , ϕ j i | i ≤ j in I  , it follows from (4- 2.1) that there exists a unique map ψ : A → B such that ψ i = ψ ◦ ϕ i for each i ∈ I . Then, using the deﬁnition of A , it is straig htforward to v er ify that ψ is a morphism from A to B in MInd . This completes the proo f of (4-2.2). ⊓ ⊔ Deﬁnition 4 - 2.2. Let κ b e an inﬁnite cardina l. A ﬁrs t- order structure A is • κ -smal l if ca rd A < κ ; • c ompletely κ - s m al l if card A + card Lg A < κ . Now that w e know what directed c o limits loo k lik e in MInd we can prove the following result. 4-2 Dir ected colimits of monotone-indexe d structures 95 Prop ositio n 4-2.3. L et κ b e an inﬁnite c ar di nal and let A b e a ﬁ rst-or der structur e. If A is c ompletely κ -smal l, then A is we akly κ - pr esente d in MInd . Conversely, if κ is u n c ountable, then, deﬁning Ω as the disjoi nt u nion of A and Lg ( A ) , A is a c ontinuous dir e cte d union, indexe d by [ Ω ] <κ , of c ompl etely κ -smal l structur es in M Ind , and A is we akly κ -pr esente d in M Ind iff it is c ompletely κ - s m al l. Pr o of . A sta nda rd argument, obtained by adapting the proo f of the corre- sp onding statemen ts for κ -presented structures with κ regular (cf., for exa m- ple, [2, Corolla ry 3.13]) to our deﬁnition of a weakly κ -pre s ent ed structure. Assume ﬁrst that A is co mpletely κ -small, let Ω b e a set, and c onsider a contin uous dir ected co limit c o cone in MInd o f the form  B , β X | X ∈ [ Ω ] <κ  = lim − →  B X , β Y X | X ⊆ Y in [ Ω ] <κ  , (4- 2 .3) and let ϕ : A → B b e a mor phism in MInd . It follows from the contin uity of the colimit (4-2.3) that the following rela tio n holds for each X ∈ [ Ω ] <κ :  B , β X ∪ Y | Y ∈ [ Ω ] <ω  = lim − →  B X ∪ Y , β X ∪ Z X ∪ Y | Y ⊆ Z in [ Ω ] <ω  . (4-2.4) In particular , it follows from Pro po sition 4-2 .1 that b oth re la tions B = [  β X ∪ Y “( B Y ) | Y ∈ [ Ω ] <ω  (4-2.5) Lg( B ) = [  Lg( B X ∪ Y ) | Y ∈ [ Ω ] <ω  (4-2.6) hold for each X ∈ [ Ω ] <κ . By applying (4-2.5) for X := ∅ , we obtain, for each a ∈ A , a ﬁnite subset U a of Ω suc h that ϕ ( a ) ∈ β U a “( B U a ). Likewise, by using (4- 2 .6), we obta in, for each s ∈ Lg ( A ), a ﬁnite subset V s of Ω s uch that s ∈ Lg( B V s ). The set X := [ ( U a | a ∈ A ) ∪ [ ( V s | s ∈ Lg( A )) belo ngs to [ Ω ] <κ , ϕ “( A ) is con tained in β X “( B X ), and Lg( A ) is c ontained in Lg ( B X ). The ﬁrst co n tainment implies that there exists a map (not nec- essarily a morphism) ψ : A → B X such that ϕ = β X ◦ ψ . F or ea ch f ∈ O p( A ), say o f ar ity n , and all x 1 , . . . , x n ∈ A , setting x := f A ( x 1 , . . . , x n ), we obtain, as ϕ is a morphism in MInd , that the equation ϕ ( x ) = f B  ϕ ( x 1 ) , . . . , ϕ ( x n )  is satisﬁed, that is, ( β X ◦ ψ )( x ) = β X  f B i  ψ ( x 1 ) , . . . , ψ ( x n )  . By using (4-2.4) together with P r op osition 4-2.1, we obtain a ﬁnite subset V of Ω such that ( β X ∪ V X ◦ ψ )( x ) = β X ∪ V X  f B X  ψ ( x 1 ) , . . . , ψ ( x n )  , that is, 96 4 Larders from ﬁrst-order structures ( β X ∪ V X ◦ ψ )( x ) = f B X ∪ V  ( β X ∪ V X ◦ ψ )( x 1 ) , . . . , ( β X ∪ V X ◦ ψ )( x n )  . (4-2.7) Similarly , for ea ch R ∈ Rel( A ), say of a r ity n , a nd all ( x 1 , . . . , x n ) ∈ R A , there ex ists a ﬁnite subset V of Ω such that  ( β X ∪ V X ◦ ψ )( x 1 ) , . . . , ( β X ∪ V X ◦ ψ )( x n )  ∈ R B X ∪ V . (4-2.8) As ca rd A + car d Lg( A ) < κ , the union Y of a ll V s in (4-2.7) and (4-2.8) belo ngs to [ Ω ] <κ . As the relations (4-2.7) a nd (4-2 .8), with V repla ced b y Y , are satisﬁed for all p os sible choices of f , R , and x 1 , . . . , x n , the map ϕ ′ := β X ∪ Y X ◦ ψ is a mor phism from A to B X ∪ Y in MInd . Observe that ϕ = β X ∪ Y ◦ ϕ ′ . This mea ns that A is weakly κ -presented in MInd . Now assume that κ is uncoun table, let A b e an ar bitrary ﬁr st-order struc- ture, a nd set Ω := A ∪ L g( A ) (disjoint union). Pick a ∈ A . F or each X ⊆ Ω , there exists a leas t (with resp ect to co nt ainment) subset X o f Ω c o ntain- ing X ∪ { a } such that X ∩ A is closed under all op er a tions (a nd constants) from X ∩ Lg( A ). F urthermore, a s κ is uncountable, if X is κ -small, then so is X . F or each X ∈ [ Ω ] <κ , denote by A X the ﬁrst-o rder structure with uni- verse X ∩ A a nd lang uage X ∩ Lg( A ). By using Pr op osition 4-2.1, we obtain the contin uous directed c olimit A = lim − → X ∈ [ Ω ] <κ A X in MInd , (4-2.9) with a ll transitio n maps and limiting maps being the cor resp onding inclusion maps. In particular, if A is weakly κ -presented in MInd , it follows from the contin uit y of the dire c ted colimit (4- 2.9) that A = A X for some X ∈ [ Ω ] <κ , so card A + card Lg ( A ) < κ . ⊓ ⊔ There ar e many examples o f inﬁnite structur e s that are ﬁnitely presented in MInd . F or example, let A := ω , Lg( A ) := { f } where f is a unary op era - tion symbo l, and deﬁne f A as the successor map on ω . This s hows that the conv erse part of Prop osition 4-2.3 doe s no t extend to the case where κ = ℵ 0 . 4-3 The relative congruence latt ice functor with resp ect to a generalized quasiv ariety F or a member A of a class V of structur es, the only cong ruences of A we will be in terested in will mostly b e V -c ongruenc es , that is, congrue nce s θ of A such that A / θ b elongs to V . This appro ach is also widely us ed in Go r - bunov [26]. How ever, here a ll the structures in V need not have the same language. This will b e useful for our extension of the Gr¨ atzer-Schmidt Theo- rem sta ted in Theorem 4-7.2. The relev ant classes V will be called gener alize d quasivarieties (Deﬁnition 4-3 .4). 4-3 Congruence lattices in generalized quasiv arieties 97 Deﬁnition 4 - 3.1. F or a morphism ϕ : A → B in MInd and a congr uence β of B , we deﬁne (Res ϕ )( β ) :=  α, ( R α | R ∈ Rel( A ))  , where α := { ( x, y ) ∈ A × A | ( ϕ ( x ) , ϕ ( y )) ∈ β } , R α := { − → x ∈ A ar( R ) | ϕ ( − → x ) ∈ R β } , for each R ∈ Rel( A ) . The veriﬁcation o f the following result is r o utine. Prop ositio n 4-3.2. (i) The assi gnment Res ϕ maps Con B to Con A , for any morphism ϕ : A → B in MInd . F ur t hermor e, this assignment pr eserves al l me ets and al l nonempty dir e ct e d joins in Con B . (ii) The assignment ( A 7→ Con A , ϕ 7→ Res ϕ ) deﬁnes a c ontr avariant functor fr om MInd t o t he c ate gory of al l algebr aic lattic es with maps pr e- serving al l me ets and al l nonempty dir e cte d joins. Lemma 4 - 3.3. L et A , B , and C b e ﬁrst -or der struct ur es, to gether with morphisms ϕ : A → B and ψ : B → C in MInd . Then the e quation Ker( ψ ◦ ϕ ) = (Res ϕ )(Ker ψ ) is satisﬁe d. Pr o of . As Ker ψ = (Res ψ )( 0 C ) and by the contrav aria nce of the functor Res, (Res ϕ )(Ker ψ ) = (Res ϕ ) ◦ (Res ψ )( 0 C ) = Res( ψ ◦ ϕ )( 0 C ) = Ker( ψ ◦ ϕ ) . This concludes the pro o f. ⊓ ⊔ Deﬁnition 4 - 3.4. A gener alize d quasivariety is a full s ubca tegory V of M Ind such that: (i) F or ev ery morphism ϕ : A → B in V , the quotient A / Ker ϕ is a mem b er of V . (ii) F or any ﬁrs t-order structur e s A and B , if A embeds into B , Lg( A ) = Lg( B ), a nd B ∈ V , then A ∈ V . (iii) Any dire c t pro duct of a nonempt y collection of ﬁrst-order str uctures in V , all with the s a me la ng uage, b elong s to V . (iv) The q uo tient A / 1 A belo ngs to V , for each A ∈ V . (v) V is clos e d under dir ected co limits in MInd . Then we s et Con V A := { α ∈ Con A | A / α ∈ V } , for any ﬁrst-o rder struc tur e A , and we call the elements of Con V A the V -c ongruenc es of A . The ( ∨ , 0)-semi- lattice of all compact elements of that lattice will be denoted by Co n V c A . W e shall us ually o mit the sup ers cript V in case V is clo sed under homomor phic images. The meaning of Con V A that we use her e is the same a s the meaning of Con K A introduced in [26, Section 1 .4.2]. 98 4 Larders from ﬁrst-order structures Example 4-3. 5. A quasivariety in a given langua ge L is a class V of ﬁr st- order s tructures on L which is c lo sed under substructures, dir e ct pro ducts, and directed colimits within the class of all mo dels of L (cf. Gorbunov [26]). Equiv alently , V is the class o f all models, for a g iven language L , that satisfy a g iven set of fo r mulas each of the fo rm ( ∀ − → x )  ^ ^ i 0 and atomic formulas E i , F . It is straightforward to verify that an y qua siv ariety is also a gener alized quasiv ariety (for (i), obse rve that A / Ker ϕ embeds into B ; for (iv), observe that A / 1 A is the terminal o b ject of V , and so it is the pro duct, index ed by the empt y set, of a family o f ob jects in V .) Example 4-3. 6. T he catego ry MInd is a genera lized quasiv ariety. Mo r e generally , let C b e a cla ss of languages which is closed under directed unio ns. Then the full s ub ca tegory MInd C consisting of all ﬁrs t- order structures A such that Lg( A ) b elongs to C is a genera lized quasiv ariety. Particular ly in- teresting examples are the following: (i) C is the class of all languages witho ut relatio n sym b ols. Then the ob jects of MInd C are all the algebr as (in the sense of universal alg ebras). (ii) C is the class of all languages containing neither rela tion symbols nor constant symbols, and all whose op eration symbols ha ve ar ity one. Then the ob jects of M Ind C are all the unary algebr as . (iii) C is the class o f a ll la nguages co n taining neither c onstant symbols nor op era tion s y m b ols. Then the o b jects of MInd C are all the r elational systems . Observe that for a gener alized quasiv a riety V and a ﬁrst-or der structure A , the set Con V A may very well be empty . Because o f Deﬁnition 4-3.4(iv), this cannot oc c ur in case A ∈ V . The following result is analogue to [26, Corollar y 1.4.11]. Lemma 4 - 3.7. L et V b e a gener alize d quasivariety and let A ∈ V . Then Con V A is an algebr aic subset of Con A , with the same b ounds. In p articular, it is an algebr aic lattic e. Pr o of . As A b elo ngs to V , its z e ro co ng ruence 0 A belo ngs to Con V c A . Moreov er , it follows from Deﬁnition 4-3.4(iv) that 1 A belo ngs to Con V A . If ( α i | i ∈ I ) is a no nempt y family of elements in Co n V A and α := V ( α i | i ∈ I ) , then the diagonal map embeds, with the sa me languag e, the structure A / α in to the pro duct Q ( A / α i | i ∈ I ); hence, by (ii) and (iii) in Deﬁnition 4-3.4, A / α b elongs to V , and so α ∈ Con V A . Therefo re, Con V A is closed under arbitrar y meets. Now let I be a no ne mpty directed po set and let ( α i | i ∈ I ) b e a n iso to ne family of elemen ts in Con V A ; set α := W ( α i | i ∈ I ). By using the explicit 4-3 Congruence lattices in generalized quasiv arieties 99 description of the co limit given in Prop osition 4-2.1, it is not hard to ver- ify that A / α = lim − → i ∈ I ( A / α i ), with the transition morphisms and limiting morphisms b eing the pro jection maps (cf. [26, P r op osition 1.4.9 ]). As all the structures A / α i belo ng to V and V is closed under directed colimits in MInd , the structure A / α b elongs to V , and so α ∈ Con V A . ⊓ ⊔ In particular, as Con V A is an algebraic subset of Co n A , which is an algebraic s ubset o f P ( A × A ) × Q  P ( A ar( R ) ) | R ∈ Rel( A )  , the following notation is well-deﬁned. Notation 4-3. 8. Le t V b e a ge ne r alized quas iv ariety and let A ∈ V . • F or elements x, y ∈ A , we denote by k x = y k V A the least co ng ruence α ∈ Con V A such that x ≡ y (mo d α ). • F or R ∈ Rel( A ), say of arity n , and − → x = ( x 1 , . . . , x n ) ∈ A n , we denote b y k R ( x 1 , . . . , x n ) k V A , abbreviated k R − → x k V A , the least congr uence α ∈ Co n V A such that R ( x 1 , . . . , x n ) (mod α ). W e re fer to Notation 4-1.2 for the (mo d α ) notation. W e sha ll call the V - congruences of the form either k x = y k V A or k R − → x k V A princip al V -c ongruenc es of A . Again, w e sha ll usually omit the supers cript V in ca se V is closed under homomorphic ima ges. Lemma 4 - 3.9. L et V b e a gener alize d quasivariety and let ϕ : A → B b e a morphism in V . Then the image of Con V B under Res ϕ is c ontaine d in Con V A . Pr o of . Let β ∈ Con V B and deno te by π : B ։ B / β the canonical pro jec- tion. Set α := Ker( π ◦ ϕ ). By applying Deﬁnition 4-3.4(i) to the comp osite map π ◦ ϕ , w e obtain that A / α belong s to V , that is, α ∈ Con V A . How ever, it follows from Lemma 4-3.3 that α = (Res ϕ )( β ). ⊓ ⊔ Let V be a generalized qua s iv ariet y. It follows from Lemmas 4-3.7 and 4-3 .9 that the assig nment ( A 7→ Co n V A , ϕ 7→ Res ϕ ) deﬁnes a contrav aria nt functor fro m V to the category of all alg ebraic lattices with maps that preserve arbitrar y meets and nonempty directed joins. By the g eneral theory of Galois connections [16, Section 0.3], for e very morphism ϕ : A → B in V , ther e exists a unique map Con V ϕ : Con V A → Co n V B such that (Con V ϕ )( α ) ≤ β iff α ≤ (Res ϕ )( β ) , for all ( α , β ) ∈ (Con V A ) × (Con V B ) . As the map Res ϕ preser ves arbitrary meets and nonempty directed joins , the map Con V ϕ preserves ar bitrary joins and it sends co mpact elemen ts to compac t elements. Hence the assignment ( A 7→ Con V A , ϕ 7→ C o n V ϕ ) 100 4 Larders from ﬁrst-order structures deﬁnes a (cov ariant) functor fro m V to the category of all algebraic lattices with compactness-pr eserving complete join-homo morphisms. Lemma 4 - 3.10. L et θ b e a c ongruenc e of a ﬁrst-or der structure A and de- note by π : A ։ A / θ the c anonic al pr oje ction. Then (Con V π )( α ) = α ∨ θ / θ ( wher e t he join α ∨ θ is evaluate d in Con V A ) , for e ach α ∈ Con V A . Pr o of . Set β ′ := (Res π )( β / θ ), for ea ch β ∈ (Con V A ) ↑ θ . It s uﬃces to pro ve that β = β ′ . F or x, y ∈ A , x ≡ y (mo d β ′ ) ⇔ π ( x ) ≡ π ( y ) (mod β / θ ) ⇔ x ≡ y (mo d β ) , while for each R ∈ Rel( A ) and each − → x ∈ A ar( R ) , R − → x (mod β ′ ) ⇔ R π ( − → x ) (mo d β / θ ) ⇔ π ( − → x ) ∈ R β / θ ⇔ R − → x (mod β ) . The conclusion follows immediately . ⊓ ⊔ By the Second Isomorphis m Theo rem (Lemma 4 -1.6), it follows that Con V ( A / θ ) is isomo r phic to the interv al (Con V A ) ↑ θ . W e will o ften ﬁnd it more convenien t to work with c omp act congr uences. F or a ny A ∈ V , a s Con V A is an alge br aic lattice, it is canonica lly isomorphic to the ideal lattice of the ( ∨ , 0)-semilattice Con V c A of a ll compa ct elements of Co n V A . F or a mor phism ϕ : A → B in V , we sha ll denote by Con V c ϕ the restriction of Con V ϕ from Con V c A to Con V c B . Ther efore, the assignment ( A 7→ Con V c A , ϕ 7→ Con V c ϕ ) deﬁnes a functor from V to the categor y Sem ∨ , 0 of all ( ∨ , 0)-semilattices with ( ∨ , 0 )-homomorphisms . W e shall also write Con c instead o f Con V c in cas e V is closed under homo morphic imag e s. In the sta tement o f the following lemma we use No tation 4-3.8. Lemma 4 - 3.11. The fol lowing statements hold, for any morphism ϕ : A → B in a gener alize d quasivarie ty V : (i) The V -c ongruenc es of A ar e exactly the joins ( in Con V A ) of princip al V -c ongruenc es of A . (ii) The c omp act V -c ongruen c es of A ar e exactly the ﬁnite joins ( in Con V A ) of princip al V -c ongruenc es of A . (iii) The e qu ation (Con V c ϕ )  k x = y k V A  = k ϕ ( x ) = ϕ ( y ) k V B is satisﬁe d, for al l x, y ∈ A . (iv) The e quation (Con V c ϕ )  k R ( x 1 , . . . , x n ) k V A  = k R ( ϕ ( x 1 ) , . . . , ϕ ( x n )) k V B is s atisﬁe d, for any R ∈ Rel( A ) , say of arity n , and al l x 1 , . . . , x n ∈ A . Pr o of . (i). It is obvious that the following equation is satisﬁed (in Co n V A ), for any α ∈ Con V A : 4-4 Preserv ation of directed colimits for con gruence lattices 101 α = _  k x = y k V A | ( x, y ) ∈ α  ∨ _  k R − → x k V A | R ∈ Rel( A ) , − → x ∈ R α  . (ii). Let D b e a nonempty directed subset of Con V A . The join θ o f D in Con A is deﬁned by the for mu las θ = [ ( α | α ∈ D ) , (4-3.1) R θ = [ ( R α | α ∈ D ) . (4-3.2) As Con V A is clos ed under directed jo ins (cf. Le mma 4 -3.7), θ b elongs to Con V A . Now let x, y ∈ A such tha t k x = y k V A ≤ θ . T his means that ( x, y ) ∈ θ , th us, by (4- 3.1), ( x, y ) ∈ α for some α ∈ D , so k x = y k V A ≤ α . Likewise, us ing (4-3.2), we ca n prov e that for eac h R ∈ Rel( A ), sa y of arity n , and each x 1 , . . . , x n ∈ A , k R − → x k V A ≤ θ implies that k R − → x k V A ≤ α for some α ∈ D . Therefore , al l princip al V -c ongruenc es of A ar e c omp act in Con V A , and thus s o are all their ﬁnite joins. The con verse follows immediately from (i). (iii). Se t α := k x = y k V A . By the deﬁnition of the Con V c functor, the fol- lowing equiv alences hold for ea ch β ∈ Con V B : (Con V c ϕ )( α ) ≤ β ⇐ ⇒ α ≤ (Res ϕ )( β ) ⇐ ⇒ k x = y k V A ≤ (Res ϕ )( β ) ⇐ ⇒ x ≡ y (mo d (Res ϕ )( β )) ⇐ ⇒ ϕ ( x ) ≡ ϕ ( y ) (mo d β ) ⇐ ⇒ k ϕ ( x ) = ϕ ( y ) k V B ≤ β ; the desired conclusion follows. The pro of of (iv) is simila r to the pr o of o f (iii). ⊓ ⊔ 4-4 Preserv ation of small directed colimits b y the relativ e compact congruence functor The prese nt section will b e devoted to the proo f o f the following result, which do es not seem to have app eared anywhere in prin t yet, even for the spe cial case of a lgebraic systems as in Gorbunov [26]. Theorem 4-4. 1. The functor Con V c : V → Sem ∨ , 0 pr eserves al l smal l di- r e cte d c olimits, for any gener alize d quasivarie ty V . Pr o of . W e are g iven a directed co limit co cone of the form (4-2 .2) in V . As V is closed under directed colimits in M Ind , (4-2.2) is also a dire c ted colimit in MInd , so we can take a dv an tage of the ex plicit description o f the colimit given in Prop o s ition 4-2.1. 102 4 Larders from ﬁrst-order structures It follows from L e mma 4-3 .1 1 (ii) that every element α ∈ Con V c A is a ﬁnite join of principa l V -congr uences of A (cf. Nota tio n 4-3.8). As each suc h principal V -congr uence inv olves only a ﬁnite num b er of element s from A , it follows that α in volv es only a ﬁnite n umber of parameters fr om A . As A is the directed union of all the subse ts ϕ i “( A i ), there ex is ts i ∈ I such that ϕ i “( A i ) contains all those par ameters. No w an immediate use of Lemma 4-3 .11(iii,iv) shows tha t α b elongs to the r ange of Con V c ϕ i . W e hav e prov ed that Con V c A is the union of all the rang es of the maps Con V c ϕ i . By vir tue of the c hara c terization of directed colimits pre sented in Sec- tion 1-2.5 (applied to the category o f a ll ( ∨ , 0)-semilattices with ( ∨ , 0)-homo- morphisms), it remains to prov e tha t for each i ∈ I and all α , β ∈ Con V c A i , (Con V c ϕ i )( α ) ≤ (Con V c ϕ i )( β ) (4-4.1) implies that there exists j ∈ I ↑ i such that (Con V c ϕ j i )( α ) ≤ (Con V c ϕ j i )( β ) . As α is a ﬁnite join o f principal V -congruences (cf. Lemma 4 -3.11), w e may assume in turn that α is principa l. W e set θ j := (Co n V c ϕ j i )( β ) and we denote b y π j : A j ։ A j / θ j the canonical pro jection, for each j ∈ I ↑ i . F rom θ k = (Con V c ϕ k j )( θ j ) we get the inequality θ j ≤ (Res ϕ k j )( θ k ), hence, by using Lemma 4-3.3, θ j ≤ Ker( π k ◦ ϕ k j ). Therefor e , b y Lemma 4-1.5, there exists a unique morphism ψ k j : A j / θ j → A k / θ k such that π k ◦ ϕ k j = ψ k j ◦ π j . Clearly , the family  A j / θ j , ψ k j | j ≤ k in I ↑ i  is an ( I ↑ i )-indexed diagr am in V . W e form the directed colimit ( B , ψ j | j ∈ I ↑ i ) = lim − →  A j / θ j , ψ k j | j ≤ k in I ↑ i  in the category MInd . As V is closed under directed colimits in MInd , the co cone ( B , ψ j | j ∈ I ↑ i ) is contained in V . Denote b y π : A → B the unique morphism in MInd such that π ◦ ϕ j = ψ j ◦ π j for each j ∈ I ↑ i . The situation is illustrated on Figure 4.1. A i ϕ j i / / π i     A j ϕ j / / π j     A π   A i / θ i ψ j i / / A j / θ j ψ j / / B Fig. 4. 1 A commutat ive diagram in a generalized quasiv ari ety V Now w e set θ := Ker π , a nd we compute 4-5 Ideal-induced morphisms and pro jectability witnesses 103 β = Ker π i ⊆ Ker( ψ i ◦ π i ) (use the ea s y dir ection o f L e mma 4- 1.5) = Ker( π ◦ ϕ i ) (cf. Figure 4.1) = (Res ϕ i )(Ker π ) (use Lemma 4-3 .3) = (Res ϕ i )( θ ) , th us (Con V c ϕ i )( β ) ⊆ θ , and thus, by (4-4.1), (Con V c ϕ i )( α ) ⊆ θ , and hence, by using part of the calculation ab ov e, α ⊆ (Res ϕ i )( θ ) = Ker( ψ i ◦ π i ). As α is principal, it has one of the forms k x = y k V A i (with x, y ∈ A i ) or k R − → x k V A i (with R ∈ Rel( A i ) and x 1 , . . . , x ar( R ) ∈ A i ). In the ﬁrst case, α ⊆ Ker( ψ i ◦ π i ) means that ψ i ◦ π i ( x ) = ψ i ◦ π i ( y ). Thu s, by the des c r iption of the directed colimit given in Prop ositio n 4-2.1, there exists j ∈ I ↑ i such that ψ j i ◦ π i ( x ) = ψ j i ◦ π i ( y ), that is, ϕ j i ( x ) ≡ ϕ j i ( y ) (mo d θ j ), which means, by the deﬁnition of θ j together with Lemma 4-3.11(iii), that (Con V c ϕ j i )( α ) ⊆ (Con V c ϕ j i )( β ). The pro of in the second ca se, that is, α = k R − → x k V A i , is similar. ⊓ ⊔ 4-5 Ideal-induced morphism s and pro jectabilit y witnesses in generalized quasiv ariet ies In view of further mo noid-theoretica l applications, we s hall need to expand our sc op e s lightly beyond the one of ( ∨ , 0)-semilattices, and w e shall th us int ro duce the notion o f an ide al-induc e d homomorphism in the con text of c ommutative m onoids . Then, in The o rem 4-5.2, we shall relate ideal-induced homomorphisms o f ( ∨ , 0 )-semilattices a nd pro jecta bilit y witnesses with re- sp ect to the relative congruence semilattice functor. W e endow ev er y commu tative monoid M with its algebr aic preor der ing, deﬁned by x ≤ y ⇐ ⇒ ( ∃ z ∈ M )( x + z = y ) , for all x, y ∈ M . F or commutativ e monoids M a nd N with N c onic al , we shall say that a monoid ho momorphism ϕ : M → N is — ide al-induc e d if ϕ is surjective and ( ∀ x, y ∈ M )  ϕ ( x ) = ϕ ( y ) ⇒ ( ∃ u, v ∈ ϕ − 1 { 0 } )( x + u = y + v )  ; (4-5.1 ) — we akly distributive if ( ∀ z ∈ M )( ∀ u, v ∈ N )  ϕ ( z ) ≤ u + v ⇒ ( ∃ x, y ∈ M )( z ≤ x + y and ϕ ( x ) ≤ u and ϕ ( y ) ≤ v )  . 104 4 Larders from ﬁrst-order structures An o-ide al in a commutativ e monoid M is a no nempt y subset I of M suc h that x + y ∈ I iff { x, y } ⊆ I , for all x, y ∈ M . ( In p articular, in a ( ∨ , 0) -semi- lattic e, the o-ide als ar e exactly the ide als in the usual sense .) Then w e denote by M /I the q uotient of M under the monoid congr uence ≡ I deﬁned by x ≡ I y ⇐ ⇒ ( ∃ u, v ∈ I )( x + u = y + v ) , for all x, y ∈ M . F urthermore, we shall wr ite x/I instead of x/ ≡ I , for any x ∈ M . Obviously , the quotien t monoid M /I is conical. The ideal-induced homomorphisms fro m a co mmu tative mono id M to a co nical c ommut ative mono id ar e exactly , up to iso morphism, the ca nonical pro jections M ։ M /I , for o-ideals I of M . The following result pr ovides a la rge supply of ideal-induced ( ∨ , 0 )-homomor- phisms. Lemma 4 - 5.1. L et V b e a gener alize d quasivariety, let A , B ∈ V with the same language, and let f : A ։ B b e a surje ctive homomorph ism. Then the c anonic al homomorphism Con V c π : Con V c A → Co n V c B is ide al-induc e d. Pr o of . By the First Iso morphism Theor em (Lemma 4- 1.5), we may a s sume that B = A / θ , with f : A ։ A / θ the cano nical pro jection, for some θ ∈ Con V A . The surjectivity of Con V c f follows immediately from the sur- jectivit y of f together with Lemma 4- 3.11. Let α , β ∈ Co n V c A such that (Con V c f )( α ) = (Con V c f )( β ). By Lemma 4-3.10, this mea ns that α ∨ θ = β ∨ θ , th us, by the compactness of bo th α a nd β , there exists a co mpact V - congruence ξ ≤ θ suc h that α ∨ ξ = β ∨ ξ . As (Con V c f )( ξ ) = 0, this proves that the map Con V c f satisﬁes (4 - 5.1). ⊓ ⊔ Theorem 4-5. 2. L et S b e a ( ∨ , 0) -semilattic e, let V b e a gener alize d quasi- variety, let A ∈ V , and let S b e a ( ∨ , 0) -semilattic e. Then every ide al-induc e d ( ∨ , 0) - homomorphism ϕ : Con V c A → S has a pr oje ctability witness with r e- sp e ct to the C o n V c functor. Pr o of . W e set θ := W  α ∈ Con V c A | ϕ ( α ) = 0  . As the join deﬁning θ is nonempty directed, it follows from Lemma 4-3.7 that it may be ev aluated as well in Co n A as in Con V A , so θ ∈ Co n V A . F urthermore, ϕ ( α ) = 0 ⇐ ⇒ α ≤ θ , for each α ∈ Con V c A . (4-5.2) As θ ∈ Con V A , the structure A := A / θ belo ngs to V . The canonical pr o jec- tion a : A ։ A / θ is obviously an epimorphism. It follows from Lemma 4-1 .6 that Con V A = { α / θ | α ∈ (Con V A ) ↑ θ } , and thus Con V c A = { α ∨ θ / θ | α ∈ Con V c A } , where the joins α ∨ θ a re ev aluated in Con V c A . F urthermore, for all α , β ∈ Con V c A , ( α ∨ θ ) / θ ≤ ( β ∨ θ ) / θ iff α ≤ β ∨ θ , iff (by the compactness 4-5 Ideal-induced morphisms and pro jectability witnesses 105 of α ) there exists γ ≤ θ in Con V c A such that α ≤ β ∨ γ ; this implies, by (4-5.2), that ϕ ( α ) ≤ ϕ ( β ). Co nversely , if ϕ ( α ) ≤ ϕ ( β ), then, as ϕ is ideal-induced, there exists γ ∈ Con V c A suc h that α ≤ β ∨ γ and ϕ ( γ ) = 0 ; then from ϕ ( γ ) = 0 it follows that γ ≤ θ , thus ( α ∨ θ ) / θ ≤ ( β ∨ θ ) / θ . As ϕ is surjective, this makes it p oss ible to deﬁne a semilattice isomorphism ε : Con V c A → S by the rule ε  α ∨ θ / θ  := ϕ ( α ) , for each α ∈ Con V c A . It follo ws immediately from Lemma 4-3.10 that ϕ = ε ◦ (Con V c a ). In order to prov e that ( a, ε ) is a pr o jectability witness for ϕ : Con V c A → S , it r emains to prov e item (iv) of Deﬁnition 1-5.1. Let f : A → X b e a morphism in V and let η : Con V c A → Co n V c X such that Con V c f = η ◦ (Co n V c a ) . (4 -5.3) F or all x, y ∈ A , x ≡ y (mo d θ ) ⇔ a ( x ) = a ( y ) ⇔ k a ( x ) = a ( y ) k V A = 0 A (beca use A ∈ V ) ⇔ (Con V c a )  k x = y k V A  = 0 A (use Lemma 4-3.11(iii)) ⇒ (Con V c f )  k x = y k V A  = 0 X (use (4-5.3)) ⇔ k f ( x ) = f ( y ) k V X = 0 X (use Lemma 4-3.11(iii)) ⇔ f ( x ) = f ( y ) (beca use X ∈ V ) , while for each R ∈ Rel( A ) and each − → x ∈ A ar( R ) , R − → x (mod θ ) ⇔ a ( − → x ) ∈ R A ⇔ k Ra ( − → x ) k V A = 0 A (beca use A ∈ V ) ⇔ (Co n V c a )  k R − → x k V A  = 0 A (use Lemma 4-3.11(iv)) ⇒ (Co n V c f )  k R − → x k V A  = 0 X (use (4-5.3)) ⇔ k Rf ( − → x ) k V X = 0 X (use Lemma 4-3.11(iv)) ⇔ f ( − → x ) ∈ R X (beca use X ∈ V ) . This prov es that θ ≤ K er f . Therefore, by Lemma 4-1.5, there exists a unique morphism g : A → X in MInd such that f = g ◦ a . B y using (4-5.3), it follows that η ◦ (Con V c a ) = Con V c f = (Con V c g ) ◦ (Con V c a ), thus, as Con V c a is surjective, η = Co n V c g . ⊓ ⊔ 106 4 Larders from ﬁrst-order structures 4-6 An extension of t he L¨ owenh eim-Sk olem Theorem The main result of the presen t section, namely Pr op osition 4-6.2, will b e used for verifying the L¨ owenheim-Sk olem Condition in the pro of of Theorem 4-7.2. Roughly spea king, it says that in many situations, if ϕ : A ⇒ B is a double arrow, th en there ar e enough small substructures U of A suc h that ϕ ↾ U : U ⇒ B . W e shall use the following easy mo del-theoretica l lemma. Lemma 4 - 6.1. L et λ b e an inﬁnite c ar dinal, let L b e a λ -smal l ﬁrst -or der language, and let I b e a λ -dir e cte d monotone σ - c omplete p oset. Consider a σ -c ontinuous dir e cte d c olimit c o c one ( A , ϕ i | i ∈ I ) = lim − →  A i , ϕ j i | i ≤ j in I  , (4-6.1) of mo dels for L and L -homomorph isms, with car d A i < λ for e ach i ∈ I . Then t he set J := { i ∈ I | ϕ i is an elementary emb e dding } is a σ -close d c oﬁnal s ubset of I . In p art icular, J is nonempty. Pr o of . W e set I := { i ∈ I | ϕ i is a n embedding } . Claim. The set I is σ -closed coﬁnal in I . Pr o of . It is o b vio us that I is σ -closed in I . F or elements i , j ∈ I , let i ⋖ j hold if i ≤ j and for each a to mic formula F ( x 1 , . . . , x n ) of L and all e lement s x 1 , . . . , x n ∈ A i , A | = F ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) implies that A j | = F ( ϕ j i ( x 1 ) , . . . , ϕ j i ( x n )). It follows from (4-6.1) that for each i ∈ I , ea ch atomic formula F ( x 1 , . . . , x n ) of L , and a ll elements x 1 , . . . , x n ∈ A i , there exists j ∈ I ↑ i such that A | = F ( ϕ i ( x 1 ) , . . . , ϕ i ( x n )) implies that A j | = F ( ϕ j i ( x 1 ) , . . . , ϕ j i ( x n )). As both L and A i are λ -small, we obtain, using the λ -directedness assumption on I a nd repeating the ar g ument ab ov e for all atomic form ulas of L and all lists of elements of A i , that for each i ∈ I there exists j ∈ I suc h that i ⋖ j . Hence, for each i ∈ I there exists a sequence ( i m | m < ω ) of elements o f I such that i 0 = i and i m ⋖ i m +1 for each m < ω . Set j := W ( i m | m < ω ). Let F ( x 1 , . . . , x n ) b e an atomic formula of L and let x 1 , . . . , x n ∈ A j such that A | = F ( ϕ j ( x 1 ) , . . . , ϕ j ( x n )). As the colimit (4- 6 .1) is σ - contin uous, A j = lim − → m<ω A i m , thus there a re m < ω and y 1 , . . . , y n ∈ A i m such that x s = ϕ j i m ( y s ) for each s ∈ { 1 , . . . , n } ; he nce A | = F ( ϕ i m ( y 1 ) , . . . , ϕ i m ( y n )). As i m ⋖ i m +1 , we obtain tha t A i m +1 | = F ( ϕ i m +1 i m ( y 1 ) , . . . , ϕ i m +1 i m ( y n )) , 4-6 An exte nsion of the L¨ ow enheim-Skolem Theorem 107 and hence, a pply ing the homomorphism ϕ j i m +1 , we o btain A j | = F ( x 1 , . . . , x n ). This completes the pro o f that j belo ngs to I . Therefo r e, I is co ﬁnal in I . ⊓ ⊔ Claim As all ϕ i , for i ∈ I , are embeddings , w e may assume that they are inclusio n maps a nd th us that A = S  A i | i ∈ I  . Now the statement that J is a σ - closed subset of I follows immediately fro m the Element ar y Chain Theo rem [8, Theo rem 3.1.9]. F urther, let L ∗ be a Sk olem ex pansion o f L and let A ∗ be a corresp onding Skolem expa nsion of the mo del A (cf. [8, Section 3.3]). F or ea ch i ∈ I , we con- struct a seq uence ( i m | m < ω ) of indices by setting i 0 := i a nd letting i m +1 be any index in I ↑ i m such that A i m +1 contains the Skolem h ull of A i m . The element j := W ( i m | m < ω ) belongs to I and A j = S ( A i m | m < ω ) is its own Sk olem h ull, th us it is an elementary submo del of A ; so j ∈ J . This prov es that J is coﬁnal in I , and thus also in I . ⊓ ⊔ W e now pres ent a few s imple monoid-theoretica l a pplications of Lemma 4- 6.1. Prop ositio n 4-6.2. L et λ b e an inﬁnite c ar dinal, let I b e a λ -dir e cte d mono- tone σ -c omplete p oset. L et M and N b e c ommutative monoids and let ϕ : M → N b e a monoid homomorphism. Conside r a σ - c ontinu ous dir e cte d c olimit c o c one ( M , τ i | i ∈ I ) = lim − →  M i , τ j i | i ≤ j in I  , (4-6.2) with card M i < λ for e ach i ∈ I and car d N < λ . Then the fol lowing state- ments hold : (i) If ϕ is ide al-induc e d, then I id := { i ∈ I | ϕ ◦ τ i is ide al-induc e d } is σ -close d c oﬁnal in I ; (ii) If ϕ is we akl y di st ribut ive, then I wd := { i ∈ I | ϕ ◦ τ i is we akly distributive } is σ -close d c oﬁnal in I . Note. It is obvious that if ϕ is surjective, then the set I surj := { i ∈ I | ϕ ◦ τ i is surjective } is σ -clo sed coﬁna l in I . Indeed, as I is λ -directed and N is λ -small, there exists i ∈ I suc h that ϕ ◦ τ i is surjectiv e, and then ϕ ◦ τ j is surjectiv e for each j ≥ i . Pr o of . As (4-6 .2) is a σ -co nt inuous directed colimit co cone, it is straig ht for- ward to verify that the sets I id and I wd are both σ -clo sed in I . Now, in order to apply Lemma 4- 6 .1, we enco de every monoid homomorphism ψ : X → Y (for comm utative monoids X and Y ) b y a single ﬁr st-order structure A ψ , in such a wa y that ψ b eing ideal-induced (resp., w eakly dis tributive) c a n be expressed by a ﬁrs t-order sentence. In order to do this, we deﬁne a new language L by 108 4 Larders from ﬁrst-order structures L := { + , 0 X , 0 Y , X , Y , R } , where + is a binary op er ation symbol, X and Y are b oth (unary) predica te symbols, R is a binary relatio n sy m b ol, and 0 X and 0 Y are bo th constant symbols. Deﬁne the universe of A ψ as ( X × { 0 } ) ∪ ( Y ×{ 1 } ), int er pret X by X × { 0 } , Y by Y × { 1 } , and R b y the set of all pairs of the fo r m (( x, 0) , ( ψ ( x ) , 1)) for x ∈ X ; furthermore, deﬁne the a ddition on A ψ by setting ( x 0 , 0 ) + ( x 1 , 0 ) := ( x 0 + x 1 , 0 ) , for all x 0 , x 1 ∈ X , ( y 0 , 1 ) + ( y 1 , 1 ) := ( y 0 + y 1 , 1 ) , for all y 0 , y 1 ∈ Y , ( x, 0) + ( y , 1) = ( y , 1) + ( x, 0) := (0 X , 0 ) , for all ( x, y ) ∈ X × Y . Finally , we in terpr et 0 X by (0 X , 0 ) and 0 Y by (0 Y , 1 ). Then ψ b eing sur jective is equiv alent to A ψ satisfying the ﬁrst-or der statement ( ∀ y )  Y ( y ) ⇒ ( ∃ x )( X ( x ) and R ( x , y ))  , while ψ b eing ideal-induced is equiv alent to ψ b eing surjective together with the ﬁrst-or der statement ( ∀ x 0 , x 1 , y )   X ( x 0 ) and X ( x 1 ) and Y ( y ) a nd R ( x 0 , y ) and R ( x 1 , y )  ⇒ ( ∃ u 0 , u 1 )  X ( u 0 ) and X ( u 1 ) and R ( u 0 , 0 Y ) and R ( u 1 , 0 Y ) and x 0 + u 0 = x 1 + u 1   . Likewise, it is straightforw ard to verify that ψ b eing w eakly distributive is also a ﬁrst-or der prop erty of A ψ . Now assume, fo r example, that ϕ is ideal-induced. As A ϕ = lim − → i ∈ I A ϕ ◦ τ i (with canonical trans ition morphisms and limiting morphisms), it follows from Lemma 4-6.1 that the set J of all i ∈ I suc h that τ i deﬁnes an element ar y embedding from A ϕ ◦ τ i to A ϕ is coﬁnal in I . By the discussion above, this set contains I id ; hence the latter is also co ﬁnal in I . The a rgument for I wd is similar. ⊓ ⊔ Of cour s e, it w ould hav e been ab out as ea sy to prov e Prop os ition 4-6.2 directly , in each of the cases (i) a nd (ii). How ever, we ho p e that the ab ov e approach clearly illustrates the fact that man y such statement s can be ob- tained immediately from Lemma 4-6 .1. 4-7 A diagram v ersion of the Gr¨ atz er -Schmid t Theorem In the present section we shall esta blish a diagra m extension o f the Gr¨ atzer- Schm idt Theo r em, na mely Theo rem 4 -7.2. W e remind the r eader that Sem ∨ , 0 denotes the category of all ( ∨ , 0 )-semi- lattices with ( ∨ , 0)-homomorphisms. W e also denote by Sem surj ∨ , 0 ( Sem idl ∨ , 0 , 4-7 A diagram version of the Gr¨ atzer-Sc hmidt Theorem 109 Sem wd ∨ , 0 , respec tively) the sub catego ry o f Sem ∨ , 0 whose ob jects ar e all ( ∨ , 0)- semilattices and whose mor phisms are all surjective (idea l-induced, weakly distributive, resp ectively) ( ∨ , 0)-homo morphisms. The pro o f of the following lemma is a straightforward exercise. (W e r e fer to Deﬁnition 1-2.5 for sub c at- egories closed under small directed colimits.) Lemma 4 - 7.1. The sub c ate gories Sem surj ∨ , 0 , Sem idl ∨ , 0 , and Sem wd ∨ , 0 ar e close d under al l smal l dir e cte d c olimits within Sem ∨ , 0 ( cf. Deﬁnition 1- 2.5) . An algebr a is a ﬁrst-o rder structure A s uch that Rel( A ) = ∅ . The Gr ¨ atzer- Schm idt Theo rem [31, Theorem 10] states that every ( ∨ , 0)-semilattice is isomorphic to Co n c A , for some a lgebra A . As the algebra A has the same congruences as the algebr a consisting o f the universe of A endowed with the unary p olyno mials of A , we may ass ume that A is a unary a lgebra. Denote by MAlg 1 the full sub category of MInd consisting of all unary a lgebras. Hence MAlg 1 is a genera liz ed quas iv ariet y (cf. Example 4-3.6(ii)). W e sha ll call it the categ o ry of unary monotone-indexe d algebr as . The following result is a diagra m extension of the Gr¨ atzer -Schmidt Theorem. Theorem 4-7. 2. L et P b e a p oset and let S =  S p , σ q p | p ≤ q in P  b e a P - indexe d diagr am of ( ∨ , 0) -semilattic es and ( ∨ , 0) -homomorphisms. If either P is ﬁnite or ther e ar e c ar dinals κ and λ , with λ r e gular, such that P and al l S p , for p ∈ P , ar e λ -smal l and the r elation ( κ, <ω , λ ) → λ holds, then every P - indexe d diagr am of ( ∨ , 0) -semilattic es and ( ∨ , 0) -homomorphisms c an b e lifte d, with r esp e ct to the Con c functor, by some diagr am of unary monotone- indexe d algebr as. Pr o of . Denote by P the ( ∨ , 0)-semilattice of all ﬁnitely genera ted low er sub- sets of P . As the ca teg ory Sem ∨ , 0 has all (not necessar ily directed) colimits, we can extend the diagra m − → S to a P - indexed dia gram − → S ∗ in Sem ∨ , 0 by setting S ∗ X := lim − → ( S p | p ∈ X ) , for e a ch X ∈ P , with the canonical tra nsition maps a nd limiting maps. F urthermor e, P is ﬁnite in case P is ﬁnite, and card P ≤ card P + ℵ 0 . In particular, the assump- tions ma de for P , − → S remain v alid for P , − → S ∗ , and so w e ma y assume from the start that P is a ( ∨ , 0)-semilattice. Set λ := ℵ 1 in ca se P is ﬁnite (in that case we require no relation of the form ( κ, <ω , λ ) → λ ). W e denote by MAlg ( λ ) 1 (resp., Se m ( λ ) ∨ , 0 ) the class of all completely λ -small unar y a lgebras (resp., the class of all λ -small ( ∨ , 0)-semilattices). Claim 1. Denote by Φ the identit y functor on Sem ∨ , 0 . Then the q uadruple ( Sem ∨ , 0 , Se m ∨ , 0 , Se m idl ∨ , 0 , Φ ) is a left larder. 110 4 Larders from ﬁrst-order structures Pr o of . The only not completely trivia l statement that w e need to verify is (PROJ( Φ, Sem idl ∨ , 0 )), that is, every extended pro jection of Sem ∨ , 0 is ideal- induced. As every pro jection in Sem ∨ , 0 is, trivia lly , ideal- induced, this follo ws immediately from Lemma 4-7.1. ⊓ ⊔ Claim 1 Claim 2. The 6-uple ( MAlg 1 , M Alg ( λ ) 1 , Se m ∨ , 0 , Se m ( λ ) ∨ , 0 , Se m idl ∨ , 0 , Co n c ) is a pro jectable right λ -larder. Pr o of . F or each A ∈ MAlg ( λ ) 1 , the semilattice Con c A is λ -small, th us, by Prop ositio n 4-2.3, it is w eakly λ -presented in Sem ∨ , 0 . Hence the condition (PRES λ ( MAlg ( λ ) 1 , Co n c )) is satisﬁed. Let B b e a unary algebra, we m ust verify that (LS r λ ( B )) is satisﬁed. Le t S be a λ -small ( ∨ , 0 )- s emilattice, let I b e a λ -sma ll s et, and let ( u i : U i ⇒ B | i ∈ I ) be a family of double a rrows with a ll the U i completely λ -small. As a t the end o f the pro o f of Prop osition 4- 2.3, we set Ω := B ∪ Lg( B ) (disjoin t union) and we express B a s a contin uous directed colimit B = lim − → X ∈ [ Ω ] <λ B X in MAlg 1 , with all maps in the co cone being the corresp o nding inclusion maps. Then applying Theorem 4-4 .1 to the gener alized q uasiv ariety V := MAlg 1 yields a co ntin uous dire cted colimit Con c B = lim − → X ∈ [ Ω ] <λ Con c B X in Sem ∨ , 0 . (4-7.1) (Due to an earlier introduced conv ention, we omit the V sup erscript in the Co n V c notation, a s, here, V is closed under homomor phic images.) De- note by β X : B X ֒ → B the inclusion map, for ea ch X ∈ [ Ω ] <λ . It follows from (4 -7.1) together with Pr op osition 4- 6.2 that the set J := { X ∈ [ Ω ] <λ | ϕ ◦ Con c β X is idea l-induced } is σ - closed coﬁnal in [ Ω ] <λ . As there exists X ∈ [ Ω ] <λ such that B X contains (as well for the universes as for the langua ges) all ima g es u i “( U i ), it follows that such an X can b e chosen in J . Then β X is monic, u i E β X for each i ∈ I , and ϕ ◦ Con c β X is idea l-induced. The pro jectability statemen t follows immediately from Theor em 4-5.2. ⊓ ⊔ Claim 2 By the tw o claims ab ove, it follows from Pro po sition 3-8.3 (with A † := Sem ( λ ) ∨ , 0 ) that the 8-uple ( Sem ∨ , 0 , M Alg 1 , Se m ∨ , 0 , Se m ( λ ) ∨ , 0 , M Alg ( λ ) 1 , Se m idl ∨ , 0 , Φ, Con c ) is a λ -larder. 4-7 A diagram version of the Gr¨ atzer-Sc hmidt Theorem 111 Now we ﬁrst assume that P is ﬁnite. By Corollary 3-5.10, there are n < ω and a λ -lifter ( X , X ) of P suc h that card X = λ + n . By the Gr¨ atzer-Schmidt Theorem, there exists B ∈ MAl g 1 such that Con c B ∼ = F ( X ) ⊗ − → S . By CLL (Lemma 3- 4.2), ther e exists a P -indexed diagram − → B in MAlg 1 such that Con c − → B ∼ = − → S ; that is, − → S can b e lifted with res pe c t to the Con c functor. Finally a ssume that P is inﬁnite. W e do not have any theorem tha t en- sures the exis tence o f a λ -lifter of P , so ins tea d of using CLL w e inv oke Corollar y 3-7 .2, which yields again a morphis m − → χ : Con c − → B . ⇒ − → S for some diagram − → B in MAlg 1 . W e conclude the pro of as ab ov e, b y using pro jectabil- it y . ⊓ ⊔ By using the comments preceding Co rollary 3-7.2, we obtain, in pa rticu- lar, the following extensio n of the Gr¨ atzer- Schm idt Theo r em to diagrams of algebras . Corollary 4 - 7.3. Assume that the class of al l Er d˝ os c ar dinal s is pr op er. Then every p oset-indexe d diag r am of ( ∨ , 0) -s emilattic es and ( ∨ , 0) - homomor- phisms c an b e lifte d, with r esp e ct to the Co n c functor, by some diagr am of monotone-indexe d un ary algebr as. A similar proo f, with MAlg 1 replaced by the generalized quasiv ariety of all group oids, yields the following result, that extends Lamp e’s result from [4 2] (and its extension to one ar row in Lampe [43]) that every ( ∨ , 0 , 1 )-semilattice is isomor phic to the compac t congruence semila ttice of some gro upo id. This result was ﬁrst established by the ﬁrst author for dia grams indexed by ﬁnite po sets, s ee Gillib ert [18, Cor ollary 7.1 0]. Prop ositio n 4-7.4. Every diagr am of ( ∨ , 0 , 1) -semilattic es and ( ∨ , 0 , 1 ) -ho- momorphisms, indexe d by a ﬁnite p oset, c an b e lif te d, with r esp e ct to the Con c functor, by some diagr am of gr oup oids. F urt hermor e, if the class of al l Er d˝ os c ar dina ls is pr op er, then the ﬁniteness assu m ption on t he indexing p oset is not ne e de d. Remark 4- 7 .5. By replacing MAlg 1 and the category of all gr oup oids by their full sub ca tegories of members with 4 -p ermutable congr uence lattices (i.e., α ∨ β = αβ αβ for all cong ruences α and β of the str ucture; the def- inition of m -p ermutable c ongruenc e lattic e , for an in teger m ≥ 2, is simi- lar), it is fur thermore p ossible to stre ngthen Theorem 4 -7.2, Coro llary 4-7.3, and Prop osition 4-7.4 b y requiring all the unary a lgebras (resp., group oids) to ha ve 4- per mut able congruence lattices. W e need the ea sily v eriﬁe d fact that every homomor phic image of a co ngruence 4-p ermutable alg e bra is con- gruence 4-p er m utable, tog ether with the statement that ev ery congr uence 4-p ermutable algebr a is a cont inuous direc ted colimit, in MInd , indexed by 112 4 Larders from ﬁrst-order structures some [ Ω ] <λ , of λ -small congruence 4-p ermutable a lg ebras. T he latter fact is established by proving an easy reﬁnement of the pro of of the ﬁnal s ta temen t of P rop osition 4 -2.3. This in turn is used to prov e the L ¨ owenheim-Sk olem Condition in an a nalogue of Theorem 4-7 .2 for co ngruence 4 -p ermutable al- gebras. W e believe that the re a der who be a red with us up to now will hav e no diﬃculty in supplying the missing details . 4-8 Right ℵ 0 -larders from congruence-prop er quasiv arieties One of the caveats in extending Theor em 4 - 7.2 to the cas e wher e λ = ℵ 0 is the impo ssibility to extend Prop ositio n 4 -6.2 to that case. Hence, in order to obtain right ℵ 0 -larders , we shall need to strengthen the assumptions ov er the generalized quasiv ariet y V . As the applications that w e are having in mind deal with s tructures in a ﬁxed language, we shall deal with quasivarietie s . This will lead to Theo r em 4-8.2. W e prov e in Prop os itions 4-8.6 and 4-8.7 that the ass umption of that theor em, namely that V be congruence-pr op er and lo ca lly ﬁnite, is veriﬁed in case V is a ﬁnitely g enerated qua siv ariety . Deﬁnition 4 - 8.1. W e say that a quasiv ariety V is • c ongruenc e-pr op er if Con V A ﬁnite implies that A is ﬁnite (i.e., it has ﬁnite universe), for each A ∈ V ; • str ongly c ongruenc e-pr op er if it is congruence- prop er and for each ﬁnite ( ∨ , 0)-s emilattice S there are only ﬁnitely many (up to isomorphism) A ∈ V s uch that Co n V A ∼ = S . W e denote b y V ﬁn the cla ss of all ﬁnite structures (i.e., structures with ﬁnite universe) in a quasiv ariety V . In particular, Sem ﬁn ∨ , 0 is the class o f all ﬁnite ( ∨ , 0)-semilattices. As usual, a ﬁrst-order structure A is lo c al ly ﬁn it e if the s ubuniverse of A generated b y a ny ﬁnite subset of A is ﬁnite, and a quasiv ariety V is lo ca lly ﬁnite if every member of A is lo cally ﬁnite. Theorem 4-8. 2. L et V b e a c ongruenc e-pr op er and lo c al ly ﬁn ite qu asivariety on a ﬁ r s t -or der language L with only ﬁ nitely many r elation symb ols. Then the 6 -uple ( V , V ﬁn , Se m ∨ , 0 , Se m ﬁn ∨ , 0 , Se m idl ∨ , 0 , Co n V c ) is a pr oje ctable right ℵ 0 -lar der. Pr o of . The statement (PRES ℵ 0 ( V ﬁn , Co n V c )) follows from the ﬁniteness o f the collec tio n of all r elation symbo ls of V , while the pro jectabilit y statement follows from Theorem 4-5.2. W e m ust v erify (LS r ℵ 0 ( B )), fo r B ∈ V . Let S b e a ﬁnite ( ∨ , 0)-semila t- tice, let ϕ : Con V c B → S b e ideal- induce d, let n < ω , let U 0 , . . . , U n − 1 4-8 Right ℵ 0 -larders from ﬁrst-order structures 113 be ﬁnite members o f V , and let ( u i : U i → B | i < n ) b e a ﬁnite sequence of L -mor phisms (for what follows we shall not re q uire the u i s b e monic). W e set θ := _  β ∈ Con V c B | ϕ ( β ) = 0  . As in the proo f of Theore m 4 -5.2, θ is a V -congruence of B and, setting C := B / θ , there exists a unique iso morphism ε : Con V c C → S such that ε ( β ∨ θ / θ ) = ϕ ( β ) for e ach β ∈ Co n V c B . In pa r ticular, as Con V c C is ﬁnite and V is congruence-pr op er, C is ﬁnite, thus so is B /θ . It follows that there exists a ﬁnite subset F of B suc h tha t B / θ = { x/θ | x ∈ F } . (4-8.1) As V is lo cally ﬁnite, the subuniverse V of B genera ted by the subset F ∪ S ( u i “( U i ) | i < n ) is ﬁnite. Denote by V the corresp onding mem b er of V and by v : V ֒ → B the inclusion map. In particular, v is monic a nd u i E v for ea ch i < n . Set η := (Res v )( θ ), a nd denote b y π : B ։ B / θ and ρ : V ։ V / η the canonical pro jections. It follows from Lemma 4-3.3 that Ker( π ◦ v ) = (Res v )(Ker π ) = (Res v )( θ ) = η , and th us, b y the Fir st Iso morphism Theo- rem (Lemma 4 - 1.5), there exists a unique em b edding v : V / η ֒ → B / θ such that π ◦ v = v ◦ ρ , a s illustr ated o n the left hand side o f Figure 4.2. V   v / / ρ     B π     Con V c V Con V c v / / Con V c ρ   Con V c B Con V c π   ϕ ( H H H H H H H H H H H H H H H H H H H H V / η   v / / B / θ Con V c ( V / η ) Con V c v / / Con V c ( B / θ ) ∼ = ε / / S Fig. 4. 2 Proving that ϕ ◦ Con V c v is ideal- induced Now we consider the c ommut ative diagra m represented o n the right hand side of Figure 4.2. It follows from Lemma 4-5 .1 that both morphisms Co n V c ρ and Con V c π are ideal-induced; we ma rk idea l-induced morphisms b y double arrows on Figur e 4.2. F urthermore, for ea ch b ∈ B , there exists, by (4- 8.1), x ∈ F suc h that x/θ = b /θ . As F is contained in V , the function v is deﬁned at x/η and v ( x/η ) = v ( x ) /θ = x/θ = b/θ , and hence v is s ur jective. As v is also an embedding, it is an iso morphism. Hence the map Con V c v is also an isomor phism. As ε is als o a n isomor phis m and Con V c ρ is ideal-induced, it follows that the ( ∨ , 0)-homomor phism ϕ ◦ Con V c v = ε ◦ (Con V c v ) ◦ (Co n V c ρ ) 114 4 Larders from ﬁrst-order structures is ideal-induced. ⊓ ⊔ W e sha ll now show a large class o f quasiv arie ties (cf. Example 4-3 .5) that satisfy the assumptions of Theor em 4-8.2. Deﬁnition 4 - 8.3. Let V be a quasiv ariety on a ﬁr st-order langua ge L . A mem b er A o f V is sub dir e ctly irr e ducible (with respec t to V ) if Con V A has a smallest nonzer o element. W e say that V is ﬁnitely gener ate d if it is the smallest qua siv ariety containing a given ﬁnite set of ﬁnite structures (not necessarily one ﬁnite str uctur e—for example, A may not b elong to the qua- siv ariety generated by A × B ). Lemma 4 - 8.4 (fol klore). A quasivariety V on a ﬁrst-or der language L is ﬁn itely gener ate d iff it c ont ains only ﬁ nitely many sub dir e ctly irr e ducible memb ers ( up t o isomorphi sm ) and al l these structu r es ar e ﬁn ite. Pr o of . Supp ose ﬁr s t that V is gener ated by a ﬁnite set { A i | i < n } of ﬁnite structures. It follows from [2 6, Corollary 3.1.6] that ev ery relatively s ubdi- rectly irreducible member of V embeds into one of the A i . Hence, rega rdless of the cardinality of L , the quasiv a riety V has only ﬁnitely man y subdirectly irreducible members up to iso mo rphism, and these structur es a re a ll ﬁnite. Conv ersely , if V has only ﬁnitely ma n y sub direc tly ir reducible members and all these structures are ﬁnite, then it follows from the quasiv ariety ana- logue o f Birkho ﬀ ’s subdirect decompositio n Theor e m [2 6, Theo rem 3.1.1] that each member o f V embeds int o a pr o duct of those subdir ectly irreducibles, and thus V is ﬁnitely generated. ⊓ ⊔ Remark 4- 8 .5. It follows easily from J ´ onsson’s Lemma that Every ﬁn itely gener ate d variety of lattic es (or, more generally , every ﬁnitely g enerated congruence- distributive v ariety) has only ﬁnitely many s u b dir e ctly irr e ducible memb ers; that is, it is also ﬁnitely gener ate d as a quasivariety . This situa tion is quite unt ypical: while there are ma ny exa mples o f ﬁnitely generated qua- siv arieties, the ﬁnitely generated v arieties that a r e also ﬁnitely g enerated as quasiv arieties a r e no t so widespr e ad. Let us for example co nsider the situation for gr oups . A simple use o f the fundamental structure theorem for ﬁnite abe lian gro ups shows easily that Every ﬁ nitely gener ate d variety of ab elian gr oups is also a ﬁnitely gener ate d quasivariety . On the other ha nd, co nsider the quaternion gr oup Q := { 1 , − 1 , i, − i, j, − j, k , − k } with i 2 = j 2 = k 2 = i j k = − 1. The cen ter of Q is Z := { 1 , − 1 } . Now the s ubgroup Z n := { ( x 1 , . . . , x n ) ∈ Z n | x 1 · · · x n = 1 } is normal in Q n , and it is not hard to v erify that Z n /Z n is the least non- trivial normal subgro up of Q n /Z n ; hence Q n /Z n is sub dire ctly irr educible. It b e longs to the v ariety of gr oups generated b y Q , and it has order 2 · 4 n . In particular, the variety of gr oups gener ate d by Q c ontains inﬁnitely many ﬁn ite 4-9 Relative critical points betw een quasiv arieties 115 sub dir e ct ly irr e ducible memb ers . This exa mple is inspired b y the one, giv en in Neumann [47, Example 51.33 ], of a mo nolithic (i.e., s ubdir ectly ir reducible), ﬁnite, non-critical gr oup. The following r esult ex tends [18, Lemma 3.8], with a similar pro o f. Prop ositio n 4-8.6. L et V b e a ﬁnitely gener ate d quasivariety on a ﬁrst -or der language L . Then V is str ongly c ongruenc e-pr op er. Pr o of . Denote b y K a ﬁnite set of representativ es of the subdir ectly irr e - ducible members of V mo dulo isomo rphism. Let S b e a ﬁnite lattice and let ι : S → Con V A b e an isomorphism, with A ∈ V . W e start with a stan- dard arg umen t giving Birk ho ﬀ ’s sub direct decomp os ition theorem in qua - siv arieties, s ee, for example, [26, Theorem 3.1.1]. It is well-kno wn that an y element in a n alg ebraic lattice is a meet o f completely meet-irreducible ele- men ts (see, for example, [16, Theo r em I.4.25]). By applying this to the zero element in the lattice S , we see that if P denotes the set o f all meet-irr educ- ible elements o f S , the subset ι “( P ) of Con V A meets to the zero congruence of A , thus the diagonal map A → Y ( A /ι ( p ) | p ∈ P ) , x 7→ ( x/ι ( p ) | p ∈ P ) is a n embedding. As all subdirec tly irreducible member s of V are ﬁnite, all structures A /ι ( p ) are ﬁnite, thus A ∗ := Q ( A /ι ( p ) | p ∈ P ) is ﬁnite, and th us A is ﬁnite. F urthermor e, if A denotes the class of all pro ducts of the form Q ( S p | p ∈ P ) where all S p ∈ K , A ∗ embeds into some mem b er of A , hence so do es A . As A is ﬁnite and eac h mem b er of A has only ﬁnitely man y substructures, there are only ﬁnitely many p ossibilities for A . ⊓ ⊔ Recall tha t a variety is a qua s iv ariety closed under homomor phic images. Prop ositio n 4-8.7 (folklore). L et V b e a ﬁnitely gener ate d ( quasi ) variety on a ﬁrst -or der language L . Then V is lo c al ly ﬁnite. Pr o of . Of course, the result for quas iv arie ties follows from the result for v ari- eties. If a v ariet y V is genera ted by a str ucture A , then the free V - o b ject on n generator s is the s ubstructure of A A n generated by the pro jectio ns , for ea ch po sitive in tege r n (the usual pro of of this fact is not a ﬀected by the p ossible presence of relations); in particular , if A is ﬁnite, then this structure is ﬁnite. ⊓ ⊔ Other classes of lo cally ﬁnite, stro ngly c o ngruence-pr op er (quasi)v arieties will b e given in Sec tion 4-10. 4-9 Relative c ritical p oin ts b et ween quasiv arieties In this section w e shall use the results of Section 4-8 in order to relate, in Theorem 4-9 .2, the liftability of ﬁnite diagrams of ( ∨ , 0)-semilattices, with 116 4 Larders from ﬁrst-order structures resp ect to the relative compact congruence semilattice functor, in quasiv a- rieties A and B , with the relative critical p oint cr it r ( A ; B ) intro duced in Deﬁnition 4-9.1. It w ill turn out that the latter lies b elow a n aleph of ﬁ- nite index, the latter b e ing smaller than the restricted Kura towski index (cf. Deﬁnition 3-5.3) of the shap e of a diagram liftable in A but not in B . As a consequence, w e o btain, under some ﬁniteness as sumptions on A and B , that crit r ( A ; B ) is always either smaller than ℵ ω or equa l to ∞ (Theorem 4-9 .4). Here, the symbol ∞ denotes a “num b er” greater than all cardinal num b ers. This extends to quasiv arieties of algebra ic systems a result obtained earlier by the ﬁr st author o n v arieties of a lg ebras (cf. [1 8, Co rollar y 7.13]). W e ca ll the r elative c omp act c ongruenc e class of a generalized quasiv ari- ety V the class of a ll ( ∨ , 0)-semilattices that are isomorphic to Co n V c A for some A ∈ V . W e deno te this cla s s by Con c , r V . Deﬁnition 4 - 9.1. The r elativ e critic al p oint b etw een quasiv arieties A and B is deﬁned a s crit r ( A ; B ) := min { card S | S ∈ (Con c , r A ) \ (Con c , r B ) } if Con c , r A 6⊆ Con c , r B , and ∞ otherwise. In particular, in case both A and B are v arieties, crit r ( A ; B ) is equal to the critical p oint crit( A ; B ) a s intro duced in [1 8, Deﬁnition 7.2]. Theorem 4-9. 2. L et A and B b e quasiva rieties on ( p ossibly distinct ) ﬁrst- or der languages with only ﬁnitely many r elation symb ols such that B is b oth c ongruenc e-pr op er and lo c al ly ﬁnite, and let P b e an almost join-semilattic e. Assume that t her e exists a P -indexe d diagr am − → A =  A p , α q p | p ≤ q in P  of ﬁnite obje cts of A such that the diagr am Con A c − → A has no lifting, with r esp e ct to Con B c , in B . The fol lowing statements hold: (i) L et ( X , X ) b e an ℵ 0 -lifter of P . Then crit r ( A ; B ) ≤ car d X + ℵ 0 . (ii) If P is ﬁnite nontrivial with zer o, then crit r ( A ; B ) ≤ ℵ kur 0 ( P ) − 1 . W e refer to Deﬁnition 3-5 .3 for the deﬁnition of the restric ted Kur atowski index kur 0 ( P ). Pr o of . W e set A † := { A p | p ∈ P } , B † := { B ∈ B | B is ﬁnite } , S := Sem ∨ , 0 , and we deﬁne S † as the class of all ﬁnite ( ∨ , 0)-semilattices. F urthermore, we deﬁne S ⇒ as the catego ry of all ( ∨ , 0)-semilattices with ideal-induced homomorphisms. It follows from Theorem 4-8.2 together with the assumptions on B that ( B , B † , S , S † , S ⇒ , Co n B c ) is a pro jectable right ℵ 0 -larder . (4 - 9.1) F urthermore, ( A , S , S ⇒ , Co n A c ) is a left larder. (As the empt y structure is not allow ed in A , pro jections in A a re sur jective homomorphisms, th us so are ex- tended pro jections of A , and thus, by Lemma 4- 5.1, Con A c sends pr o jections 4-9 Relative critical points betw een quasiv arieties 117 of A to ideal-induced ho momorphisms in S . That Con A c preserves all sma ll directed colimits follows from Theorem 4-4.1.) Therefore, by using Prop osi- tion 3-8.3, we obtain that ( A , B , S , A † , B † , S ⇒ , Co n A c , Co n B c ) is a pro jectable ℵ 0 -larder. Now suppose that the assumptions of (i) are satisﬁed, set κ := card X + ℵ 0 and A := F ( X ) ⊗ − → A . As the la nguage of A has only ﬁnitely many relations, it follows from Lemma 4-3.11(i) that Co n A U is ﬁnite for each ﬁnite U ∈ A . As A is the colimit of a diagram of at most κ ﬁnite o b jects of A , we get card C o n A c A ≤ κ . Hence, in order to conclude the pro of, it suﬃces to prove that Con A c A is no t isomorphic to Co n B c B , for any ob ject B of B . Suppo se otherw is e, and let χ : Con B c B → Con A c A b e an iso morphism. By CLL (Lemma 3-4 .2), there ex is ts a P -indexed diagram − → B of B such that Con B c − → B ∼ = Con A c − → A . This contradicts the ass umption on − → A . Now supp ose that the assumptions o f (ii) are satisﬁed, and set n := kur 0 ( P ). As ( ℵ n − 1 , < ℵ 0 ) ❀ P , it follows from L emma 3-5.5 that there exists a norm-c ov ering X of P s uch that X is a low er ﬁnite almost join-semilattice with zero , car d X = ℵ n − 1 , and X , together with the collection of all principa l ideals of X , is a n ℵ 0 -lifter of P . Now the conclusion follows from (i) ab ov e. ⊓ ⊔ In order to pro ceed, w e prov e a simple c ompactness result, similar, in formulation and pro of, to the one of [1 8, Theorem 7.11]. Although this result can b e extended to diag r ams indexe d b y structures far mor e gener a l than po sets, we choose, for the sak e of simplicit y , to formulate it only in the la tter context. Lemma 4 - 9.3. L et B and S b e c ate gories, let Ψ : B → S b e a functor, let P b e a p oset, and let − → S =  S p , σ q p | p ≤ q in P  b e a diagr am in S . We make the fol lowing assumptions: (L1) F or e ach p ∈ P , ther e exists a ﬁnite set B p of obje cts of B such t hat ( ∀ X ∈ Ob B )  Ψ ( X ) ∼ = S p ⇒ ( ∃ Y ∈ B p )( X ∼ = Y )  . (L2) F or al l p ≤ q in P , al l X ∈ B p , and al l Y ∈ B q , ther e ar e only ﬁnitely many morphi sms fr om X to Y in B . (L3) Each S p , for p ∈ P , has only ﬁnitely m any automorphisms. If the r estriction − → S ↾ X has a lifting with r esp e ct to Ψ , for e ach ﬁnite subset X of P , then − → S has a lifting with r esp e ct to Ψ . Pr o of . F or ea ch X ∈ [ P ] <ω , there ar e − → B X =  B X,p , β X,q X,p | p ≤ q in X  in B X and a natural equiv alence ( ε X,p | p ∈ X ) : Ψ − → B X . → − → S ↾ X . It follows from (L1 ) that we may assume that B X,p ∈ B p for each p ∈ X . F urthermore, we set Q X := [ P ] <ω ↑ X , and we ﬁx an ultra ﬁlter U on [ P ] <ω such that Q X ∈ U for each X ∈ [ P ] <ω . Let p ∈ P . As Q { p } belo ngs to U a nd is the disjoin t union of all sets 118 4 Larders from ﬁrst-order structures { X ∈ Q { p } | B X,p = B } , for B ∈ B p , and a s, by (L1), B p is ﬁnite, ther e exists a unique B p ∈ B p such that R p := { X ∈ Q { p } | B X,p = B p } b elo ng s to U . Let p ≤ q in P . As R p ∩ R q belo ngs to U and is the disjoint union of all s ets { X ∈ R p ∩ R q | β X,q X,p = β } , for β : B p → B q in B , and as, by (L2), there are o nly ﬁnitely many such β , there exists a unique morphism β q p : B p → B q in B such that { X ∈ R p ∩ R q | β X,q X,p = β q p } b elo ng s to U . Let p ∈ P . As R p belo ngs to U and is the disjoint union of all se ts { X ∈ R p | ε X,p = ε } , for isomo rphisms ε : Ψ ( B p ) → S p , and as, b y (L3), there are only ﬁnitely many such ε , there exists a unique isomorphism ε p : Ψ ( B p ) → S p such that { X ∈ R p | ε X,p = ε p } b elo ng s to U . It is str a ightforw ard to verify that − → B :=  B p , β q p | p ≤ q in P  is a P -indexed diagram in B and that ( ε p | p ∈ P ) is a natural equiv alence fro m Ψ − → B to − → S . ⊓ ⊔ Now we can improv e the pictur e by the following dichotom y result, that extends [18, Corolla r y 7.14 ] from v arieties to qua siv arieties. Theorem 4-9. 4 (Dic hotom y Theorem). L et A and B b e lo c al ly ﬁnite quasivarieties on ( p ossibly distinct ) ﬁrst- or der languages with only ﬁnitely many r elation symb ols su ch that B is st ro ngly c ongruenc e-pr op er. If Con c , r A 6⊆ Co n c , r B , then crit r ( A ; B ) < ℵ ω . Pr o of . Supp ose that crit r ( A ; B ) ≥ ℵ ω . Given a structure A ∈ A , we m ust ﬁnd B ∈ B such that Con A c A ∼ = Con B c B . Pick a ∈ A and denote by P the set of all ﬁnite substructures of A containing a a s an ele ment. Observe that P is a ( ∨ , 0)-semilattice. Set A p := p a nd denote by α q p the inclusion em b edding from A p int o A q , for all p ≤ q in P . Likewise, denote by α p the inclusion embedding fro m A p int o A . Set − → A :=  A p , α q p | p ≤ q in P  . It follows from the lo cal ﬁniteness of A that ( A , α p | p ∈ P ) = lim − → − → A in A . (4-9.2) F or each ﬁnite ( ∨ , 0)-subsemilattice Q of P , the res tricted Kuratowski in- dex kur 0 ( Q ) o f Q is a no n-negative integer, hence it follows from The- 4-9 Relative critical points betw een quasiv arieties 119 orem 4 - 9.2 that the diagra m Co n A c ( − → A ↾ Q ) can b e lifted with resp ect to the functor Con B c . As B is strongly congr uence-prop er, it follows fr o m Lemma 4 -9.3 that the full dia gram Con A c − → A can b e lifted with res pec t to Con B c , that is , there are a P -index ed diagra m − → B of B and a natural equiv alence Con A c − → A . → Con B c − → B . ( Observe t hat this c omp actness ar gument r e quir es the stro ng c ongruenc e-pr op erness assumption on B .) Therefore, tak- ing B := lim − → − → B , we obtain, using (4-9.2) together with Theo rem 4-4.1 (preserv ation of directed colimits b y the Con A c and Con B c functors), that Con A c A ∼ = Con B c B . ⊓ ⊔ Remark 4- 9 .5. As stated in Pr op ositions 4 -8.6 and 4-8.7, the assumptions on B in the statemen ts of both Theorems 4-9.2 and 4-9 .4, that is, being bo th stro ngly congr uence-prop er and lo cally ﬁnite, are sa tisﬁed in case B is a ﬁnitely generated quasiv ariety . Remark 4- 9 .6. The pro of of Theore m 4-9.2 shows that w e can weaken the assumptions of that theorem, by requiring car d Con A c A ≤ ℵ kur 0 ( P ) − 1 , for each nonempty ﬁnite pro duct A of struc tur es of the form A p (indeed, if this holds, then the inequality card Co n A c  F ( X ) ⊗ − → A  ≤ ℵ kur 0 ( P ) − 1 holds as well). In that case we no longer need the ﬁniteness ass umption a b o ut the relational part of A . This holds, in par ticular, in case Con A c A is ﬁnite for each nonempt y ﬁnite pr o duct A of str uctures of the form A p . This holds, for example, in case A is a v arie ty of mo dular lattices in which ev ery ﬁnitely generated member has ﬁnite le ngth. A similar r emark applies to Theo r em 4-9.4, with a s lightly more a wkward formulation. The ass umption on A now states that ea ch A ∈ A is a directed colimit of a diagram, indexed by some ( ∨ , 0)-semila ttice, of structures in A any nonempty ﬁnite pro duct of which has a ﬁnite congruence lattice (r el- atively to A ). In pa rticular, this holds again in case A is a quasiv ariety of mo dular lattices in which every ﬁnitely gener ated member has ﬁnite length. W e conc lude this sec tion by the following analo gue of [18, Co rollary 7 .12] for quasiv arieties. O bserve the diﬀerent placement of the ﬁniteness assump- tion: instea d of dea ling with ﬁnite semilattices that can b e lifted from A , we are dealing with the more restricted class of the Con A c A for ﬁnite A ∈ A . Corollary 4 - 9.7. L et A and B b e lo c al ly ﬁnite quasiva rieties on ( p ossibly distinct ) ﬁrst-or der languages with only ﬁ nitely many r elation symb ols s uch that B is str ongly c ongruenc e-pr op er. The fol lowing statements ar e e qu ivalent: (i) crit r ( A ; B ) > ℵ 0 . (ii) F or every dia gr am − → A of ﬁnite memb ers of A , indexe d by a tr e e, ther e exists a diagr am − → B of B su ch that Con A c − → A ∼ = Con B c − → B . (iii) F or every diagr am − → A of ﬁnite memb ers of A , indexe d by a ﬁ nite cha in, ther e exist s a diagr am − → B of B su ch that Con A c − → A ∼ = Con B c − → B . 120 4 Larders from ﬁrst-order structures Pr o of . (i) ⇒ (ii). A ss ume that (i) holds. By using aga in L e mma 4-9.3, it suﬃces to prove that for every diagr a m − → A of ﬁnite members of A , index ed by a ﬁn ite tree P , ther e exists a diagram − → B o f B such that Con A c − → A ∼ = Con B c − → B . By Prop ositio n 3-5.6, P has a n ℵ 0 -lifter ( X , X ) with X at mo st co un table. The conclusion follows immediately from Theorem 4-9 .2(i). (ii) ⇒ (iii) is trivial. (iii) ⇒ (i). It follows from Lemma 4-9.3 that every diagram o f ﬁnite ( ∨ , 0)- semilattices, indexed by the chain ω of all na tur al n umbers, whic h has a lifting in A , also has a lifting in B . Now let S be an at most co un table ( ∨ , 0)-se mila ttice in Con c , r A . Ther e exists A ∈ A such that S ∼ = Con A c A . W e claim that A can be taken at mo s t countable. W e use the arg ument of the second part of the pro of of Prop o- sition 4-2 .3. Fix a ∈ A and denote by A X the substruc tur e of A genera ted by X ∪ { a } , for eac h X ⊆ A . By using the description o f direc ted colimits given in Section 1-2.5, we obtain the directed colimit re presentation A = lim − → X ∈ [ A ] 6 ℵ 0 A X in A , with all maps in the co co ne being the corre s po nding inclusion maps. By applying Theorem 4-4 .1, we obtain Con A c A = lim − → X ∈ [ A ] 6 ℵ 0 Con A c A X in Sem ∨ , 0 . (4-9.3) As A is loca lly ﬁnite, each A X is at most countable. As the la nguage of A has only ﬁnitely man y relation symbols, it follo ws that eac h Con A c A X is at mo st countable. Denote by α X : A X ֒ → A the inclusio n ma p, fo r each X ∈ [ A ] 6 ℵ 0 . By applying Lemma 4-6.1 (with λ := ℵ 1 , I := [ A ] 6 ℵ 0 , and L := {∨ , ∧ , 0 } ) to (4-9.3), we obtain that the set J := { X ∈ [ A ] 6 ℵ 0 | Co n A c α X is an embedding } is coﬁnal in [ A ] 6 ℵ 0 . On the o ther hand, as Co n A c A ∼ = S is at most countable, there exists X ∈ [ A ] 6 ℵ 0 such that Co n A c α X is surjective. There exists Y ∈ J containing X . As Con A c α Y is a n isomorphism, it follows that S ∼ = Con c A Y , which completes the pro of of our claim. F rom now on until the end of the pro o f w e ﬁx A ∈ A at most count- able such that Con A c A ∼ = S . W e must prove that Con A c A is isomorphic to Con B c B for some B ∈ B . W e argue as at the e nd of the pr o of o f [1 8, Cor o l- lary 7.1 2 ]: as A is at mos t coun table and lo cally ﬁnite, it is the directed union o f a chain ( A n | n < ω ) of ﬁnite substructures, g iving cano nically an ω -indexed diagram − → A . Let a n ω -index ed diagra m − → B in B lift, with resp ect to the functor Con B c , the diagr am Con A c − → A . Setting B := lim − → − → B , it follows 4-10 Finitely generated v arieties of algebras 121 from Theorem 4-4.1 (preserv ation of dir ected colimits b y the Con A c and Con B c functors) that Con A c A ∼ = Con B c B . ⊓ ⊔ 4-10 Strong congruence-prop erness of certain ﬁnitely generated v arieties of algebras In view o f Pr o p o sitions 4- 8.6 and 4-8.7, it would have lo o ked easier to for- m ulate the Dichotom y T heo rem (Theorem 4- 9.4) under the more res trictive assumption where B is a ﬁnitely gener ate d quasivari ety , arguing that there are no “natural” examples of (quasi)v arieties that are both strongly congruence - prop er and lo ca lly ﬁnite without b eing ﬁnitely generated. This impress io n (that the Dic hotomy Theor em is desig ne d for ﬁnitely generated quasiv ari- eties o nly) might b e reinforced by the comments in Remark 4-8.5, showing an example of a ﬁnitely gener a ted v ariety of groups which is not ﬁnitely generated a s a quasiv ariety . W e shall now show some direc tio ns suggesting that this is not so. In what follows we shall focus on varieties of algebr as of ﬁ nite t yp e : there are only ﬁnitely many oper ation (or consta nt ) symbols a nd no rela tion symbols . A v ariet y V of algebras is c ongruenc e-mo dular if the congruence la ttice o f every mem b er o f V is mo dular. F or examples, v arieties of gr oups (or ev en lo op s ) or o f mo dules ar e congruence-mo dular . By using their commutator theory for congruence-mo dular v arieties, F r eese and McKenzie pr ov e in [13, Theo- rem 1 0.16] the following remark able res ult: Theorem 4-10. 1. Le t A b e a ﬁ n ite algebr a such t hat the variety V ar ( A ) gener ate d by A is c ongruenc e-mo dular, and let B ∈ V ar ( A ) . If Con B has ﬁnite length n , then card B ≤ (card A ) n . It follows immediately that Every ﬁnitely gener ate d c ongruenc e-mo dular variety of ﬁn ite t yp e is str ongly c ongruenc e-pr op er . Even for v arieties of groups, this result is not trivial, in particular in view of the example in Remark 4- 8.5. In Hobby and McKenzie [33, Theorem 14 .6 ], Theorem 4 -10.1 is extended to the case where V ar ( A ) omits the tame c ongruenc e the ory typ es 1 and 5 . This holds, in particular, in ca se there is a nontrivial lattice ident ity sa tis ﬁed by the congruence lattices of a ll members of V ar ( A ), cf. [33, Theore m 9.18 ]. Therefore we o btain the following consequence of that result together with the Dichotom y Theorem (Theorem 4-9 .4). Theorem 4-10. 2. Le t A b e a lo c al ly ﬁnite quasivariety on a ﬁrst -or der language with only ﬁnitely many r elation symb ols and let B b e a ﬁnitely gener ate d variety of algebr as with ﬁnite similarity typ e, omitting typ es 1 and 5 ( this holds in c ase B satisﬁes a nontrivial c ongruenc e identity ) . If Con c , r A 6⊆ Co n c , r B , then crit r ( A ; B ) < ℵ ω . 122 4 Larders from ﬁrst-order structures In particular, in case b oth A and B are ﬁnitely generated v arieties of either lattices, gr oups, lo ops, or mo dules (the latter on ﬁnite rings), Con c , r A 6⊆ Con c , r B implies that c rit r ( A ; B ) < ℵ ω . An example of a ﬁnitely genera ted v ariety o f algebra s with a r bitrarily large (ﬁnite or inﬁnite) simple alg e br as (and th us non congruenc e -prop er), attributed to C. Shallon, is given in [33, Exer cise 14 .9 (4)]. 4-11 A p ot en tial use of larders on non-regular cardinals Although we do not know of any “concre te” pr oblem s o lved b y λ -lar der s for non-regula r λ , we sha ll outline in the pres ent section a p otential use which do es not seem to follow from CLL applied to larder s indexed by regular cardinals. F or a qua siv ariety V and an inﬁnite cardinal λ , we denote by V ( λ ) the class of all λ -small members of V . W e star t with the following la rderho o d r esult. Theorem 4-11. 1. Le t λ b e an u nc ountable c ar dinal and let V b e a qu asiva- riety on a λ -smal l ﬁ rst-or der language L . Then the 6 -uple ( V , V ( λ ) , Se m ∨ , 0 , Se m ( λ ) ∨ , 0 , Se m idl ∨ , 0 , Co n V c ) is a pr oje ctable right λ -lar der. Pr o of . Similar to the pro of of Claim 2 within the pro of of Theorem 4 -7.2. F or each A ∈ V ( λ ) , the semilattice Con V c A is λ -small, th us, b y Prop ositio n 4 -2.3, it is weakly λ -pr esented in Sem ∨ , 0 . Hence the s tatement (PRES λ ( V ( λ ) , Co n V c )) is satisﬁed. Let B ∈ V , we must verify that (LS r λ ( B )) is satisﬁed. Let S be a λ -small ( ∨ , 0)-s emilattice, let I b e a cf ( λ )-small set, and let ( u i : U i ⇒ B | i ∈ I ) b e an I -indexed family o f double arr ows with all the U i being λ -small. There exists a successor (th us regula r uncountable) car dinal µ ≤ λ such that L , S , and all U i are µ -small. As at the end of the pr o of o f Prop o sition 4-2 .3, we express B as a co ntin uous dire cted colimit B = lim − → X ∈ [ B ] <µ B X in V (here B X is simply the V -substructure of B generated by X together with some ﬁxed ele ment of B ), with all maps in the co c o ne b eing the corr e sp o nd- ing inclus io n maps. Then applying Theorem 4-4.1 to V yields a contin uous directed colimit Con V c B = lim − → X ∈ [ B ] <µ Con V c B X in Sem ∨ , 0 . (4-11.1 ) 4-11 A poten tial use of larders on non-regular cardinals 123 Denote b y β X : B X ֒ → B the inclus ion map, for each X ∈ [ B ] <µ . It follows from (4 -11.1) to gether with Prop os ition 4-6.2 that the set J := { X ∈ [ B ] <µ | ϕ ◦ Con V c β X is idea l-induced } is σ -c lo sed coﬁnal in [ B ] <µ . As there exists X ∈ [ B ] <µ such that B X contains all images u i “( U i ), it follows that such an X can be chosen in J . Then β X is monic, u i E β X for each i ∈ I , and ϕ ◦ Co n c β X is idea l-induced. The pro jectability statemen t follows immediately from Theor em 4-5.2. ⊓ ⊔ By applying Theorem 4-1 1 .1 to the case λ := ℵ ω , w e o btain the following. Corollary 4 - 11.2. L et V b e a quasivariety on an ℵ ω -smal l ﬁrst -or der lan- guage L and supp ose t hat ther e exists a diagr am − → S = ( S m , σ n m | m ≤ n < ω ) , indexe d by the chai n ω of al l n atur al numb ers, of ℵ ω -smal l ( ∨ , 0) - s emilattic es, that c annot b e lifte d, with r esp e ct to the Con V c functor, by any diagr am in V . Then ther e exists a ( ∨ , 0) -semilattic e with at most ℵ ω elements that is not isomorphi c to Con V c B for any B ∈ V . Pr o of . Our pro o f will follow the lines of the one of Theo rem 4-7 .2. Claim. Denote by Φ the iden tity functor on Sem ∨ , 0 . Then the quadruple ( Sem ∨ , 0 , Se m ∨ , 0 , Se m idl ∨ , 0 , Φ ) is a left larder. Pr o of . As the pro of of Claim 1 within the pro o f o f Theorem 4-7.2. ⊓ ⊔ Claim By the claim above together with Theorem 4-1 1.1 and Pr op osition 3-8.3 (with A † := Sem ( ℵ ω ) ∨ , 0 ), we obtain that the 8-uple ( Sem ∨ , 0 , V , Sem ∨ , 0 , Se m ( ℵ ω ) ∨ , 0 , V ( ℵ ω ) , Se m idl ∨ , 0 , Φ, Con V c ) is an ℵ ω -larder. By a pplying Propo sition 3-5.6 to T := ω , w e obtain an ℵ ω -lifter ( X, X ) of ω such that X is a low er ﬁnite almos t join-semilattice with zero a nd card X = ℵ ω . The ( ∨ , 0)-semilattice S := F ( X ) ⊗ − → S has at most ℵ ω elements. If S is isomorphic to Con V c B for some B ∈ V , then, b y CLL (Lemma 3-4.2), there ex ists an ω -indexed diag ram − → B in V such that Co n c − → B ∼ = − → S , a contra- diction. ⊓ ⊔ Chapter 5 Congrue nce-p erm utable, congrue nce-pres erving exten sions of lattices Abstract. This chapter is intended to illustrate how to use CLL for so lving an o pe n problem of lattice theory , although the statement of that pr o blem do es no t inv olve lifting diag rams with r esp ect to functors. W e s et αβ := { ( x, z ) ∈ X × X | ( ∃ y ∈ X )(( x, y ) ∈ α and ( y , z ) ∈ β } , for a ny bina r y r elations α and β on a set X , a nd w e say that an alg ebra A is c ongruenc e-p ermu table if αβ = β α for a n y cong ruences α and β o f A . The problem in question is: Do es every lattic e of c ar dinality ℵ 1 have a c ongruenc e-p ermutable, c ongruenc e-pr eserving extension? This problem is part of Problem 4 in the survey paper T ˚ uma and W eh- rung [59] but it was certainly known b efore . Due to ear lier w ork by Plo ˇ sˇ cica, T ˚ uma , and W ehrung [49], the co rresp onding nega tive r esult for free lattices on ℵ 2 generator s was alre ady known. The co unt able case is still op en, al- though it is pr ov ed in Gr¨ a tzer, Lakser, and W ehrung [30] that every c ount- able, lo cally ﬁnite (or even “loca lly congruence-ﬁnite”) lattice has a relatively complemented (thus co ngruence-p ermutable) co ngruence-pre serving exten- sion. The ﬁnite case is solved in Tischendorf [57], where it is prov ed that every ﬁnite lattice em b eds congruence-prese r vingly into some ﬁnite atom- istic (thus congruence- p er mu table) lattice. This result is improv ed in Gr¨ atzer and Sc hmidt [32], wher e the authors prov e that every ﬁnite lattice e m b eds congruence- preserving ly into some sectionally complemented ﬁnite la ttice. Most o f Chapter 5 will consist of chec king one after another the v arious assumptions that need to b e satisﬁed in order to b e able to use CLL. Most of these v eriﬁcatio ns are element ar y . In that s ense (i.e., considering the veriﬁca- tion of the assumptions under lying CLL a s tedious but elementary), the hard core o f the solution to the pro blem ab ove consists of the unliftable family o f squares presented in Lemma 5-3.1. 125 126 5 Congruence-preserving ext ensions 5-1 The categor y of semilattice-metric spaces In this section we sha ll in tro duce the category that will play the role o f the S of the statement of CLL (Lemma 3-4.2) and state its go o d b ehavior in terms of directed colimits and ﬁnitely presen ted o b jects. W e re mind the reader that semilattic e-value d distanc es have be e n in tro duced in Exa mple 1- 1.5. Deﬁnition 5 - 1.1. F or a set A , a ( ∨ , 0)-semila ttice S , and an S -v alued dis- tance δ : A × A → S , we shall say that the triple A := ( A, δ, S ) is a semila tt ic e- metric sp ac e , and we shall often write δ A := δ , ˜ A := S . W e say that A is ﬁnite if b oth A and ˜ A are ﬁnite. F or s e milattice-metric spaces A a nd B , a morphism from A to B is a pair f = ( f , ˜ f ), where f : A → B , ˜ f : ˜ A → ˜ B is a ( ∨ , 0)-homomorphism, and the equation δ B ( f ( x ) , f ( y )) = ˜ f  δ A ( x, y )  is satisﬁed for all x, y ∈ A . The comp osition of morphisms is deﬁned by g ◦ f := ( g ◦ f , ˜ g ◦ ˜ f ). W e denote by Metr the categor y of all semilattice-metric s paces. The fo llowing lemma implies that b oth for getful functor s fro m Me tr to Set and Sem ∨ , 0 preserve all small direc ted colimits. Its pro of, although somewhat tedious, is straig ht for ward, a nd we shall omit it. Lemma 5 - 1.2. L et  A i , f j i | i ≤ j in I  b e a dir e cte d p oset-indexe d diagr am in Metr . We form the c olimits ( A, f i | i ∈ I ) = lim − →  A i , f j i | i ≤ j in I  in Set ,  ˜ A, ˜ f i | i ∈ I  = lim − →  ˜ A i , ˜ f j i | i ≤ j in I  in Sem ∨ , 0 . Then ther e exists a unique ˜ A -value d distanc e δ on A such that f i := ( f i , ˜ f i ) is a morphism fr om A i to A := ( A, δ, ˜ A ) for e ach i ∈ I . F u rt hermor e, ( A , f i | i ∈ I ) = lim − →  A i , f j i | i ≤ j in I  in Metr . The pro of of the following result go es along the lines of the pr o of of Prop o - sition 4-2 .3. Prop ositio n 5-1.3. Every semilattic e-metric sp ac e A is a monomorphic di- r e cte d c olimit of a diagr am of ﬁnite semilattic e-m et ric sp ac es. F urthermor e, A is ﬁnitely pr esente d in Me tr iff it is ﬁn ite. Pr o of . Denote b y I the s e t of all pair s i = ( X, ˜ X ) suc h that X is a ﬁnite subset of A , ˜ X is a ﬁnite ( ∨ , 0)-subsemilattice o f ˜ A , and { δ A ( x, y ) | x, y ∈ X } ⊆ ˜ X ; order I by comp onent wise containmen t. Then set X i := ( X, δ A ↾ X × X , ˜ X ), and denote by ξ i (resp., ˜ ξ i ) the inclusion map from X i int o A (r esp., from ˜ X i int o ˜ A ). The pair ξ i := ( ξ i , ˜ ξ i ) is a mo no morphism from X i to A in Metr . 5-2 The categ ory of all semilattice-metric cov ers 127 Deﬁne likewise a morphis m ξ j i : X i → X j in Metr , for i ≤ j in I . As every ﬁnitely gener ated ( ∨ , 0)-semila ttice is ﬁnite, it is s traightforw ar d to v erify the statement ( A , ξ i | i ∈ I ) = lim − →  X i , ξ j i | i ≤ j in I  . (5-1.1) The ﬁrst part of Pro po sition 5-1 .3 follows. The proof that A ﬁnite implies A ﬁnitely presented is similar to the pro of of the ﬁr st part o f Prop os itio n 4- 2 .3. Conv ersely , supp ose that A is ﬁnitely presented. As (5-1.1) is a monomorphic directed colimit, we infer, as at the end of pro o f of Corollar y 2- 4.7, that A is isomo rphic to one of the X i , th us A is ﬁnite. ⊓ ⊔ 5-2 The categor y of all semilattice-metric cov ers In this section we shall introduce the categor y that will play the role of B and the functor tha t will play the ro le o f Ψ in the statement of CLL (Lemma 3-4 .2), and state their go o d b ehavior in terms of direc ted co lim- its and ﬁnitely presented ob jects. Our Ψ is deﬁned as a forgetful functor. The right ℵ 0 -larder in question will b e stated in Pr op osition 5- 2.2. Deﬁnition 5 - 2.1. A semilattic e-metric c over is a q uadruple A := ( A ∗ , A, δ, S ), where (i) A is a se t and A ∗ is a subset of A ; (ii) S is a ( ∨ , 0)-semilattice; (iii) δ is an S -v alued distance on A ; (iv) ( Par al lelo gr am Rule ) F or all x, y , z ∈ A ∗ , there exists t ∈ A s uch that δ ( x, t ) ≤ δ ( y , z ) and δ ( t, z ) ≤ δ ( x, y ) . W e shall often write δ A := δ and ˜ A := S . Observe that the triple ( A, δ A , ˜ A ) is then a s emilattice-metric space. W e say that A is ﬁn ite if b oth A and ˜ A are ﬁnite. F or semilattice-metric cov ers A and B , a morphism from A to B is a morphism f : ( A, δ A , ˜ A ) → ( B , δ B , ˜ B ) in Metr such that f “( A ∗ ) ⊆ B ∗ . W e denote by Metr ∗ the categor y of all semilattice-metr ic covers. Our nex t r esult sho ws how to construct a right ℵ 0 -larder from semilattice- v alued s pa ces and covers. Prop ositio n 5-2.2. Denote by Metr ﬁn ( r esp., Metr ∗ ﬁn ) the class of al l ﬁ- nite obje cts in Metr ( r esp., Metr ∗ ) , by Metr ⇒ the sub c ate gory of Metr c on- sisting of al l morphisms ( f , ˜ f ) with f surje ctive, and by Ψ : Me tr ∗ → M etr the for getful functor A 7→ A ♭ := ( A ∗ , δ A ↾ A ∗ × A ∗ , ˜ A ) . Then the o ctuple ( Metr ∗ , M etr ∗ ﬁn , M etr , M etr ﬁn , M etr ⇒ , Ψ ) is a right ℵ 0 -lar der. 128 5 Congruence-preserving ext ensions Pr o of . The statement (PRE S ℵ 0 ( Metr ∗ ﬁn , Ψ )) follows from the second part of Prop o sition 5-1 .3. It r emains to c heck (LS r ℵ 0 ( B )), for eac h semilattice-metric cov er B . Let A be a ﬁnite semilattice-metr ic s pace. The triple Ψ ( B ) = ( B ∗ , δ B ↾ B ∗ × B ∗ , ˜ B ) is a semilattice-metric space. Let ϕ : Ψ ( B ) ⇒ A be a morphism in Metr ⇒ ; hence ϕ : B ∗ → A is surjective. Let n < ω and let u i : U i → B , for i < n , be ob jects in the comma ca tegory Metr ∗ ﬁn ↓ B . This mea ns that each U i is a ﬁnite semilattice-metric cov er while u i : U i → B in M etr ∗ . As ϕ “( B ∗ ) = A and b oth A and u i “( U ∗ i ) are ﬁnite, with u i “( U ∗ i ) ⊆ B ∗ (for all i < n ), there exis ts a ﬁnite subset V ∗ of B ∗ containing S ( u i “( U ∗ i ) | i < n ) such that ϕ “ ( V ∗ ) = A . As B is a s e milattice-metric cover and b oth V ∗ and S ( u i “( U i ) | i < n ) are ﬁnite subsets o f B , with V ∗ ⊆ B ∗ , ther e exists a ﬁnite subset V of B con taining V ∗ ∪ S ( u i “( U i ) | i < n ) such that ( ∀ x, y , z ∈ V ∗ )( ∃ t ∈ V )  δ B ( x, t ) ≤ δ B ( y , z ) and δ B ( t, z ) ≤ δ B ( x, y )  . (5-2.1) As b oth S  ˜ u i “( ˜ U i ) | i < n  and V ar e ﬁnite, there exists a ﬁnite ( ∨ , 0)-sub- semilattice ˜ V of ˜ B such that [  ˜ u i “( ˜ U i ) | i < n  ∪ { δ B ( x, y ) | x, y ∈ V } ⊆ ˜ V . Denote by δ V the res triction o f δ B from V × V to ˜ V . It follows from (5-2.1) that the quadruple V := ( V ∗ , V , δ V , ˜ V ) is a semilattice-metric co ver. F urther, denote by f i : U i → V the r estriction of u i from U i to V , for each i < n , by v (resp., ˜ v ) the inclusion map from V into B (res p., from ˜ V into ˜ B ), and set v := ( v , ˜ v ). As v and ˜ v are b oth one-to- o ne, v is monic. Obviously , u i = v ◦ f i , so f i : u i → v is an a rrow in the comma ca tegory Metr ∗ ﬁn ↓ B . F urthermore, from ϕ “( V ∗ ) = A it follows that ϕ ◦ Ψ ( v ) ∈ Metr ⇒ . ⊓ ⊔ 5-3 A family of unliftable squares of semilattice-metric spaces In this sec tio n we shall intro duce a family of dia grams in Metr , indexed by the ﬁnite p oset { 0 , 1 } 2 , which are unliftable with resp ect to the functor Ψ deﬁned in Pro p o s ition 5-2.2. W e shall say that a squar e from a ca tegory C is a { 0 , 1 } 2 -indexed dia- gram in C , that is, an ob ject o f the category C { 0 , 1 } 2 . A typical square in the category Metr is repre s ent ed o n Figur e 5 .1. Say tha t elemen ts x and y in a p oset with zero are ortho gonal if there is no no nzero element z such that z ≤ x and z ≤ y . The follo wing lemma is a sp ecial case of a more genera l statement, ho wev er it will b e suﬃcient for our purp oses. 5-3 A family of unliftable squares of semilattice-metric space s 129 A A 1 g 1 = = { { { { { { { { A 2 g 2 a a C C C C C C C C A 0 f 1 a a C C C C C C C C f 2 = = { { { { { { { { Fig. 5. 1 A square in the cat egory Metr Lemma 5 - 3.1. L et − → A b e a squar e in Metr lab ele d as on Figur e 5.1 , with elements 0 , 1 ∈ A 0 , a i ∈ A i for i ∈ { 0 , 1 , 2 } , and α, β ∈ ˜ A 0 , satisfying the fol lowing c onditions: (i) Al l maps f 1 , f 2 , g 1 , g 2 ar e inclusion maps. (ii) The elements ˜ f i ( α ) and ˜ f i ( β ) ar e ortho gonal in ˜ A i , for e ach i ∈ { 1 , 2 } . (iii) δ A 0 (0 , a 0 ) ≤ α , while δ A i ( a i , 1 ) ≤ ˜ f i ( α ) for e ach i ∈ { 1 , 2 } . (iv) δ A 0 ( a 0 , 1 ) ≤ β , while δ A i (0 , a i ) ≤ ˜ f i ( β ) for e ach i ∈ { 1 , 2 } . If ther e exists a s quar e − → B of semilattic e- m etric c overs t o gether with a morphism − → χ : Ψ − → B . ⇒ − → A in ( Metr ⇒ ) { 0 , 1 } 2 , t hen δ A ( a 1 , a 2 ) = 0 . W e illustrate the assumptions of Lemma 5 -3.1 on Figur e 5.2. Unlike what is suggested on the top part of that illustration, it will turn out that the liftabilit y as sumption implies that a 1 and a 2 are iden tiﬁed by the dista nce function on A . a 0 a 0 a 0 a 0 a 1 a 1 a 2 a 2 A 0 A 1 A 2 A 0 0 0 0 1 1 1 1 f 1 f 2 g 1 g 2 α β α ( i ) := ˜ f i ( α ) β ( i ) := ˜ f i ( β ) α ( i ) α ( i ) α ( i ) α ( i ) β ( i ) β ( i ) β ( i ) β ( i ) Fig. 5. 2 A square in Metr unl i ftable wi th resp ect to Ψ and Metr ⇒ 130 5 Congruence-preserving ext ensions Pr o of (Pr o of of L emma 5-3.1 ). Repre s ent − → B as on the left ha nd side of Fig- ure 5.3, and s et w := v 1 ◦ u 1 = v 2 ◦ u 2 . Hence Ψ − → B is represented on the right hand s ide of Figure 5.3 (re call the ﬂat no tation, Ψ ( X ) = X ♭ ). B B ♭ B 1 v 1 > > } } } } } } } } B 2 v 2 ` ` A A A A A A A A B ♭ 1 v ♭ 1 > > } } } } } } } } B ♭ 2 v ♭ 2 ` ` A A A A A A A A B 0 u 1 ` ` A A A A A A A A u 2 > > } } } } } } } } w O O B ♭ 0 u ♭ 1 ` ` A A A A A A A u ♭ 2 > > } } } } } } } w ♭ O O Fig. 5. 3 Squares in the categories Metr ∗ and Metr A morphism − → χ : Ψ − → B . ⇒ − → A in ( Metr ⇒ ) { 0 , 1 } 2 consists of a collection of double arrows χ i : B ♭ i ⇒ A i , for i ∈ { 0 , 1 , 2 } , and χ : B ♭ ⇒ A , all in Metr ⇒ , such tha t the diagram represe nted in Fig ur e 5.4 comm utes. Actually our pro of will not r equire the top ar row χ : B ♭ → A to b e a double ar row. A A 1 g 1 = = { { { { { { { { B ♭ χ K S A 2 g 2 a a C C C C C C C C B ♭ 1 χ 1 K S v ♭ 1    E E         A 0 f 1 3 3 3 3 Y Y 3 3 3 3 3 3 3 3 3 f 2     E E          B ♭ 2 v ♭ 2 3 3 3 Y Y 3 3 3 3 3 3 3 3 χ 2 K S B ♭ 0 u ♭ 1 ` ` A A A A A A A χ 0 K S u ♭ 2 > > } } } } } } } Fig. 5. 4 A cub e of semilattice-metric space s As χ 0 is surjective, there are ˙ 0 , ˙ 1 , ˙ a 0 ∈ B ∗ 0 such that χ 0 ( ˙ 0) = 0 , χ 0 ( ˙ 1) = 1, and χ 0 ( ˙ a 0 ) = a 0 . F or ea ch i ∈ { 1 , 2 } , as χ i is sur jective, there exis ts ˙ a i ∈ B ∗ i such that χ i ( ˙ a i ) = a i . As B 0 is a semilattice-metric co ver, it follo ws from the Parallelogra m Rule tha t there e x ists ˙ x ∈ B 0 such that δ B 0 ( ˙ 0 , ˙ x ) ≤ δ B 0 ( ˙ a 0 , ˙ 1) and δ B 0 ( ˙ x, ˙ 1) ≤ δ B 0 ( ˙ 0 , ˙ a 0 ) . (5-3.1) 5-3 A family of unliftable squares of semilattice-metric space s 131 Let i ∈ { 1 , 2 } . As χ i u i ( ˙ 0) = f i χ 0 ( ˙ 0) = f i (0) = 0, we ge t ˜ χ i δ B i ( ˙ a i , u i ( ˙ 0)) = δ A i ( χ i ( ˙ a i ) , χ i u i ( ˙ 0)) (because { ˙ a i , u i ( ˙ 0) } ⊆ B ∗ i ) = δ A i ( a i , 0 ) ≤ ˜ f i ( β ) . (5-3.2) On the o ther ha nd, ˜ χ i ˜ u i δ B 0 ( ˙ a 0 , ˙ 1) = ˜ f i ˜ χ 0 δ B 0 ( ˙ a 0 , ˙ 1) = ˜ f i δ A 0 ( a 0 , 1 ) (beca use { ˙ a 0 , ˙ 1 } ⊆ B ∗ 0 ) ≤ ˜ f i ( β ) . (5- 3.3) As the following inequalities hold (we use (5-3 .1)), δ B i ( u i ( ˙ 0) , u i ( ˙ x )) = ˜ u i  δ B 0 ( ˙ 0 , ˙ x )  ≤ ˜ u i  δ B 0 ( ˙ a 0 , ˙ 1)  , we obtain the inequalities δ B i ( ˙ a i , u i ( ˙ x )) ≤ δ B i ( ˙ a i , u i ( ˙ 0)) ∨ δ B i ( u i ( ˙ 0) , u i ( ˙ x )) ≤ δ B i ( ˙ a i , u i ( ˙ 0)) ∨ ˜ u i  δ B 0 ( ˙ a 0 , ˙ 1)  . (5-3.4) Therefore, by apply ing the ( ∨ , 0)-homomor phism ˜ χ i to (5-3.4) and b y us- ing (5- 3.2) and (5 -3.3), we obtain the inequalit y ˜ χ i δ B i ( ˙ a i , u i ( ˙ x )) ≤ ˜ f i ( β ) . (5-3.5) A similar pro of, exchanging the roles of the elements 0 and 1 of A 0 and o f the elements α and β of ˜ A 0 , in the arg umen t a b ove, leads to the ineq uality ˜ χ i δ B i ( ˙ a i , u i ( ˙ x )) ≤ ˜ f i ( α ) . (5-3.6) As ˜ f i ( α ) and ˜ f i ( β ) are orthogona l in ˜ A i , it follows fr om (5-3.5) and (5-3 .6) that the following equation is satisﬁed: ˜ χ i δ B i ( ˙ a i , u i ( ˙ x )) = 0 . (5-3.7) By applying ˜ g i to the equation (5-3.7) and using the equatio n ˜ g i ◦ ˜ χ i = ˜ χ ◦ ˜ v i , we obtain the equation ˜ χ ˜ v i δ B i ( ˙ a i , u i ( ˙ x )) = 0 , that is (cf. Figure 5.3 for the deﬁnition o f w ), ˜ χδ B ( v i ( ˙ a i ) , w ( ˙ x )) = 0 . This holds for each i ∈ { 1 , 2 } , hence, by using the triangula r inequality , 132 5 Congruence-preserving ext ensions ˜ χδ B ( v 1 ( ˙ a 1 ) , v 2 ( ˙ a 2 )) = 0 . As b oth elements v 1 ( ˙ a 1 ) and v 2 ( ˙ a 2 ) be long to B ∗ , we obtain the equatio n δ A ( χv 1 ( ˙ a 1 ) , χv 2 ( ˙ a 2 )) = 0 . As ( χ ◦ v i )( ˙ a i ) = ( g i ◦ χ i )( ˙ a i ) = g i ( a i ) = a i , fo r each i ∈ { 1 , 2 } , the conclusion of the lemma follows. ⊓ ⊔ 5-4 A left larder inv olving algebras and semilattice-metric spaces The original problem that we wish to solve in volv es congruence-pres e rving extensions of algebr as , that is, ﬁr s t-order s tr uctures without r elation s ymbols (cf. Exa mple 4-3.6). In the present section, w e sha ll asso ciate functorially , to every algebra A , a semilattice-metric space A ♮ , thus creating a left la r der (cf. Prop ositio n 5-4.3). F urthermore, the a ssignment ( A 7→ A ♮ ) turns quotients of algebra s to quotients of semilattice-metric spaces (Pr op osition 5- 4.5). Notation 5-4. 1. W e denote by MAlg the full sub categ ory o f MInd whose ob jects are all the algebr as , that is, the ﬁrst-order structures without rela- tion symbo ls. F urthermor e, for an algebra A , we set A ♮ := ( A, δ A , Co n c A ), where δ A ( x, y ) is deﬁned as the congruenc e of A g e nerated by the pa ir ( x, y ), for all x, y ∈ A . That is, using the notation of Section 4-3, δ A ( x, y ) := k x = y k A . F or a mor phis m f : A → B in MAlg , we set f ♮ := ( f , Co n c f ), where Con c f is the natural ( ∨ , 0)-homomorphism from Con c A to Con c B (cf. Sec- tion 4-3). Lemma 5 - 4.2. The c ate gory MAlg has arbitr ary smal l pr o ducts ( indexe d by n onempty set s ) . Pr o of . Let I b e a nonempty set and let ( A i | i ∈ I ) be an I -indexed family of algebra s. W e form the intersection L := T (Lg( A i ) | i ∈ I ), the cartesia n pro duct A := Q ( A i | i ∈ I ), and for each f ∈ L , say o f ar it y n , we deﬁne f A : A n → A the usual wa y , that is, f A  ( x 1 i | i ∈ I ) , . . . , ( x n i | i ∈ I )  :=  f A i ( x 1 i , . . . , x n i ) | i ∈ I  , for all elements  x k i | i ∈ I  (for 1 ≤ k ≤ n ) in A . The in terpr etations of the consta n ts a re deﬁned similarly . This deﬁnes a ﬁrst-order str ucture A :=  A,  f A | f ∈ L  . It is straightforw ar d to verify tha t A , together with the canonical pro jections A ։ A i , for i ∈ I , is the pro duct of the family ( A i | i ∈ I ) in M Al g . ⊓ ⊔ 5-4 A left larder in volving algebras and semilattice-metric spaces 133 Prop ositio n 5-4.3. L et Φ : MAlg → Metr b e t he X 7→ X ♮ functor int ro - duc e d in Notation 5-4 .1 . Then the quadruple ( MAlg , Metr , Metr ⇒ , Φ ) is a left lar der. Pr o of . The c o ndition (CLOS( MAlg )) follows from the closure o f MAlg under directed colimits within MInd (this follows trivia lly from Prop osi- tion 4-2.1). The condition (PROD( MAlg )) follows fr om Lemma 5-4.2. The condition (CONT( Φ )) follows easily fro m the descr iption of the directed col- imits given in P rop osition 4 -2.1 a nd Lemma 5-1.2, together with the pr eser- v ation of dire c ted colimits b y the Con c functor (Theorem 4-4 .1). The pro jection X × Y ։ X is surjective, for all nonempty s ets X a nd Y . It follows ea s ily that the map f is s urjective for ea ch extended pro jection f : X → Y in MAlg . The condition (PROJ( Φ, Metr ⇒ )) fo llows. ⊓ ⊔ Our next lemma will make it poss ible to deﬁne quotients of semilattice- metric spaces . The pr o of is str a ightforw ard and we shall omit it. W e r efer to Section 4-5 for the deﬁnition of the quotient of a comm utative monoid b y an o-ideal. Lemma 5 - 4.4. L et A b e a semilattic e-metric sp ac e and let I b e an ide al of ˜ A . Then t he binary r elation α := { ( x, y ) ∈ A × A | δ A ( x, y ) ∈ I } is an e quivalenc e r elation on A , and ther e exists a un ique ˜ A/I - value d dis- tanc e δ on A/α su ch t hat δ ( x/α, y /α ) = δ A ( x, y ) /I , for al l x, y ∈ A . Denote by A /I the semilattic e-metric sp ac e ( A/α, δ, ˜ A/I ) . Then the p air ( π , ˜ π ) , wher e π ( r esp., ˜ π ) denotes the c anonic al pr oje ction fr om A onto A/α ( r esp., fr om ˜ A ont o ˜ A/I ) , is a double arr ow fr om A onto A /I in M etr . In the context of Lemma 5-4.4, we shall say that the semilattice-metric space A /I is the quotient of A by the ideal I , and that the pa ir ( π , ˜ π ) is the c anonic al pr oje ction from A onto A /I . Quotients of s emilattice-metric spaces and quotients of alg ebras ar e related by the following consequence of Lemma 4-3.1 0. Prop ositio n 5-4.5. L et α b e a c ongru en c e of an algebr a A , and set I := (Con c A ) ↓ α . Th en the semilattic e-metric sp ac es ( A /α ) ♮ and A ♮ /I ar e iso- morphic. Pr o of . It follows fro m the deﬁnition o f the ♮ op erator that ( A /α ) ♮ =  A/α, δ A /α , Co n c ( A /α )  , where, by using the Second Isomor phism Theorem (Lemma 4- 1.6), 134 5 Congruence-preserving ext ensions δ A /α ( x/α, y /α ) = k x/α = y /α k A /α =  α ∨ k x = y k A  /α =  α ∨ δ A ( x, y )  /α , for any x, y ∈ A . On the other hand, the equiv alence α ′ on A deter mined, as in Lemma 5 -4.4, by the ideal I and the distance δ A satisﬁes ( x, y ) ∈ α ′ ⇔ δ A ( x, y ) ∈ I ⇔ k x = y k A ≤ α ⇔ ( x, y ) ∈ α , for any x, y ∈ A ; thus α = α ′ . Therefore, A ♮ /I =  A/α, δ, (Con c A ) /I  , where δ ( x/α, y /α ) := δ A ( x, y ) /I , for all x, y ∈ A . Now it follows from Lemma 4- 3 .10 together with the Seco nd Isomor phism Theorem (Theor em 4 -1.6) that the ass ignment ( ξ /I 7→ ( α ∨ ξ ) /α ) de- ﬁnes an isomorphism ϕ : (Con c A ) /I → Co n c ( A /α ). As, by deﬁnition, ϕ  δ A ( x, y ) /I  =  α ∨ δ A ( x, y )  /α for a ny x, y ∈ A , the pa ir (id A/α , ϕ ) is an is o morphism from A ♮ /I onto ( A /α ) ♮ . ⊓ ⊔ 5-5 CPCP -retracts and CPCP-extensions In this section we ﬁnally solve the problem stated at the b eginning of Chap- ter 5, in the nega tive (Coro llary 5-5 .6). In fact, we prov e a stronger neg ative statement (Theorem 5-5 .5). This statement inv olves the notion of a CP CP- retract. Deﬁnition 5 - 5.1. A semilattice-metric s pace A is a CPCP-r etr act if ther e are a semilattice-metric cov er B and a double ar row χ : B ♭ ⇒ A . Prop ositio n 5-5.2. L et A b e a semilattic e-metric sp ac e and let I b e an id e al of ˜ A . If A is a CPCP-r etr act, then so is A /I . Pr o of . The canonical pro jection π : A → A /I is a double a rrow in Me tr . It follows immediately that if B is a semila ttice-metric cov er and χ : B ♭ ⇒ A , then π ◦ χ : B ♭ ⇒ A /I . ⊓ ⊔ Deﬁnition 5 - 5.3. An alg ebra B is a CP-extension of an algebra A if A ⊆ B , Lg( A ) ⊆ Lg( B ), the inclusion ma pping e : A ֒ → B is a morphism in MAl g , and Con e is an isomorphism from Con A onto Con B . In other words, ev ery congruence of A extends to a unique co ng ruence of B . W e s ay that a CP-extensio n B of A is a CPCP-extension of A if for all x, y , z ∈ A there exists t ∈ B such that ( x, t ) ∈ k y = z k B and ( t, z ) ∈ k x = y k B . 5-5 CPCP- retracts and CPCP-extensions 135 In particular , if B is a CP-extension of A and an y tw o cong ruences of B per mut e (i.e., αβ = β α for all α, β ∈ Co n B ), then B is a CPCP -extension of A . Prop ositio n 5-5.4. If an algebr a A has a CPCP-extension, then the semi- lattic e-metric sp ac e A ♮ is a CPCP-r etr act. Pr o of . Let B b e a CP CP-extension of A and denote b y e : A ֒ → B the inclusion mapping. Set δ B ( x, y ) := k x = y k B , for a ll x, y ∈ B . Then C := ( A, B , δ B , Co n c B ) is a semilattice- metric co ver. By assumption, Con c e is an isomorphism from Con c A onto Con c B . Then (id A , (Co n c e ) − 1 ) : C ♭ ⇒ A ♮ . ⊓ ⊔ Now w e reach the main theor e m of Chapter 5. Theorem 5-5. 5. L et V b e a n ondistributive variety of lattic es and let F b e a fr e e b ounde d lattic e on at le ast ℵ 1 gener ators within V . Then F ♮ is not a CPCP-r etr act. Conse quently, F has no CPCP-extension. Consequently , F has no cong r uence-preser ving, congruenc e - pe rmutable extension. Pr o of . It suﬃces to pro ve that there exists a bounded lattice L of car dinality at most ℵ 1 within V such that L ♮ is not a CP CP-retra ct. Indeed, L is a quotient o f F , thus, by P rop osition 5-4 .5, L ♮ is a quotient of F ♮ . Hence, b y Prop ositio n 5 -5.2, if F ♮ were a CPCP -retract, then so would b e L ♮ . This prov es o ur cla im. As V is a nondistr ibutive lattice v ariety , either the ﬁv e-element nonmo dula r lattice N 5 or the ﬁve-elemen t mo dula r nondistributive lattice M 3 belo ng to V . La b e l the ele ments of M 3 and N 5 as on Fig ure 5 .5. a 0 a 0 a 1 a 1 a 2 a 2 0 0 1 1 M 3 N 5 Fig. 5. 5 Labeling the lattices M 3 and N 5 Now w e s hall co nstruct a squar e of la ttices A 0 , A 1 , A 2 , A with inclusion maps f i : A 0 ֒ → A i , g i : A i ֒ → A , as follows. If M 3 belo ngs to V , let − → A b e the diagra m on the left hand s ide of Figure 5.6. If M 3 do es not b elong to V , let − → A b e the diagr am on the r ight ha nd side of Figur e 5.6. Obser ve that − → A is alwa ys a squar e o f ﬁnite lattices in V . It follo ws from t he comments following D eﬁnition 3-5 .2 that kur( { 0 , 1 } 2 ) ≤ 2 (in fa c t k ur( { 0 , 1 } 2 ) = 2), th us the relation ( ℵ 1 , < ℵ 0 ) ❀ { 0 , 1 } 2 holds. As 136 5 Congruence-preserving ext ensions a 0 a 0 a 0 a 0 a 1 a 1 a 2 a 2 A 0 A 1 A 2 A 0 0 0 0 1 1 1 1 a 0 a 0 a 0 a 1 a 1 a 2 a 2 A 0 A 1 A 2 A 0 0 0 1 1 a 0 0 1 1 f 1 f 1 f 2 f 2 g 1 g 1 g 2 g 2 Fig. 5. 6 Two squares of ﬁnite lattices this argument relies on results from our pap er [23], we shall present a direct pro of. Let F : [ ω 1 ] <ω → [ ω 1 ] <ω be an isotone map. It follo ws easily fro m K ura- towski’s characterization of ℵ 1 given in Kuratowski [41] (see also [10, The - orem 46.1]) that there are dis tinct α 0 , α 1 ∈ ω 1 \ F ( { 0 } ) such that α 0 / ∈ F ( { 0 , α 1 } ) and α 1 / ∈ F ( { 0 , α 0 } ). Pick α ∈ ω 1 \  { α 0 , α 1 } ∪ F ( { α 0 } ) ∪ F ( { α 1 } )  . Then the map f : { 0 , 1 } 2 → ω 1 , (0 , 0 ) 7→ 0 , (1 , 0) 7→ α 0 , (0 , 1) 7→ α 1 , (1 , 1) 7→ α is one-to- o ne, and F ( f “( { 0 , 1 } 2 ↓ p )) ∩ f “( { 0 , 1 } 2 ↓ q ) ⊆ f “ ( { 0 , 1 } 2 ↓ p ) , for all p , q ∈ { 0 , 1 } 2 such that p ≤ q . This co mpletes the pro of of our claim. Hence, by Lemma 3-5.5, the square { 0 , 1 } 2 has an ℵ 0 -lifter ( X , X ) where X has ca rdinality ℵ 1 . As − → A is a diagr am of bounded lattices and 0 , 1-lattice embeddings in V , the condensate L := F ( X ) ⊗ − → A is a b ounded lattice in V . It has cardina lit y at most ℵ 1 —one c a n prov e that it is exactly ℵ 1 but this will not ma tter here. W e apply CLL (Lemma 3- 4.2) to the ℵ 0 -larder given by Prop ositions 5- 2.2 and 5 -4.3 (via the tr ivial Prop ositio n 3-8.3), namely ( MAlg , Metr ∗ , M etr , M Alg ﬁn , M etr ∗ ﬁn , M etr ⇒ , Φ, Ψ ) , where MAlg ﬁn denotes the class of all ﬁnite alg ebras and Φ and Ψ ar e the functors int ro duced in Sections 5-4 and 5-2, resp ectively . Supp ose that L ♮ is 5-5 CPCP- retracts and CPCP-extensions 137 a CP C P -retra c t. By deﬁnition, this means that there are a semilattice-metric cov er B and a double a rrow χ : Ψ ( B ) ⇒ Φ ( F ( X ) ⊗ − → A ). Now it fo llows from CLL (Lemma 3-4 .2) that there ar e a square − → B from Metr ∗ and a double arrow − → χ : Ψ − → B ⇒ Φ − → A . How ever, Φ − → A is a diagram of semilattice-metr ic spaces of the sor t de- scrib ed in the a ssumptions of Lemma 5-3.1, with α := k 0 = a 0 k A 0 and β := k a 0 = 1 k A 0 : for exa mple, b oth f i = Con c f i are is omorphisms, with Con c A 0 ∼ = Con c A 1 ∼ = Con c A 2 ∼ = { 0 , 1 } 2 , and α and β corres p o nd, throug h those isomorphisms, to the atoms (1 , 0) and (0 , 1 ) of the squar e { 0 , 1 } 2 , which are orthog onal in { 0 , 1 } 2 . On the o ther ha nd, δ A ( a 1 , a 2 ) 6 = 0 in b oth cases : a contradiction. ⊓ ⊔ As the free b ounded lattice on ℵ 1 generator s within V is a homomor phic image of th e free lattice on ℵ 1 generator s within V , t he result of Theo r em 5- 5 .5 applies to the latter lattice as well. This commen t also applies to the follo wing immediate coro lla ry . Corollary 5 - 5.6. L et V b e a n ondistributive variety of lattic es. Then the fr e e lattic e ( r esp. fr e e b ounde d lattic e ) on ℵ 1 gener ators within V has no c ongruenc e-p ermutable, c ongruenc e-pr eserving exten sion. Remark 5- 5 .7. It is not har d to v erify that the class of all lattices with 3-p ermutable congruences is clo sed under directed colimits and ﬁnite di- rect pro ducts (the veriﬁcation for ﬁnite direct pro ducts requires co ngruence- distributivity). As A 0 , A 1 , A 2 , a nd A are all b ounded lattices with 3- per mut able cong ruences while the f i and g i are 0 , 1-la ttice homomorphisms, it follows that the condensate L := F ( X ) ⊗ − → A of the pro of of Theore m 5-5.5 is also a bo unded lattice with 3-p ermutable cong ruences. Nevertheless, this lattice has no CPCP-e x tension. Chapter 6 Larders from v on Neumann reg u lar rings Abstract. The assignment that sends a regular ring R to its la ttice of all principal rig ht ideals can be natura lly extended to a functor, denoted b y L (cf. Sec tion 1- 1.2). An earlier o ccur rence of a co ndensate-like construction is provided b y the pro of in W ehrung [66, Theorem 9.3]. This construction tur ns the non-liftabilit y of a certain 0 , 1-lattice endomorphism from M ω (cf. E x- ample 1-1 .9) to a non-co or dinatizable, 2-distributive co mplemen ted mo dular lattice, of car dinality ℵ 1 , with a spanning M ω . Thus the idea to adapt the functor L to our la rder co nt ext is natural. The present c hapter is desig ned for this goal. In a ddition, it will pave the categorica l wa y for solv ing, in the second author’s pap er [69], a 19 6 2 proble m by J´ o ns son. 6-1 Ideals of regular rings and of lattices In the present section w e shall establish a few basic facts a bo ut regular rings without unit and their ide a ls, some o f whic h, although they are well-kno wn in the unital c ase, hav e not a lwa ys, to o ur knowledge, appe a red anywhere in print in the non-unital case. All our r ing s will b e a s so ciative but no t necessarily unital. Lik ewise, o ur ring homo mo rphisms will not neces sarily preser ve the unit even if it exists. An element b in a ring R is a quasi-inverse of an elemen t a if a = aba . W e s ay that R is (v on Neumann) r e gular if every elemen t of R has a quasi-inv erse in R . A reference for unital r egular r ings is Go o dearl’s mono graph [2 4]. W e s ta rt with the following well-known fact, which is co ntained, in the regular case, in [4 5, Section VI.4]. How ever, the result is v alid for general rings. W e include a pro o f for conv enience. Lemma 6 - 1.1 (fol klore). L et a and b b e idemp otent elements in a ring R . Then t he fol lowing ar e e quivalent: (i) aR and bR ar e isomorphic as ( non ne c essarily unital ) right R -mo dules. 139 140 6 Larders from v on Neumann r egular rings (ii) Ther e ar e mutual ly quasi-inverse elements x, y ∈ R such that a = y x and b = xy . (iii) Ther e ar e x, y ∈ R such that a = y x and b = xy . Pr o of . (i) ⇒ (ii). Let f : aR → bR b e a n isomo rphism of right R -mo dules . The element x := f ( a ) belong s to bR and x = f ( aa ) = f ( a ) a = xa b elong s to Ra , so x ∈ bRa . Likewise, the element y := f − 1 ( b ) b elongs to aRb . F urthermo re, a = f − 1 f ( a ) = f − 1 ( x ) = f − 1 ( bx ) = f − 1 ( b ) x = y x , likewise b = xy . Finally , xy x = bx = x (we use xy = b and x ∈ bR ), likewise y xy = y . (ii) ⇒ (iii) is trivia l. Finally , assume that (iii) ho lds. F or each t ∈ aR , the element xt = xat = xy xt = b xt be longs to bR . It follows that the assignment ( t 7→ xt ) deﬁnes a homomor phism f : aR → bR of rig ht R -mo dules. Likewise, the assignment ( t 7→ y t ) deﬁnes a homomo rphism g : bR → aR of right R - mo dules. F rom gf ( a ) = y xa = a 2 = a it follows that g ◦ f = id aR ; likewise f ◦ g = id bR , so f and g are mutually inv erse. ⊓ ⊔ Corollary 6 - 1.2. L et I b e a two-side d ide al of a ring R and let a and b b e idemp otent element s of R such that aR and bR ar e isomorphic as right R -mo dules. Then a ∈ I iff b ∈ I . Pr o of . It follows from Lemma 6-1.1 that there are mutually quasi- inv erse elements x, y ∈ R such that a = y x and b = xy . If a ∈ I , then y = y xy = ay also b elongs to I , hence b = xy be lo ngs to I . ⊓ ⊔ F or a regular ring R , we set L ( R ) := { xR | x ∈ R } = { xR | x ∈ R, x 2 = x } . Although our rings may not b e unital, it is the case that ev ery elemen t a in a regula r ring R b elongs to aR : indeed, if b is a quasi-inv erse of a , then a = aba ∈ aR . The following r esult is prov ed in F ryer and Halp erin [14, Sectio n 3.2 ]. Prop ositio n 6-1.3. L et R b e a r e gular ring and let a , b b e idemp otent ele- ments in R . F urthermor e, let u b e a quasi-inverse of b − ab . Then t he fol lowing statements hold: (i) Put c := ( b − ab ) u . Then aR + bR = ( a + c ) R . (ii) Supp ose that b 2 = b and put d := u ( b − ab ) . Then aR ∩ bR = ( b − bd ) R . (iii) If aR ⊆ bR , then bR = aR ⊕ ( b − ab ) R . Conse quently, L ( R ) , p artial ly or der e d by c ontainment, is a se ctional ly c om- plemente d sublattic e of the lattic e of al l right ide als of R . In p articular, it is mo dular. 6-1 Ideals of regular rings and of lattices 141 A lattice is c o or dinatizable if it is isomorphic to L ( R ) for some regular ring R . Hence every co ordinatiza ble lattice is sectionally co mplement ed a nd mo dular. The conv erse is false, the sma lle st counterexample b eing the nine- element lattice of length tw o M 7 . The following co nsequence of Prop ositio n 6-1.3 is obse r ved in Micol’s the- sis [4 6]. Corollary 6 - 1.4. L et R and S b e r e gular rings and let f : R → S b e a ring homomorph ism. Then t her e exists a unique map L ( f ) : L ( R ) → L ( S ) such that L ( f )( xR ) = f ( x ) S for e ach x ∈ R , and L ( f ) is a 0 - lattic e homomor- phism. F urthermor e, the assignment ( R 7→ L ( R ) , f 7→ L ( f )) deﬁnes a fun ctor fr om the c ate gory Reg of al l r e gular rings and ring homomo rphisms to the c ate gory SCML of al l se ct ional ly c omplemente d mo dular la tt ic es and 0 -lattic e homomorph isms. The following us eful result is folklor e . Lemma 6 - 1.5. L et R b e a ( not ne c essarily unital ) ring, let M b e a right R -mo dule, and let A and B b e submo dules of M . If A and B ar e p ersp e ctive in the submo dule latt ic e Sub M of al l s u bmo dules of M , then A and B ar e isomorphi c. If A ∩ B = { 0 } , t hen the c onverse holds. Pr o of . If C is a submo dule o f M suc h that A ⊕ C = B ⊕ C , then the map f : A → B that to each a ∈ A ass o ciates the unique element of ( a + C ) ∩ B is a n iso morphism. Conv er s ely , if A ∩ B = { 0 } and f : A → B is an isomor- phism, then, setting C := { x − f ( x ) | x ∈ A } , we obtain that A ⊕ C = B ⊕ C . ⊓ ⊔ Observe, in the second part of the pr o of ab ov e, that if A and B ar e ﬁnitely generated, then so is C . This will be used (for unital R ) in case M = R R , that is, R viewed as a right mo dule ov er itself. The following result is observed in the unital case in [62, Lemma 4.2], how ever the a rgument present ed there works in gener al. Lemma 6 - 1.6. L et R b e a r e gular ring. Then an ide al I of L ( R ) is neutr al iff it is close d under isomorp hism, that is, X ∈ I and X ∼ = Y ( as right R -mo dules ) implie s t hat Y ∈ I , for al l X, Y ∈ L ( R ) . Pr o of . If I is closed under isomorphism, then it is closed under p ers pec tivit y (cf. Le mma 6- 1.5), thus, as L ( R ) is sectionally co mplement ed mo dular, I is neutral (cf. Section 1-2 .3). Conv ersely , assume that I is neutral and let X , Y ∈ L ( R ) such that X ∼ = Y and X ∈ I . It follows fro m Prop osition 6-1.3 that there exists Y ′ ∈ L ( R ) such that ( X ∩ Y ) ⊕ Y ′ = Y . As X belongs to I , so do es X ∩ Y . As X ∼ = Y and Y ′ is a right ideal of Y , there exists a right idea l X ′ of X such that X ′ ∼ = Y ′ . As X ′ ∩ Y ′ ⊆ X ∩ Y ∩ Y ′ = { 0 } , it follows from Lemma 6- 1.5 that X ′ ∼ Y ′ . 142 6 Larders from v on Neumann r egular rings F rom X ′ ⊆ X and X ∈ I it follows that X ′ ∈ I , but X ′ ∼ Y ′ and I is a neutral ideal of L ( R ), thus Y ′ ∈ I . Therefo r e, Y = ( X ∩ Y ) + Y ′ belo ngs to I . ⊓ ⊔ The following result follows from the pro of of [11, Le mma 2]. It enables to reduce many pr oblems ab out regular r ings to the unital case. W e include a pro of for conv enience. Lemma 6 - 1.7 (F aith and Utumi). L et R b e a r e gular ring. Then every ﬁnite su bset X of R is c ontaine d into eRe , for some idemp otent e ∈ R . Pr o of . It follows from Prop os ition 6-1 .3(i), applied to the oppos ite ring of R , that there exists an idemp otent f ∈ R suc h that R X ⊆ Rf . It follows from Prop ositio n 6-1.3(i), applied to R , tha t there exists an idemp otent g ∈ R such that X R + f R ⊆ g R . Set e := f + g − f g . Then e 2 = e while f e = f and eg = g , so X ⊆ R f = Rf e ⊆ Re and X ⊆ g R = eg R ⊆ eR , and s o X ⊆ eRe . ⊓ ⊔ The following result is o bs erved, in the unital case, in the pro of of [62, Theorem 4.3]. The non-unital case requires a non-trivial use of Lemma 6-1.7, that we present here. Lemma 6 - 1.8. L et R b e a r e gular ring and let I b e a neutr al ide al of L ( R ) . Then xR ∈ I implies that y xR ∈ I , for al l x, y ∈ R . Pr o of . Let y ′ be a quasi-inv er s e of y . It follows from Lemma 6-1.7 that there exists an idemp otent element e o f R such tha t { x, y } ⊆ eR e . It follows fro m Prop ositio n 6-1.3 that Y := xR ∩ ( e − y ′ y ) R belo ngs to L ( R ). As Y ⊆ xR and L ( R ) is sectionally complemented, ther e exists Z ∈ L ( R ) such that Y ⊕ Z = xR . Denote by f : xR ։ y xR the le ft multiplication by y , and let t be an elemen t of ker f := f − 1 { 0 } . Then ( e − y ′ y ) t = et − y ′ y t = et − 0 = t , thus t ∈ Y . Conv ersely , let t ∈ Y . T her e ex ists t ′ ∈ R such that t = ( e − y ′ y ) t ′ , so y t = y ( e − y ′ y ) t ′ = ( y e − y y ′ y ) t ′ = ( y − y ) t ′ = 0, th us t ∈ ker f . Consequently , Y = ker f , and so f induces an isomorphism fro m Z onto y xR . F rom Z ⊆ xR it follows that Z ∈ I , and thus, by Lemma 6-1 .6, y xR ∈ I . ⊓ ⊔ The following r esult is stated, in the unital case, in [62, Theorem 4.3]. The pro of prese nted there, ta king fo r granted the results of Coro llary 6 -1.2, Lemma 6-1 .6, and Lemma 6-1.8, trivially extends to the non-unital cas e. W e denote by Id R the la ttice of all t wo-sided ideals of a ring R , and b y NId L the la ttice of all neutral idea ls of a lattice L (that it is indeed a lattice follows from [2 8, T he o rem II I.2 .9], s ee also [29, Theorem 259 ]). Prop ositio n 6-1.9. L et R b e a r e gular ring. Then one c an deﬁne mutual ly inverse lattic e isomorphisms ϕ : NId L ( R ) → Id R and ψ : Id R → NId L ( R ) by t he fol lowing rule: ϕ ( I ) := { x ∈ R | xR ∈ I } , for e ach I ∈ NId L ( R ) ; ψ ( I ) := L ( R ) ↓ I , for e ach I ∈ Id R . 6-1 Ideals of regular rings and of lattices 143 Lemma 6 - 1.10. L et R b e a re gular ring and let I b e a neutr al ide al of L ( R ) . Set I := { x ∈ R | xR ∈ I } . Then I is a two-side d ide al of R and ther e ex ists a u nique map ψ : L ( R ) / I → L ( R/I ) such that ψ ( xR/ I ) = ( x + I )( R /I ) , for e ach x ∈ R , (6-1.1) and ψ is a lattic e isomorphism. Pr o of . It follows from Prop os ition 6-1.9 that I is a tw o-sided ideal of R . Denote by p : R ։ R/ I and π : L ( R ) ։ L ( R ) / I the resp ective ca nonical pro jections. Then the required condition (6-1 .1) is equiv alent to ψ ◦ π = L ( p ). The uniqueness s ta temen t o n ψ follows fro m the surjectivity of π . As L ( R ) is sectionally complemented, every congr uence of L ( R ) is determined b y the congruence class of zero, which is a neutr a l ideal; hence, in order to prove that there exists a map ψ satisfying (6-1.1) and that this map is a lattice embedding, it suﬃces to prov e that π − 1 { 0 } = L ( p ) − 1 { 0 } . F or ea ch x ∈ R , π ( xR ) = 0 iff xR ∈ I , iff x ∈ I , iff ( x + I )( R/I ) = 0, that is, L ( p )( x R ) = 0, whic h prov es our c la im. Fina lly , the surjectivit y of ψ follows from the surjectivity of L ( p ) together with the equa tion ψ ◦ π = L ( p ). ⊓ ⊔ The following r esult is folklore. W e include a pro o f for convenience. Prop ositio n 6-1.11. The funct or L pr eserves al l smal l dir e cte d c olimits. Pr o of . Let ( R , f i | i ∈ I ) = lim − →  R i , f j i | i ≤ j in I  be a direc ted colimit c o- cone in Reg . It follows fro m the results of Section 1-2.5 that the follo wing statements hold: R = [ (rng f i | i ∈ I ) , (6-1.2) ker f i = [  ker f j i | j ∈ I ↑ i  , for each i ∈ I . (6-1.3) It follows from (6 -1.2) that L ( R ) = S (rng L ( f i ) | i ∈ I ). F urther, for each i ∈ I , as the la ttice L ( R i ) is sectionally complemen ted, every co ngruence of L ( R i ) is determined b y the congruence c lass of zero (cf. [28, Section I I I.3], see a lso [29, Theorem 272]), thus, in or der to verify the equation Ker L ( f i ) = [  Ker L ( f j i ) | j ∈ I ↑ i  (where we set Ker g := { ( x, y ) ∈ D × D | g ( x ) = g ( y ) } , fo r e ach function g with domain D ), it is suﬃcient to verify the equa tion L ( f i ) − 1 { 0 } = [  L ( f j i ) − 1 { 0 } | j ∈ I ↑ i  . How ever, this follows immediately from (6-1 .3). By the re s ults of Sec- tion 1-2.5, we obtain that 144 6 Larders from v on Neumann r egular rings ( L ( R ) , L ( f i ) | i ∈ I ) = lim − →  L ( R i ) , L ( f j i ) | i ≤ j in I  . This concludes the pro o f. ⊓ ⊔ 6-2 Right larders fr om regular rings In the present se c tion we shall construct many rig ht lar de r s fr om ca tegories of r egular rings satisfying a few additional clos ure pro pe rties (cf. Theo- rem 6-2 .2). All the la rders consider ed in this section will b e pro jectable, and our ﬁrst lemma deals with pro jectabilit y witness e s . Lemma 6 - 2.1. L et R b e a r e gular ring and let L b e a se ctional ly c om- plemente d mo dular lattic e. Then every surje ctive 0 -lattic e homomorphism ϕ : L ( R ) ։ L has a pr oje ctabil ity witn ess of the form ( a, ε ) , wher e a is a surje ctive ring homomorphism with domain R . Pr o of . As the ideal I := ϕ − 1 { 0 } is neutra l in L ( R ) and L ( R ) is section- ally complemented mo dular , ϕ induces an iso morphism β : L ( R ) / I ։ L . By Pr op osition 6-1.9, I := { x ∈ R | xR ∈ I } is a t wo-sided ideal of R . Denote by π : L ( R ) ։ L ( R ) / I and a : R ։ R/I the resp ective canonical pro jections. By Lemma 6- 1.10, there exists a unique la ttice iso mo rphism α : L ( R/ I ) ։ L ( R ) / I such that π = α ◦ L ( a ). Hence ε := β ◦ α is an isomorphism fr om L ( R/I ) onto L , and ϕ = ε ◦ L ( a ). In order to prove that ( a, ε ) is a pro jectabilit y witness for ϕ : L ( R ) ։ L , it rema ins to prov e that for e very r e gular ring X , every ring homomo rphism f : R → X , and every 0-la ttice ho momorphism η : L ( R /I ) → L ( X ) such that L ( f ) = η ◦ L ( a ), there exists a ring ho momorphism g : R/I → X such that f = g ◦ a a nd η = L ( g ). The assumption L ( f ) = η ◦ L ( a ) mea ns that η  ( x + I )( R/I )  = f ( x ) X , for each x ∈ R . In particular, k er a = I ⊆ ker f , th us there exists a unique ring homomor- phism g : R/I → X such that f = g ◦ a . It follows that η ◦ L ( a ) = L ( f ) = L ( g ) ◦ L ( a ) , but L ( a ) is sur jective, th us η = L ( g ). ⊓ ⊔ In Theorem 6-2.2, for a class C of s tructures and an inﬁnite ca rdinal λ , we denote by C ( λ ) the clas s of all mem b ers of C with λ -small universe. Our next result will provide us with a large cla s s of rig h t λ -larder s. Theorem 6-2. 2. L et R b e a ful l sub c ate gory of the c ate gory Reg of re gular rings and r ing homomorphi sms and let λ b e an inﬁ n ite c ar dinal. We assume the fol lowing: 6-2 Right larders f rom r egular rings 145 (i) R is close d u n der sm al l dir e cte d c olimits and homomorph ic ima ges. (ii) Every λ -smal l subset X of a memb er R of R is c ontaine d in some λ - smal l memb er of R c ontaine d in R . Denote by SCML ։ the su b c ate gory of SCML with the same obje cts and whose arr ows ar e the surje ctive lattic e homomorph isms. Th en the 6 -uple ( R , R ( λ ) , SCM L , SCM L ( λ ) , SCM L ։ , L ) is a pr oje ctable right λ -lar der. Pr o of . F or each R ∈ R ( λ ) , the lattice L ( R ) is λ -small, thus, by Prop o- sition 4-2.3 applied within the categor y of a ll lattices with zero with 0 - lattice homomorphisms, L ( R ) is weakly λ -presented. This completes the pro of o f (PRES λ ( R ( λ ) , L )). The pro jectabilit y statemen t follows trivially from Lemma 6 -2.1 tog ether with the closure of R under homomor phic imag es. It remains to verify (LS r cf ( λ ) ( R )), for each ob ject R ∈ R . Let L be a λ - small sectionally complemented modula r lattice, let I b e a cf ( λ )-small set, let ( u i : U i → R | i ∈ I ) b e a n I -indexed family of r ing ho mo morphisms with all card U i < λ , and le t ϕ : L ( R ) ։ L b e a surjectiv e 0-lattice homo mo rphism. W e s ha ll c o nstruct a monomor phism v : V ֌ R , with a λ -small ob ject V of R , with a ll u i E v and ϕ ◦ L ( v ) surjective. The ﬁrst parag raph o f the pro of of Lemma 6- 2.1 shows that we ma y assume that L = L ( R/J ), for some tw o- sided idea l J of R , and ϕ = L ( p ), wher e p : R ։ R/J denotes the canonica l pro jection. F rom card I < c f ( λ ) and all card U i < λ it follows that the set U := S ( u i “( U i ) | i ∈ I ) is λ -small. As L ( R/J ) = { ( x + J )( R /J ) | x ∈ R } is also λ -small, there exists a λ -small subs e t V of R containing U such that L ( R/J ) = { ( x + J )( R/J ) | x ∈ V } . (6-2.1) By assumption, there exis ts a λ -small R ′ ∈ R contained in R such that V ⊆ R ′ . Denote by v : R ′ ֒ → R the inclus io n ma p. F r om u i “( U i ) ⊆ R ′ it follows that u i E v , for each i ∈ I . As the ra nge of v contains V , the range of ϕ ◦ L ( v ) co ntains all elements of the form ( x + J )( R /J ), where x ∈ V , and th us, by (6-2.1), ϕ ◦ L ( v ) is s urjective. ⊓ ⊔ The as sumptions o f Theor em 6- 2.2, ab out the categor y R , are satisﬁed in t wo noteworthy cases: • R is the category of a ll reg ular r ing s. W e obtain that the 6 -uple ( Reg , Reg ( λ ) , SCM L , SCM L ( λ ) , SCM L ։ , L ) is a pro jectable right λ -larder for eac h unco unt able cardinal λ . This is used, for λ := ℵ 1 , in the second author’s pap er [69], to solv e a 1962 problem 146 6 Larders from v on Neumann r egular rings due to J´ onsson, b y ﬁnding a non-coo rdinatizable sec tio nally complemen ted mo dular lattice, o f cardinality ℵ 1 , with a large J´ onsso n four -frame. • R is the categor y Lo cMat F of all rings that a re lo c al ly matricial ov er some ﬁeld F . By deﬁnitio n, a ring is matricial over F if it is a ﬁnite dir ect pro duct of full matrix rings over F , and lo cally ma tr icial ov er F if it is a directed colimit o f matricial r ings ov er F . Then Theo rem 6-2.2 yields that the 6-uple ( Lo cMat F , Lo cMat ( λ ) F , SCM L , SCM L ( λ ) , SCM L ։ , L ) is a pro jectable r ight λ -larder , for every inﬁnite cardinal λ s uch that cf ( λ ) > ca r d F (in pa rticular, in cas e F is ﬁnite, ev ery inﬁnite car dinal λ works). Chapter 7 Discuss ion The discussio n undertaken, in Chapter 6, about the functor L o n regula r rings, c an be mimick ed for the functor V ( nonstable K-the ory ) introduced in Ex ample 1-1.3, restr icted to (von Neuma nn) regular rings. It is a funda- men tal op en problem in the theory o f regular rings whether ev ery conical reﬁnement monoid, o f c a rdinality at most ℵ 1 , is isomor phic to V ( R ) for some regular ring R , cf. Go o dearl [25], Ara [3]. Due to counterexamples dev elop ed in W ehrung [61], the s ituation is hop eless in ca rdinality ℵ 2 or ab ov e. Recent adv ances on those matters, for more general clas ses o f r ing s suc h as exchange rings but also for C*-algebr as , can b e found in W ehrung [7 1]. Now let us discuss some open problems. In our mind, the most fundamental op en problem raised by the statement of CLL is the extension of the class of p osets P , or even categorie s that are not po sets, for which a weak form of CLL could hold. F or ex a mple, w e establish in Corolla ry 3 -7.2 a w ea k for m of CLL, v alid for an y almos t join-semilattice (assuming la rge enoug h cardinals), regar dless of the existence of a lifter. Problem 1. Is there a weak form of CL L that would work for an arbitrary po set, or even just the rightmost p oset of Figure 2.1, pag e 47? This problem is rela ted to the repr e s ent atio n problem of distributive alg e- braic lattices as co ngruence lattices of member s o f a congr uence-distributive v ariet y—for ex ample, now that lattices are r ule d out [6 7], majority algebr as . Is every distributive algebraic lattice isomo r phic to the co ngruence lattice of so me ma jor ity algebr a? If this w ere the ca s e, then, beca use o f the results of [60], the somewhat lo osely form ulated Problem 1 could hav e no reasona ble po sitive answer. In our next tw o problems, we ask whether the assumptions of lo cal ﬁnite- ness and strong cong ruence-pro per ness can b e removed fr om the statements of b oth Theo rem 4-9 .2 a nd Theorem 4-9.4. Positive results in that direction can b e fo und in the ﬁrst author’s pap er [20]. 147 148 7 Discussi on Problem 2. Let A and B b e quas iv arieties in ﬁnite (p ossibly diﬀerent) lan- guages, a nd let P b e a nontrivial ﬁnite a lmost join-se mila ttice with zero. Prov e that if there exis ts a P -indexed diagr am − → A =  A p , α q p | p ≤ q in P  of ﬁnite members of A such that Con A c − → A has no lifting, with r e s pe c t to Con B c , in B , then crit r ( A ; B ) ≤ ℵ kur 0 ( P ) − 1 . The following problem asks for an ambitious g eneralizatio n of Theo- rem 4-9.4 (the Dichotom y Theorem): it asks not only for relaxing the as- sumptions on the quasiv arieties A and B , but a lso for an improv ement of the cardinality bo und. It can also b e view ed as an (ultimate?) recasting o f the Critical Poin t Conjectur e formulated in T ˚ uma and W ehrung [59]. F or a partial p ositive solution, see Gillib ert [2 2]. Problem 3. Let A and B b e quasiv arieties on ﬁnite (p ossibly diﬀerent) lan- guages. Prove that if Con c , r A 6⊆ Con c , r B , then crit r ( A ; B ) ≤ ℵ 2 . Our next problem is formulated in the same con text as P roblems 2 and 3. Problem 4. Is there a recursive algo rithm that, giv en ﬁnitely generated qua- siv arieties A and B in ﬁnite languag es, outputs a code for their rela tive critica l po int crit r ( A ; B ) (e.g., the pa ir (0 , n ) if the cr itical point is n and the pair (1 , n ) if the cr itical p oint is ℵ n )? T o illustrate the extent of our ignorance with resp ect to Problem 4, we do not even know whether there ar e ﬁnitely generated lattice v arieties A a nd B such that crit( A ; B ) is eq ua l to ℵ 1 in one universe of set theory but to ℵ 2 in another one. W e do not even know whether the critical point betw een t wo ﬁnitely generated lattice v arieties is absolute (in the set-theor etical sense)! Our next problem asks for a functorial extension of the Pudl´ ak-T ˚ uma Theorem [51], w hich states that every ﬁnite lattice embeds in to so me ﬁnite partition lattice. It inv olves the formulation o f lattice em b eddings into parti- tion lattices via the semila ttice-v alued distances introduced in J ´ onsson [36]. W e r emind the reader that Metr denotes the ca teg ory of all semilattice- metric spaces (cf. Deﬁnition 5 -1.1) and Metr ﬁn denotes the full sub cate- gory of a ll ﬁnite members of M etr . F urther, we denote by Sem ﬁn ∨ , 0 (resp., Sem ﬁn , inj ∨ , 0 ) the categ ory of all ﬁnite ( ∨ , 0 )-semilattices with ( ∨ , 0)- homo- morphisms (resp., ﬁnite ( ∨ , 0 )- s emilattices with ( ∨ , 0)-emb eddings) and by Π : Metr ﬁn → Sem ﬁn ∨ , 0 the forgetful functor. Problem 5. Do es there ex ist a functor Γ : Sem ﬁn , inj ∨ , 0 → Metr ﬁn such that δ Γ ( S ) is a surjective V-distance, for ea ch ﬁnite ( ∨ , 0)-semilattice S , and Π ◦ Γ is isomorphic to the identit y? In his pa per [48], P lo ˇ sˇ cica extends the metho ds intro duced in [67] (for solving CLP) and [54] (for improving the cardinality b ound) to prov e that if F denotes the free ob ject, in the v ariety of all bounded lattices gener- ated by M 3 , on ℵ 2 generator s, then fo r every po s itive integer m and every 7 Discussi on 149 lattice L with m -p e rmutable congr uences, the congrue nce lattices o f F and of L are no t isomorphic. W e ask whether there is a similar strengthening o f Corollar y 5-5.6: Problem 6. Denote by F the free o b ject on ℵ 1 generator s in a nondis- tributive v ariety V o f (b ounded) lattices and let m b e a p ositive integer. Prov e that F has no congruence- preserving extension to a lattice with m - per mut able cong ruences. The answer to Pro blem 6 is known to b e p ositive in case V con tains a s a mem b er either M 3 or a few of the successor s of N 5 with resp ect to the v ariet y order (cf. Gillib e r t [21]), but it is not known, for example, in case V is the v ariety generated by N 5 . Our next problem a sks for a nearly as stro ng as possible diagra m extension of the Gr¨ atzer-Schmidt T he o rem [31]. A partial positive answer is given in Theorem 4-7.2. Problem 7. Do es there exist a functor Γ , from ( ∨ , 0)-semilattices with ( ∨ , 0)-e mbeddings , to the category MInd o f a ll monotone- indexed structur es, such that Con c ◦ Γ is iso morphic to the identit y? Problem 8. Let λ be a n inﬁnite car dinal. Is every λ -liftable p oset well- founded? By Le mma 3-6.2, Problem 8 amounts to determining whether ( ω + 1 ) op has a λ -lifter. By Co rollary 3-6.4, this does not hold for all v a lues of λ , in particular in case λ = (2 ℵ 0 ) + . F or further comments ab out Pro blem 8, see Remark 3- 7.3. A related problem is the following. Problem 9. Let P b e a p oset and let λ b e an inﬁnite cardinal. If P has a λ -lifter, do es it have a λ -lifter ( X, X ) such that X is a lower ﬁnite almost join-semilattice and X is the collectio n of a ll extre me ideals of X ? If P is well-founded, then so is the s et of all extre me ideals of X (partially ordered b y co ntainmen t). In particular, if P roblems 8 and 9 b oth hav e a po sitive answer, then every λ -liftable poset is well-founded and it has a λ - lifter ( X , X ) with X a low er ﬁnite almost join- s emilattice and X , being the collection of all extreme ideals of X , is also well-founded. Op en problems r elated to those on our list can be found in v a rious items from o ur biblio graphy , for example [18, 19, 23, 59, 60, 64, 65, 66, 6 7]. References 1. S. Abramsky and A. Jung, Domain the ory . Handb ook of logic in computer science, V ol. 3 , 1–168, Handb. Log. Comput . Sci., 3 , Oxford Univ. Press, N ew Y ork, 1994. 2. J. Ad´ amek and J. Rosic k ´ y, “Locally Present able and Accessible Categories”. London Mathematical So ciet y Lecture Note Series 189 . C ambridge University Press, Cam- bridge, 1994. xiv+316 p. ISBN: 0-521-42261-2 3. P . Ara, The r e alization pr oblem for von N e umann r e gular rings . Ring theory 2007, 21–37, W orl d Sci. Publ., Hack ensac k, NJ, 2009. 4. P . Ara and K. R. Go o dearl, L e avitt p ath algebr as of sep ar ate d gr aphs , preprin t. Av ail- able online at http://arxiv. org/abs/1004.4979 . 5. G. M. Bergman, Copr o ducts and some universal ring c onstructions , T rans. Am er . Math. So c. 200 (1974), 33– 88. 6. G. M. Bergman and W. Dic ks, Universal derivations and universal ring c onstructions , Pa ciﬁc J. Math. 79 (1978), 293 –337. 7. G. Birkhoﬀ, “Lattice Theory”. Corrected reprin t of the 1967 third edition. Ameri- can Mathematical Society Collo quium Publications, V ol. 25 . American Mathematical Society , Pro vidence, R.I., 1979. vi+418 p. ISBN: 0-8218-1025-1 8. C. 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Cam bridge Universit y Press, Cambridge, 1987. iv+227 p. IS BN: 0-521-34832-3 . Out of print, av ailable online at http://www.ma th.hawaii.edu/~ralph/Commutator/ 14. K. D. F ryer and I. Halp erin, The von Neumann co or dinatization theo r em for c omple- mente d mo dular lattic es , Acta Sci. M ath. (Szeged) 17 (1956), 203–249. 15. P . Gabriel and F. Ulmer, “Lok al Pr¨ asen tierbare Kategorien” (German). Lecture Notes in Mathematics 221 . Springer-V erlag, Berli n - New Y ork, 1971 . v+200 p. 16. G. Gierz, K. H. H ofmann, K. Keimel, J. D . Lawson, M. M islov e, and D. S. Scot t, “Con- tin uous Lattices and Domains.” Encyclopedia of Mathematics and its Applications, 93 . Camb ri dge Universit y Press, Cambridge, 2003. xxxvi+591 p. ISBN: 0-521-80338-1 151 152 References 17. P . Gillib ert, “Poin ts cri tiques de couples de v ari´ et ´ es d’alg` ebres” (F rench ). Doctorat de l’Universit ´ e de Caen, December 8, 2008. 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Sym b ol Index A ( p ) , 48 A ♭ , f ♭ , 127 , 130, 134, 135 A ♮ , f ♮ , 132 , 133–1 36 ℵ α , 7–9, 14–2 1, 24 , 25, 34, 64, 65, 71, 73, 74, 81–84, 86, 87, 89, 96, 109, 112, 116–123, 125, 127, 135–137, 139, 145–149, 153 A /I ( I ideal of A ∈ Bool P ), 56 A /I ( A ∈ Metr , I ideal of ˜ A ), 133 A / θ , where θ ∈ Con A , 91 ar( s ), 30 , 90–93 , 97, 99, 100, 103 , 105 ( κ, <ω, λ ) → ρ , 22, 74, 83 , 84, 86, 109 κ → ( θ ) <ω 2 , 84 x → ( y, z ) 5 , 74 ( κ, <λ ) ❀ P , 21, 23, 73 , 74, 76, 79, 80, 84, 87, 117, 135 Q ⇓ X , 26 Q ↓ X , 26 Q  X , 26 Q ⇑ X , 26 Q ↑ X , 26 Q ⇈ X , 26 C ′ ↓ C , 28 At A , 25 f : A ֒ → B , 28 , 56, 65, 66, 77– 79, 82, 110, 113, 120, 123, 134, 135, 145 f : A ֌ B , 27 , 64, 68, 69, 71, 73, 76–78, 82, 88, 145 f : A ։ B , 28 , 42, 43, 56, 72, 91–93, 99, 100, 102, 104, 113, 132, 133, 142– 146 f : A ⇒ B , 8 , 28 , 30, 32, 68, 71, 72, 85– 88, 106, 110, 122, 128, 130, 134, 135, 137 f : A . ⇒ B , 28 , 30, 85, 111, 129, 130 f : A . → B , 28 , 43, 72, 117, 119 β / α , 92 B m ( 6 r ), 74 Bo ol , 25 BT op , 25 Bo ol P , 21, 42, 48 , 49–56, 59, 62, 63 BT op P , 21, 48 , 49, 50, 55, 56 k x = y k V A , 99 k R ( x 1 , . . . , x n ) k V A , k R − → x k V A , 99 C ( λ ) , 144 Clop X , Clop X , Clop f , 25 , 50 , 51 (CLOS( A )), 65 , 71, 72, 87, 88, 133 (CLOS λ ( B † , B )), 69 , 70, 71, 88 (CLOS r λ ( S ⇒ )), 69 , 71, 88 cf ( α ), 24 C ′ ↓ C , 28 Con c functor, 100 , 109–111 Con c A , Con c f , 13, 14, 16–19, 97 , 109–111, 116, 132–135, 137 Con c A , 17 , 18, 89 Con V c functor, 7, 23, 100 , 101, 104, 112, 116, 117, 119–123 Con V c A , Con V c f , 97 , 98, 100, 102–105, 112, 113, 116–123 , 148 Con c , r V , 7, 116 , 118, 121, 122, 148 Con A , 16, 91 , 93, 97–99, 101, 104, 121, 134, 135 Con V A , Con V f , 97 , 98–101, 104, 112, 114, 115, 117 1 A , 91 , 97, 98 0 A , 91 , 98, 105 V V , 32 , 33, 98 (CONT( Φ )), 65 , 71, 87, 133 (CONT λ ( Ψ )), 69 , 71, 88 crit( A ; B ), 17 , 18, 89, 116, 148 crit r ( A ; B ), 7, 8, 115, 116 , 118, 119, 121, 122, 148 Cst( A ), 30 155 156 Sym b ol Ind ex D ac , 16 δ u A , 62 dim( P ), 73 dom f , 24 ε q p , 57 , 63, 66 f “( X ), f “ X , 24 f − 1 X , 24 F L ( X ), 16 F( X ), 57 , 58, 59 F ( X ), 20–23 , 46, 57, 58 , 59, 61, 64–66, 71, 72, 84, 111, 117, 119, 123, 136, 137 f Y X , 57 , 59, 65–67 Id c R , R ring, 13 , 16 NId L , L l attice, 142 Id P , P poset, 21, 26 , 45, 48–50, 54, 83 Id R , R ring, 23, 142 Id s X , X norm-cov ering, 58 , 64, 81 J( P ), P poset, 26 κ + , κ + α , 24 Ker ϕ , 91 , 92, 97–99, 102, 103, 105, 113, 143 k er f , 142 kur( P ), 73 , 74, 80, 135 kur 0 ( P ), 73 , 74, 116–119, 148 [ X ] κ , 24 [ X ] <κ , 24 [ X ] 6 κ , 24 [ X ] <λ ▽ , 65 Lg( A ), 30 Lift( − → S ), 85 , 87 L f unctor, 139, 141 , 143, 145–1 47 L ( R ), L ( f ), 16, 20, 23, 24, 139, 140 , 141 , 142–145 Lo cMat F , 146 Lo cMat ( λ ) F , 146 (LS µ ( B )), 19, 71 , 88 (LS b µ ( B )), 68 , 69, 70 (LS r µ ( B )), 19, 88 , 110, 112, 122 , 128, 145 MAlg , 132 , 133, 134 , 136 MAlg 1 , 24, 109 , 110, 111 MAlg ( λ ) 1 , 109 , 110 MAlg ﬁn , 136 M n ( R ), M ∞ ( R ), 13 Max P , 25 Metr , 126 , 127–129 , 133, 134, 136, 148 Metr ⇒ , 127 , 128–1 30 Metr ﬁn , 127 , 148 Metr ∗ , 127 , 128, 136, 137 Metr ∗ ﬁn , 127 , 128, 136 MInd , 19, 23, 89, 90 , 91–99, 101, 102, 105, 109, 111, 132, 133, 149 MInd C , 98 ▽ i

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