Stone Duality for Skew Boolean Algebras with Intersections

Stone Dualit y for Sk ew Bo olean Algebras wit h In tersections Andrej Bauer F acult y of Mathematics and Physics Universit y of Ljubljana Andrej.Bau er@andrej.com Karin Cv etko-V ah F acult y of Mathematics and Physics Universit y of Ljubljana Karin.Cvet ko@fmf.uni-lj. si No v em b er 10, 2021 Abstract W e extend S tone du alit y b etw een generalized Boolean algebras and Boolean spaces, whic h are the zero-dimensional lo cally-compact H ausdorff spaces, to a n on-commutativ e setting. W e first show that the category of righ t-h anded skew Bo olean algebras with intersections is dual to the cate- gory of surjective ´ etale maps b etw een Bo olean spaces. W e then extend the duality t o skew Boolean algebras with intersections, and consider sever al v ariations in whic h the morphisms are restricted. Finally , we use the du- alit y to construct a right-handed skew Bo olean algebra without a lattice section. 1 In tro duction The fundament al ex a mple o f the kind o f duality we ar e interested in was es - tablished by Mar shall Stone [9, 10]: every Bo olean a lgebra cor resp onds to a zero-dimensio nal co mpa ct Ha usdorff space, or a Stone sp ac e for short, a s w ell as to a Bo ole an ring , whic h is a commutativ e ring of idempotents with a unit. In mo der n la nguage the duality is stated as equiv alence of categor ies of Bo olean algebras , Boolea n rings, and Stone s pa ces, where the later equiv alence is con- trav ariant. The duality has many generaliza tions, see [3]. Already in Sto ne’s second pap er [10, Theorem 8 ] we find an extensio n of duality to Bo ole an sp ac es , which are the zero-dimensio nal lo c al ly compact Ha usdorff spa ces. They cor re- sp ond to commutativ e rings of idemp otents, p ossibly without a unit, or equiv a- lent ly to gener alize d Bo ole an alge br as , which are like Bo olean a lgebras without a top element . Our contribution to the topic is a study of the n on- c ommu tative case. Among several v ar iations of non- commutativ e Bo olea n algebras we are a ble to pr ovide duality for skew Bo ole an algebr as with interse ctions (which w e call skew alge- br as ) b ecause they hav e a well-behav ed theory of idea ls. The pap er is org anized a s fo llows. In Sectio n 2 we recall the necess ary back- ground material ab o ut skew Bo o lean algebr a s, Bo olea n spaces, and ´ etale maps. In Section 3 we sp ell o ut the well-kno wn Stone dualit y fo r comm tutative a lge- bras. In Section 4 we establish the duality betw een r ight-handed skew algebra s 1 and skew Bo olean spaces, which w e then extend to the duality betw een skew algebras and rectangular skew Bo olea n spa ces. In Section 5 we further analyze the situation and consider several v ariations of the duality in which mor phis ms are restricted. In Section 6 we use the duality to co nstruct a right-handed skew a lgebra without a lattice s e ction. This answers negatively a hither to op en question of existence of such alg ebras. Ac kno wledgment. W e thank Jeff Egger, Mai Gehr ke, Ganna Kudrya vtsev a, Jonathan Leech, a nd Alex Simpson for discussing the topic with us a nd offering v alua ble advice. 2 Preliminary definitio ns In the first part of the section we rev iew basic concepts and no tation rega rding skew B o olean algebra s . In the seco nd part we recall some basic facts a bo ut Bo olean spaces and ´ etale maps. 2.1 Sk ew Bo olean algebras A skew lattic e is an a lgebra ( A, ∧ , ∨ ) with idemp otent and a s so ciative binary op erations me et ∧ and join ∨ satis fying the absorptio n iden tities x ∧ ( x ∨ y ) = x = ( y ∨ x ) ∧ x and x ∨ ( x ∧ y ) = x = ( y ∧ x ) ∨ x . If o ne of the opera tions is commutativ e then so is the other , in which case A is a lattice, see [5]. A s kew lattice has tw o o r der structures. The natur al p artial or der x ≤ y is defined by x ∧ y = y ∧ x = x , o r equiv alently x ∨ y = y ∨ x = y . The natur al pr e or der x  y is defined by x ∧ y ∧ x = x , or equiv alently y ∨ x ∨ y = y . The po set refelection of the natural pr e order  is known as Gr e en ’s r elation D . By Leech’s First Decomp o sition Theo rem [5 ], D is the finest congruence for w hich A/ D is a lattice. In other words, the functor A 7→ A/ D is a reflection of skew lattices into ordinar y lattices . W e denote the D -eq uiv alence class of a by D a . The reflec tion can be a nalysed further into its left- a nd rig ht-handed parts. A skew lattice A is right-hande d if it sa tis fie s the identit y x ∧ y ∧ x = y ∧ x , or equiv a lently x ∨ y ∨ x = x ∨ y . W e define left-hande d la ttices analog ously . Quotients by Green’s cong ruence rela tions R and L , which a re defined by x R y ⇐ ⇒ x ∧ y = y and y ∧ x = x, x L y ⇐ ⇒ x ∧ y = x and y ∧ x = y , provide reflections of a skew lattice A in to left-handed and right-handed skew lattices, resp ectively . By Leech’s Second Decomp os itio n Theor em [5 ] the square of cano nical quotient maps A   / / A/ L   A/ R / / A/ D is a pullback in the categ ory of skew algebra s . 2 A re ctangular b and ( A, ∧ ) is an alg ebra with a binary op er ation ∧ which is idempo tent , as so ciative, and it satisfies the rectangle identit y x ∧ y ∧ z = x ∧ z . The name comes fr o m the fact that every rectang ula r ba nd is isomorphic to a cartesian pro duct X × Y with the op era tion ( x 1 , y 1 ) ∧ ( x 2 , y 2 ) = ( x 1 , y 2 ). If A is non- empt y the sets X and Y are unique up to bijection and can b e taken to be A/ L and A/ R , respectively . Eac h rectangular band is a skew lattice for ∧ and the ass o ciated o p eration ∨ defined as x ∨ y = y ∧ x . It turns out that the D - classes o f a skew lattice form rectang ular bands, with the o p er ation induced by the s kew lattice. A skew Bo ole an algebr a ( A, 0 , ∧ , ∨ , \ ) is a skew lattice which is meet-distributive, i.e., it satisfies the identit ies x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) and ( y ∨ z ) ∧ x = ( y ∧ x ) ∨ ( z ∧ x ) , has a zer o 0, which is neutral for ∨ , a nd a r elative c omplement \ satisfying ( x \ y ) ∧ ( x ∧ y ∧ x ) = 0 and ( x \ y ) ∨ ( x ∧ y ∧ x ) = x. It fo llows fro m these require ment that the principal subalgebr as x ∧ A ∧ x of a skew Bo olea n algebr a are Bo ole a n alge bras. F or example, the t wo ident ities for the relative complement s ay that \ res tr icted to a principal suba lgebra acts as the complement op era tio n. See Le ech [6] for further details on skew Boo lean alebras. W e r emark that a skew Bo olea n algebra with a top element 1 is degen- erate in the sense that it is a lready a Bo olean algebra. Also note that a sk ew Bo olean algebra who se meet a nd jo in are comm utative is the same thing as a generalized Bo o lean algebr a. Often it is the ca se tha t an y tw o element s x a nd y of a skew Bo o lean alge- bra A hav e the greatest low er b ound x ∩ y with resp ect to the na tural pa rtial order ≤ . When this is the case, w e call ∩ interse ction a nd sp eak of a skew Bo ole an interse ction algebr a . Many ex amples of skew lattices occurr ing in na- ture p os ses intersections. The significa nc e of skew Bo olea n intersection algebras is witnessed by the fact that they for m a discriminator v a riety [1], and are there- fore b oth congruence p ermutable and congr uence distributive. Moreov er, r esults by Bigna ll and Lee ch [1] imply that every algebra A in a p o inted dis criminator v ar iety is ter m equiv alent to a r ight-handed skew Bo olean intersection algebr a whose cong ruences co inc ide with those of A . In contrast, it was observed al- ready b y Cornish [2] that the congr uence lattices of skew Boo lean a lg ebras in general satisfy no par ticular lattice identit y . Henceforth we s ha ll consider exclusively s kew Bo ole an intersection algebr as, so we simply refer to them as skew algebr as . A homomo rphism o f skew algebr as preserves a ll the o p er ations, namely 0, ∧ , ∨ , \ , and ∩ . Recall tha t an ide al , which we sometimes call  - ide al , in a skew algebra A is a subset I ⊆ A which is low er with resp ect to  and is close d under finite jo ins , so in par ticular 0 ∈ I . An idea l P is prime if it is non-trivial and a ∧ b ∈ P implies a ∈ P or b ∈ P . It can b e shown easily that the prime idea ls in A coincide with non- zero maps A → 2 in to the t wo-elmen ts lattice 2 = { 0 , 1 } which pr eserve 0, ∧ , a nd ∨ (but not necessa r ily ∩ ). Becaus e 2 is commutativ e, such maps are in bijective corres p o ndence with non-zer o maps A/ D → 2. In other words, the assig nmen t P 7→ P / D = { D a | a ∈ P } 3 is a bijection from pr ime ideals in A to prime ideals in A/ D . W e write f : X ⇀ Y to indica te that f is a partial map from X to Y , defined on its domain dom( f ) ⊆ X . The r estriction f | D of f : X ⇀ Y to D ⊆ X is the map f with the doma in res tricted to do m( f ) ∩ D . W e deno te the set of all partial maps from X to Y by P ( X , Y ). Leech’s construction [6] shows how P ( X, Y ) ca n be endow ed with a rig ht -handed skew algb era s tructure by setting 0 = ∅ , f ∧ g = g | dom( f ) ∩ dom( g ) , f ∨ g = f ∪ g | dom( f ) − dom( g ) , f \ g = f | dom( f ) − dom( g ) , f ∩ g = f ∩ g , where − is set-theore tic difference, and the set-theoretic op eratio ns on the right- hand sides act on f and g viewed as functiona l relations. W e generalize this construction to algebr as which ar e not ne c essarily rig ht-handed, b ecaus e we will need one in Section 4 .4. Given a subset D ⊆ X , let P ( X , Y ) D = { f : X ⇀ Y | dom( f ) = D } b e the set of those partia l maps X ⇀ Y whos e domain is D . Suppo se we are given for each D ⊆ X a bina ry oper ation f D on P ( X , Y ) D . W e say that the family { f D | D ⊆ X } is c oher en t when it commut es with restrictions: fo r a ll E ⊆ D ⊆ X and f , g ∈ P ( X , Y ) D it holds ( f f D g ) | E = f | E f E g | E . (1) W e usually o mit the s ubscript from f D and wr ite just f . Theorem 2.1 L et ( f D ) D ⊆ X b e a c oher ent family of r e ctangular b ands. Then P ( X , Y ) is a skew algebr a for t he fol lowing op er ations, define d for f , g ∈ P ( X , Y ) with dom( f ) = F and dom( g ) = G : 0 = ∅ , f ∧ g = f | F ∩ G f g | F ∩ G , f ∨ g = ( f | F − G ) ∪ ( g | G − F ) ∪ ( g ∧ f ) , f \ g = f | F − G , f ∩ g = f ∩ g . Pr o of. Let f , g , h ∈ P ( X , Y ) b e partial maps with do mains do m( f ) = F , dom( g ) = G , and dom( h ) = H , resp ectively . Observe that the op e rations de- fined in the sta tement of the theorem all co mm ute with re strictions, for example ( f ∨ g ) | D = f | ( F − G ) ∩ D ∪ g | ( G − F ) ∩ D ∪ ( f | D ∩ F ∩ G f g | D ∩ F ∩ G ) = f | ( D ∩ F ) − ( D ∩ G ) ∪ g | ( D ∩ G ) − ( D ∩ F ) ∪ ( f | D ∩ F ∩ G f g | D ∩ F ∩ G ) = f | D ∨ g | D . Thu s a go o d strategy fo r checking a n identit y is to do it “by parts” as follows. T o chec k u = v it suffices to check u | X = v | X and u | Y = v | Y separately , pr ovided that dom( u ) = dom( v ) = X ∪ Y . Of cour se, this only works if X a nd Y ar e 4 suitably chosen so that the restric tio ns u | X , u | Y , u | Y , and v | Y simplify when we push the restr ictions by X and Y inw ards. The following prop erties ar e easily verified: 0 is neutr a l for ∨ , idemp otency of ∧ and ∨ , a sso ciativity of ∧ . That ∩ co mputes greatest low er b ounds holds bec ause the natura l pa rtial order on P ( X , Y ) is subset inclusion ⊆ of functions viewed as functional re lations. It remains to chec k a sso ciativity o f ∨ , meet distributivity , and the prop er ties of \ . Asso ciativity of ∨ is chec ked by parts. The domain of f ∨ ( g ∨ h ) and ( f ∨ g ) ∨ h is F ∪ G ∪ H , w hich is covered b y the parts ( F ∪ G ) − H , ( F ∪ H ) − G , ( G ∪ H ) − F , and F ∩ G ∩ H . On the first part we g et (( f ∨ g ) ∨ h ) | ( F ∪ G ) − H = ( f | F − H ∨ g | G − H ) ∨ h | ∅ = f | F − H ∨ g | G − H . and ( f ∨ ( g ∨ h )) | ( F ∪ G ) − H = f | F − H ∨ ( g | G − H ∨ h | ∅ ) = f | F − H ∨ g | G − H . On ( F ∪ H ) − G a nd ( G ∪ H ) − F the calculation is similar , while o n F ∩ G ∩ H bo th meets and joins turn into f and the identit y follows a s well. Next we c heck that mee t dis tr ibutivit y holds . The domain of f ∧ ( g ∨ h ) and ( f ∧ g ) ∨ ( f ∧ h ) is F ∩ ( G ∪ H ). It is covered by the parts ( F ∩ G ) − H , ( F ∩ H ) − G , and F ∩ G ∩ H . On the first part we get ( f ∧ ( g ∨ h )) | ( F ∩ G ) − H = f | ( F ∩ G ) − H ∧ ( g | ( F ∩ G ) − H ∨ h | ∅ ) = f | ( F ∩ G ) − H ∧ g | ( F ∩ G ) − H and ( f ∧ g ) ∨ ( f ∧ h )) | ( F ∩ G ) − H = ( f | ( F ∩ G ) − H ∧ g | ( F ∩ G ) − H ) ∨ ( f | ( F ∩ G ) − H ∧ h | ∅ ) = ( f | ( F ∩ G ) − H ∧ g | ( F ∩ G ) − H ) ∨ 0 = f | ( F ∩ G ) − H ∧ g | ( F ∩ G ) − H . The calcula tio n on ( F ∩ H ) − G is similar, and on F ∩ G ∩ H it ag ain trivializes bec ause all o pe r ations b eco me f . F or the other half of meet distributivity , namely ( f ∨ g ) ∧ h = ( f ∧ h ) ∨ ( g ∧ h ) we use the par ts ( F ∩ H ) − G , ( G ∩ H ) − F , and F ∩ G ∩ H . Finaly , \ satisfies the axioms of relative complementation b eca use ( f \ g ) ∧ ( f ∧ g ∧ f ) = f | F − G ∧ f | F ∩ G = ∅ = 0 and ( f \ g ) ∨ ( f ∧ g ∧ f ) = f | F − G ∨ f | F ∩ G = f | F − G ∪ f | F ∩ G ∪ ∅ = f .  2.2 Bo olean spaces and ´ etale maps W e start b y reca ling s everal standard top ologic a l no tions. A space is zer o- dimensional if its clop ens (sets which are b o th op en and closed) form a topo log- ical base. A Stone sp ac e is a compac t ze r o-dimensiona l Hausdorff spac e , while 5 a Bo ole an s p ac e is a lo cally compact zer o-dimensional Haus do rff spac e . W e call a set which is compac t and op en a c op en . In a Bo olea n spac e the copens for m toplogical base. A pa rtial map f : X ⇀ Y is said to b e contin uous when it is co nt inuous as a map defined on the subset dom( f ) ⊆ X with the induced to po logy . Unless noted otherwis e, the domain o f definition dom( f ) is alwa ys going to be an op en subset o f X . A con tinuous map is pr op er if its in verse image map tak es compact subsets to compact subsets, while a partial contin uous map with an op en domain of definitio n is pro pe r when the inv erse imag e f − 1 ( K ) of a compact subset K ⊆ Y is compact in dom( f ), or equiv alent ly in X . An ´ etale map p : E → B , also known as lo c al home omorphism , is a con tin- uous map for which E has an op en cov er such that for ea ch U in the cov er the restriction p | U : U → p ( U ) is a homeomor phism ont o the image, and p ( U ) is op en in B . W e call E the total sp ac e and B the b ase of the ´ etale map p . T he fib er ab ove x ∈ B is the subspace E x = { y ∈ E | p ( y ) = x } . A se ction of p is a contin uous map s : U → E defined on a subset U ⊆ B , usually op en, s uch that p ◦ s = id U . ´ Etale maps with a common ba se B form a ca tegory , even a top os, in which morphisms ar e comm utative triangles E f / / p   @ @ @ @ @ @ @ E ′ p ′ ~ ~ } } } } } } } B where f is a contin uous map. It follows that f is an ´ etale map, see for example [7, II.6 ]. One co nsequence of this is that a section s : U → E of a n ´ etale map p : E → B defined on an open subset U ⊆ B is itself a n ´ etale map (b ecause the inclusion U ֒ → E is an ´ etale ma p). In pa rticular, the image s ( U ) is op en in E and (images o f ) op en se ctions form a base for E . A c op en se ction of p : E → B is a section s : U → E defined on a cop en subset U ⊆ B . Its image s ( U ) is not o nly o pe n but a lso compac t in E , and the r estriction p | s ( U ) is a homeomorphism from s ( U ) onto U . Conv ersely , if S ⊆ E is cope n a nd p | S : S → p ( S ) is a ho meo morphism onto p ( S ) then ( p | S ) − 1 : p ( S ) → S is a cop en section. This is so b ecause ´ etale ma ps are op en. W e therefore have t wo v ie w s of cop en sec tions: as sections defined on cop en subsets, and as those cop en subs e ts o f the total space which cov er each po int in the base at most o nce. In Section 4 .4 we will need to know how to compute equalizers and co e qual- izers in the categ ory of ´ eta le ma ps ov er a given base. Prop ositi on 2. 2 L et f and g b e morphisms of ´ etale maps, E p   @ @ @ @ @ @ @ f / / g / / E ′ p ′ ~ ~ } } } } } } } B The e qualizer and c o e qualizer of f and g in the c ate gory of ´ etale maps with b ase B ar e c ompute d as in the c ate gory of top olo gic al sp ac es. Mor e over, the quotient map fr om E ′ to the c o e qu alizer is op en. 6 Pr o of. In the catego ry o f top olo g ical spaces the e q ualizer of f and g is the subspace I = { x ∈ E | f ( x ) = g ( x ) } with the subspace inclusion i : I → E . F or this to b e an e qualizer in the category of ´ etale maps, p ◦ i : I → B must b e ´ e tale, which is the case b ecause I is a n op en subspace o f E . T o see this, c o nsider any x ∈ I . Because f and g are ´ e ta le maps there is a n op en section U ⊆ E cont aining x such that f | U : U → f ( U ) and g | U : U → g ( U ) are ho meo morphisms on to o p e n subsets of E ′ . Thus the intersection f ( U ) ∩ g ( U ) is o pe n, from which it follows that U ∩ f − 1 ( g ( U )) ∩ g − 1 ( f ( U )) is an o pe n neighbor ho o d of x contained in I . The co equalizer of f and g is c o mputed in top ologic a l spaces as the quotient space Q = E ′ /R where R ⊆ E ′ × E ′ is the lea st equiv alence relation gener ated by the rela tion S = { ( f ( x ) , g ( x )) ∈ E ′ × E ′ | x ∈ E } . Because S is an op en subs e t of the fib ered pro duct E ′ × B E ′ so is R , from which it follows that the canonica l quotient map q : E ′ → Q is o p en. F urthermore, b eca use p ′ ◦ f = p = p ′ ◦ g the map p ′ factors through q a s p ′ = r ◦ q . T o co mplete the pro of, we need to show that r is ´ etale, but this is easy b ecause we alr eady know tha t q is op en.  3 Dualit y for comm utativ e algebras Before emb ar king on duality for skew algebr as we review the familiar commuta- tive case. The sp e ctrum St ( A ) o f a Bo olea n algebr a A is the Stone space whos e po ints are the prime idea ls o f A . The elements a ∈ A corresp ond to the ba sic clop en sets N a = { P ∈ St ( A ) | a 6∈ P } . A homomo rphism f : A → A ′ betw een Bo olean algebr as induces a co ntin uo us map f ♭ : St ( A ′ ) → St ( A ) that maps a prime ideal P to its preimage f ♭ ( P ) = f − 1 ( P ). In the other dire c tion the du- ality maps a Stone space to the Bo olea n a lgebra of its clop en subsets, with the exp ected op er ations of intersection and unio n. A s hort path to Stone duality for gener alized Bo olean algebra s go es through the observ ation that the categ o ry GBA of genera liz e d Bo olean algebr as is equiv- alent to the slice category BA / 2 of Bo o lean a lgebras over the initial a lgebra 2 . By Stone dua lit y for Bo olea n algebr as GBA is then dual to p ointe d Stone spaces and contin uo us maps whic h pres erve the chosen point. F o r our purp oses it is more c o nv enient to tak e yet another equiv alent categor y , na mely Stone s pa ces without one p oint, whic h are pre c isely the Bo olea n s paces, and suitable partial maps b etw een them. Let us describ e the dualit y explicitly . Starting from a genera lized Bo o lean algebra A , we construct its sp e ctrum St ( A ) as the Bo olean spac e of prime ideals . An element a ∈ A corres po nds to the basic c o p en s et N a = { P ∈ St ( A ) | a 6∈ P } . A homomor phism f : A → A ′ betw een generalized B o olean alg e bras induces a partial map f ♭ : St ( A ′ ) ⇀ St ( A ) that maps a prime ideal P to its preimage f ♭ ( P ), provided the preimage is no t all of A . The doma in of definition of f ♭ is op en, for if a 6∈ f ♭ ( P ) then f ♭ is defined o n N f ( a ) . The fact that the inv erse image f − 1 ♭ ( N a ) of a basic cop en is the basic cop en N f ( a ) implies that f ♭ is bo th contin uous and pro p er . Indeed, f is prop er b ecaus e the inv erse image f − 1 ( K ) of a co mpact subset K ⊆ St ( A ) is a closed subs e t of the compact set f − 1 ( N a 1 ) ∪ · · · ∪ f − 1 ( N a n ) where N a 1 , . . . , N a n is some finite cover of K by basic cop en sets. In summar y , the category of generalize d Bo olean a lgebras 7 is equiv alent to the ca tegory of Bo olean spaces and prop er contin uous partial maps with o pe n domains of definition. 4 Dualit y for sk ew algebras An y attempt at extension of Stone duality na turally leads to consideration of prime ideals . Since the o ne- p oint spa ce 1 c orresp o nds to the initial Bo olea n algebra 2 , a p oint 1 → X on the to po logical side cor resp onds to homomor- phisms A → 2 on the alg ebraic side. Ho wev er, since such homomorphisms factor through the commutativ e reflection A → A/ D , they give us insufficient information ab out the no n-commutativ e structur e of A . W e should therefore ex- pec t that on the top ologica l side we have to lo ok for s tructures that can b e rich even though they hav e few glob al po int s, while on the algebra ic side we ca nno t afford to use ideals exclusively , but m ust als o consider cong ruence rela tions. In the ca se of a comm utative algebr a the congruence relation gener ated by a prime ideal has just t wo eq uiv alence classes. In cont ra st, the least congruence relation θ P whose zero -class contains the prime ideal P ⊆ A in a skew a lgebra A generally has many equiv alence cla sses, which ought to be accounted for on the top olog ical side of duality . The following result of Bignall and Leech [1] characterizes the congruence re la tions gener ated b y ideals in a skew alg ebra. Lemma 4.1 (Bi gnall & Lee c h) F or an ide al I ⊆ A in a skew algebr a A let θ I b e the le ast c ongruenc e r elation on A whose zer o-class c ontains I . Then for al l x, y ∈ A , x θ I y if, and only if , ( x \ ( x ∩ y )) ∨ ( y \ ( x ∩ y )) ∈ I . In fact, the zero-cla ss of θ I equals I . Consequently , the ideals of a skew alg ebra are in bijective corre s p o ndence with congr uence r elations. A consequence of Lemma 4.1 is that x θ f − 1 ( I ) y is equiv alent to f ( x ) θ I f ( y ), for any skew alg ebra homomorphism f : A → A ′ and any ideal I ⊆ A ′ . F ro m this w e obtain the following pr op erties of pr ime ideals. Lemma 4.2 L et P b e a prime ide al in a s kew algebr a A and let x, y ∈ A . Then: 1. x ∈ P or y \ x ∈ P . 2. If x ≤ y and x 6∈ P then x θ P y . 3. If x ≤ y then x ∈ P is e quivalent to y θ P y \ x . Pr o of. 1. W e ha ve x ∧ ( y \ x ) = 0 ∈ P . Because P is a prime ideal it follo ws that x ∈ P or y \ x ∈ P . 2. W e need to show that ( x \ ( x ∩ y )) ∨ ( y \ ( x ∩ y )) ∈ P . Since x ≤ y it follows that x ∩ y = x a nd th us we only need y \ x ∈ P , which follows from the first s ta tement of the lemma. 3. The e le ment x ′ = y \ x is the complement of x in the Bo olean algebr a y ∧ A ∧ y . Hence x ′ ≤ y and y \ x ′ = x . Thus Lemma 4.1 implies that y θ P x ′ is equiv alent to ( y \ x ′ ) ∨ ( x ′ \ x ′ ) = x ∈ P .  8 4.1 F rom algebras to spaces F o llowing our own advice that the to p o logical side of duality should account for the eq uiv alence classes of congruences, w e define the skew sp e ctrum Sk ( A ) of a s kew algebra A to be the space whose points are pair s ( P, e ), where P is a prime ideal in A and e is a non- z e ro equiv alence class of θ P . Since every non-zero equiv alence clas s equals [ t ] θ P for some t 6∈ P , a gener al element o f the skew sp e ctrum may be written a s ( P , [ t ] θ P ). W e wr ite just [ t ] P . Thus, as a set the skew sp ectrum is Sk ( A ) = { [ t ] P | P prime idea l in A a nd t 6∈ P } , where [ t ] P = [ u ] Q when P = Q and t θ P u . The topo logy of Sk ( A ) is the one whose basic op en sets are of the form, for a ∈ A , M a = { [ a ] P | a 6∈ P } . These really form a bas is b ecause they are closed under intersections. Lemma 4.3 L et a, b ∈ A . Then M a ∩ M b = M a ∩ b . Pr o of. F or a prime ideal P and t ∈ A the statement [ t ] P ∈ M a ∩ M b amounts to ( a 6∈ P ) ∧ ( b 6∈ P ) ∧ ( t θ P a ) ∧ ( t θ P b ) , (2) while [ t ] P ∈ M a ∩ b means ( a ∩ b 6∈ P ) ∧ ( t θ P a ∩ b ) . (3) If (2) holds then a ∩ b θ P a ∩ a = a θ P t which prov es (3). T o prov e the conv erse, supp ose (3) holds. B e c ause a ∩ b 6∈ P , a ∩ b ≤ a and a ∩ b ≤ b , Lemma 4.2 implies a θ P a ∩ b θ P b which suffices for (2).  The skew sp ectrum on its o wn contains to o little information to a ct as the dual. F or example, b oth Sk ( 2 × 2 ) and Sk ( 3 ) ar e the discre te space o n tw o p oints. Recall that 3 is the r ight-handed skew algebra whose set o f elements is { 0 , 1 , 2 } and whose D -classe s ar e { 0 } and { 1 , 2 } . One part of the missing informatio n is provided by the map q A : Sk ( A ) → St ( A ) defined by q A ([ t ] P ) = P / D , where we us ed the shorthand St ( A ) = St ( A/ D ). Prop ositi on 4. 4 The m ap q A : Sk ( A ) → St ( A ) is onto and ´ etale. Pr o of. Bec ause prime ideals a re non-tr ivial the map q A is onto. T o show that q A is contin uous, we pr ov e q − 1 A ( N a ) = [ b  a M b , (4) where we us ed the shorthand N a = N D a . F or one inclusio n, observe tha t b  a implies D b ≤ D a and hence q A ( M b ) = N b ⊆ N a . F or the other inclusion, 9 suppo se q A ([ t ] P ) ∈ N a . Then t ∧ a ∧ t 6∈ P beca use a 6∈ P and t 6∈ P , and by Lemma 4.2 we get t θ P t ∧ a ∧ t from which [ t ] P ∈ M t ∧ a ∧ t follows. Finally , q A is ´ etale b ecause its restrictio n to a basic op en set M a is a (con- tin uous) bijection o nt o the ba sic op en set N a .  Corollary 4.5 The skew sp e ctru m Sk ( A ) is a Bo ole an sp ac e. Pr o of. Lo cal co mpactness a nd zero-dimensionality are lifted from St ( A ) to Sk ( A ) by the ´ eta le map q A . T o see that Sk ( A ) is Hausdorff, let [ s ] P and [ t ] Q in Sk ( A ) b e tw o dis tinct p oints in Sk ( A ). Consider the case P = Q . W e claim tha t M s \ ( s ∩ t ) and M t \ ( s ∩ t ) are dis- joint and are neighbo r ho o ds of s and t , r esp ectively . Disjoin tness follows by Lemma 4.3 from the fact tha t ( t \ ( s ∩ t )) ∩ ( s \ ( s ∩ t )) = 0. F r o m ¬ ( t θ P s ) we conclude by Le mma 4.1 that ( s \ ( s ∩ t )) ∨ ( t \ ( s ∩ t )) 6∈ P , thus s \ ( s ∩ t ) 6∈ P or t \ ( s ∩ t ) 6∈ P . In either case s ∩ t ∈ P by (1) of Lemma 4 .2, therefo r e s θ P s \ ( s ∩ t ) by (3) of Lemma 4.2, which pr ov es [ s ] P ∈ M s \ ( s ∩ t ) , as claimed. W e similarly show that [ t ] P ∈ M t \ ( s ∩ t ) . Consider the case P 6 = Q . Because St ( A ) is Hausdor ff there exist disjoint basic o pe n nehigb or ho o ds N a and N b of q A ( P ) and q A ( Q ), r esp ectively . Th us q − 1 A ( N a ) and q − 1 A ( N b ) ar e disjoint and are op en neighbor ho o ds of [ s ] P and [ t ] Q , resp ectively .  Because Boolea n space s ar e zer o -dimensional, the ´ etale map q A : Sk ( A ) → St ( A ) has more sections that one would nor mally exp ect. Prop ositi on 4. 6 The ´ etale map q A : Sk ( A ) → St ( A ) has a se ction ab ove every c op en set, p assing t hr ough a pr escrib e d p oint ab ove the c op en set. Pr o of. More precisely , the pro p osition claims that given a cop en s et U ⊆ St ( A ) and a p oint x ∈ Sk ( A ) such that q A ( x ) ∈ U , ther e is a cop en sec tion ab ov e U containing x . F or the proo f we only ne e d surjectivity o f q A and zer o - dimensionality of St ( A ). By surjectivity the c o p en set U can b e decomp osed int o pair wise disjoint cop en sets U 1 , . . . , U n , ea ch of which ha s a section s i . The sections can b e glued together in to a section s a b ov e U . T o mak e sure that s passes through a prescr ib ed point x , just change it s uitably on a small cop en neighborho o d of q A ( x ).  Prop ositi on 4. 7 The c op en se ctions of q A : Sk ( A ) → St ( A ) ar e pr e cisely the sets M a with a ∈ A . Pr o of. W e alrea dy know that ea ch M a is a cop en section b eca use q A maps it homeomorphica lly onto the co p e n set N a . C o nv ersely , let V be a cop en se c tion in Sk ( A ). It is a finite unio n of basic co p en sets V = M a 1 ∪ · · · ∪ M a n . W e show that V = M a 1 ∨···∨ a n . I f n = 1 there is nothing to prove. Consider the case n = 2. The fact that V = M a 1 ∪ M a 2 is a sectio n a mounts to: for a ll prime ideals P in A , if a 1 6∈ P a nd a 2 6∈ P then a θ P b . Let us show that this implies M a 1 ⊆ M a 1 ∨ a 2 . I f a 1 6∈ P then a 1 ∨ a 2 6∈ P . Either a 2 ∈ P or a 2 6∈ P . In the first case a 2 θ P 0 a nd so a 1 θ P a 1 ∨ 0 θ P a 1 ∨ a 2 . In the second case, 10 a 2 θ P a 1 and so a 1 θ P a 1 ∨ a 1 θ P a 1 ∨ a 2 . W e similarly show that M a 2 ⊆ M a 1 ∨ a 2 , from which we get M a 1 ∪ M a 2 ⊆ M a 1 ∨ a 2 . T o complete the cas e n = 2 w e still hav e to prov e M a 1 ∨ a 2 ⊆ M a 1 ∪ M a 2 . Suppo se a 1 ∨ a 2 6∈ P . Then either a 1 6∈ P or a 2 6∈ P . W e consider the case a 1 6∈ P , the other one is similar. E ither a 2 ∈ P o r a 2 6∈ P . In the firs t case a 2 θ P 0 so a 1 θ P a 1 ∨ 0 θ P a 1 ∨ a 2 . In the s e cond ca s e the assumption gives us a 1 θ P a 2 and so a 1 θ P a 1 ∨ a 1 θ P a 1 ∨ a 2 . The cases n > 2 follow by rep ea ted use of the case n = 2 .  Later on we will need to know how the ma p a 7→ M a int era cts with the op erations of A . F or this pur p o se we define the satu r ation op er ation σ A ( U ) = q − 1 A ( q A ( U )). Since q A is op en, the satura tion of a co p e n is a clop en. Prop ositi on 4. 8 The assignment a 7→ M a is an or der-isomorphism fr om a skew algebr a A , or der e d by the natur al p artial or der, onto t he c op en se ctions of Sk ( A ) , or der e d by subset inclusion, satisfying the identities M a ∧ b ∧ a = σ A ( M b ) ∩ M a and M a ∨ b ∨ a = M a ∪ ( M b \ σ A ( M a )) . Pr o of. If M a = M b then a θ P b for all prime ideals P . Th us for a ny prime ideal P ( a \ ( a ∩ b )) ∨ ( b \ ( a ∩ b )) ∈ P , and so b oth a \ ( a ∩ b ) a nd b \ ( a ∩ b ) b elo ng to P . Because the in terse ction of all prime ideals is { 0 } , it follows that a \ ( a ∩ b ) = 0 = b \ ( a ∩ b ) . Therefore, D a = D a ∩ b = D b , which is only p ossible if a = b . It follows that the assig nment a 7→ M a is injective, while its s urjectivity follows fr om Pr op o- sition 4.7. That a 7→ M a is an order iso morphism follows from the following chain of equiv a le nc e s, where we use Lemma 4.3 a nd injectivity of a 7→ M a in the seco nd and third s tep, res p ec tively: a ≤ b ⇐ ⇒ a ∩ b = a ⇐ ⇒ M a ∩ b = M a ⇐ ⇒ M a ∩ M b = M a ⇐ ⇒ M a ⊆ M b . It remains to prov e the tw o identites. F or the first one, let [ a ∧ b ∧ a ] P ∈ M a ∧ b ∧ a . Then a ∧ b ∧ a 6∈ P which implies b oth b 6∈ P (and th us [ a ∧ b ∧ a ] P ∈ σ A ( M b )) and a 6∈ P . W e hav e a ∧ b ∧ a ≤ a and a ∧ b ∧ a θ P a follows by Lemma 4.2. Hence [ a ∧ b ∧ a ] P ∈ σ A ( M b ) ∩ M a . T o prov e the con verse, le t [ a ] P ∈ σ A ( M b ) ∩ M a . Hence a 6∈ P and b 6∈ P . So a ∧ b ∧ a 6∈ P , a ∧ b ∧ a ≤ a a nd thus a ∧ b ∧ a θ P a by Lemma 4.2. T o prove the second equality firs t ass ume that [ a ∨ b ∨ a ] P ∈ M a ∨ b ∨ a . Hence a ∨ b ∨ a 6∈ P . If a 6∈ P then a θ P a ∨ b ∨ a by Lemma 4.2 b ecause alwa ys a ≤ a ∨ b ∨ a , and [ a ∨ b ∨ a ] P ∈ M a follows. If a ∈ P then b 6∈ P follows from a ∨ b ∨ a 6∈ P . Thus a ∨ b ∨ a θ P 0 ∨ b ∨ 0 θ P b a nd [ a ∨ b ∨ a ] P ∈ M b \ σ A ( M a ) follows. Finally , assume that [ s ] P ∈ M a ∪ ( M b \ σ A ( M a )). Then either a 6∈ P and [ s ] P = [ a ] P , or a ∈ P , b 6∈ P and [ s ] P = [ b ] P . In the fir st case it follows that 11 a ∨ b ∨ a 6∈ P and th us a θ P a ∨ b ∨ a b y Lemma 4.2. Therefore [ s ] P = [ a ] P ∈ M a ∨ b ∨ a . In the seco nd case it follows that a ∨ b ∨ a 6∈ P and a ∨ b ∨ a θ P 0 ∨ b ∨ 0 θ P b . Again, [ s ] P = [ b ] P ∈ M a ∨ b ∨ a .  The attentiv e r eader will point out that the ´ etale map q A : Sk ( A ) → St ( A ) cannot p ossibly b e the topolog ical dual of the sk ew a lgebra A b ecause A giv es the same ´ etale map as its opp os ite algebra (in which the op erations ∧ and ∨ are the mirr o r version of those in A ). Indeed, the ´ e tale map provides sufficient information only when A is right -handed (or left-handed), a s will be shown in Section 4 .3. W e shall consider the ge neral cas e in Section 4.4. Let f : A → A ′ be a homomorphism o f skew algebra s. As we explained in Section 3, the induced ho mo morphism A/ D → A ′ / D is dual to a prop er pa r tial map f ♭ : St ( A ′ ) → St ( A ) with an op en domain, which maps prime ideals to their inv ers e imag es, when defined. Ther e is a ls o a map f ♯ : Sk ( A ′ ) → Sk ( A ) o f the same kind b etw een the skew sp ectra. It is c har a cterized b y the requirement f ♯ ([ f ( a )] P ) = [ a ] f ♭ ( P ) , which uniquely determines the v alue of f ♯ , when defined, bec a use a θ f ♭ ( P ) b is equiv a le nt to f ( a ) θ P f ( b ). If f ♯ ([ f ( a )] P ) is defined then f ( a ) 6∈ P , hence f ♯ is defined on the basic co p en set M f ( a ) , which shows that the domain of definition of f ♯ is open. Next, f ♯ is cont inuous and prop er beca use f − 1 ♯ ( M a ) = M f ( a ) for all a ∈ A . The square Sk ( A ) q A   Sk ( A ′ ) f ♯ o q A ′   St ( A ) St ( A ′ ) f ♭ o (5) commutes. It is ea sy to s ee that the v alues of q A ◦ f ♯ and f ♭ ◦ q B coincide whenever they are b oth defined, so we only show that they hav e the same domain of definition. If f ♭ ( P ) is defined then ther e is a ∈ A such tha t f ( a ) 6∈ P , in whic h case f ♯ is defined a t [ f ( a )] P and q B ([ f ( a )] P ) = P . Conversely , if f ♯ ([ f ( a )] P is defined then f ♭ ( q B ([ f ( a )] P )) = f ♭ ( P ) is defined b e c ause f ( a ) 6∈ P . Lemma 4.9 The map f ♯ is a bije ction on fib ers, i.e., given any P ∈ dom( f ♭ ) , f ♯ maps Sk ( B ) P ∩ dom( f ♯ ) bije ctively onto the fib er Sk ( A ) f ♭ ( P ) . Pr o of. Cons ide r any P ∈ dom( f ♭ ). The Lemma states that f ♯ is a bijective map b etw een the sets { [ f ( a )] P | f ( a ) 6∈ P } a nd { [ a ] f ♭ ( P ) | a 6∈ f ♭ ( P ) } . F o r injectivity , supp os e f ♯ ([ f ( a )] P ) = f ♯ ([ f ( a ′ )] P ) wher e f ( a ) 6∈ P and f ( a ′ ) 6∈ P . By the definition of f ♯ it follows tha t [ a ] f ♭ ( P ) = [ a ′ ] f ♭ ( P ) , which is equiv alent to a θ f ♭ ( P ) a ′ , and hence f ( a ) θ P f ( a ′ ). This establishes the fact that [ f ( a )] P = [ f ( a ′ )] P . F o r sur jectivit y , pic k any [ a ] f ♭ ( P ) where a 6∈ f ♭ ( P ). It follows that f ( a ) 6∈ P , so f ♯ is defined at [ f ( a )] P and maps it to [ a ] f ♭ ( P ) .  In view of the previous lemma one might contemplate turning f ♯ around, so that instead of a par tial map whic h is bijective on fib ers w e w ould g et a total map 12 which is injectiv e on fib ers. The tr o uble is that the in verted map need not be contin uous, so the top olog ical nature of f ♯ m ust b e obscure d by a mor e co mplex condition. 4.2 F rom spaces to algebras As we alrea dy indicated, the original a lg ebra A ca nnot alwa ys b e reco nstructed from q A : Sk ( A ) → St ( A ). Howev er, if A is right-hande d then q A : Sk ( A ) → St ( A ) ca r ries all the information needed, s o we consider this c a se first. W e c all a surjective ´ eta le map p : E → B b etw een Bo olean spaces a skew Bo ole an sp ac e . The co rresp onding right-handed skew alg ebra A p consists of cop en sections of p , i.e., an element of A p is a copen subset S ⊆ E such tha t p | S is injective. T o des crib e the right-handed skew structure on A p , r ecall the saturation op era tion σ p ( S ) = p − 1 ( p ( S )), and define for S, R ∈ A p 0 = ∅ , S ∧ R = σ p ( S ) ∩ R , S ∨ R = S ∪ ( R − σ p ( S )) , S \ R = S − σ p ( R ) , S ∩ R = S ∩ R. It is clear that these op era tions map back into A p . F or exa mple, S ∩ R is a section, and it is cop en be c ause it is the intersection of tw o cop en subsets o f the Hausdorff space E . It is not difficult to chec k that the a b ov e o p er ations form a skew alg ebra with ba re ha nds. An a lternative, more ele gant way of establishing the skew structure is to view the element s of A p as partial maps s, r : B ⇀ E and exhibit A p as a subalge br a of the right-handed sk ew algebra P ( B , E ) as describ ed in Section 2 .1. The construc tio n of the skew alge br a A p induces the usual constructio n of a generalized Bo o lean algebr a via the lattice reflection, a s follows. Prop ositi on 4. 10 L et p : E → B b e a skew Bo ole an sp ac e and let B ∗ b e the Bo ole an algebr a of c op en subsets of B . The map A p → B ∗ define d by S 7→ p ( S ) is the lattic e r efle ction of A p . Pr o of. By Prop o sition 4.6 the map S 7→ p ( S ) is sur jective, and it is easily seen to be a lattice homomor phism. Thus we only have to check p ( S ) = p ( R ) is equiv a le nt to S D R , where S and R a re cop en sections o f p . This follows from S  R ⇐ ⇒ S ∧ R ∧ S = S ⇐ ⇒ σ p ( S ) ∩ σ p ( R ) ∩ S = S ⇐ ⇒ S ⊆ σ p ( R ) ⇐ ⇒ p ( S ) ⊆ p ( R ) .  W e turn atten tion to morphisms nex t. W e already kno w that a ho momor- phism f : A → A ′ of skew algebra s induces a commutativ e square (5) in which the horizontal pa rtial maps are prop er, contin uous and hav e open do mains of 13 definition, and that the top map is bijective on fib ers by Lemma 4.9. So we de- fine a morphism ( g , h ) b etw een skew Bo ole a n spaces p : E → B and p ′ : E ′ → B ′ to b e a commutativ e diag ram E g / p   E ′ p ′   B h / B ′ (6) in which g a nd h ar e prop er contin uous partial maps with o pe n domains of definition. F urthermor e, we requir e that g is a bijection on fib ers , in the s e ns e that g maps E x ∩ dom( g ) bijectively onto E h ( x ) for every x ∈ dom( h ). Lemma 4.11 If ( g , h ) is a morphism b etwe en skew Bo ole an sp ac es p : E → B and p ′ : E ′ → B ′ then p ( g − 1 ( S )) = h − 1 ( p ′ ( S )) for every c op en se ction S in E ′ . Pr o of. Let S b e a cop en section in E ′ . If y ∈ p ( g − 1 ( S )) then there exists x ∈ g − 1 ( S ) such that y = p ( x ). Then h ( y ) = h ( p ( x )) = p ′ ( g ( x )) ∈ p ′ ( S ), so that y ∈ h − 1 ( p ′ ( S )) and p ( g − 1 ( S )) ⊆ h − 1 ( p ′ ( S )) follows. On the other hand, if h ( y ) ∈ p ′ ( S ) then h ( y ) = p ′ ( z ) fo r some z ∈ S . Because g is surjective on fib ers there exists x ∈ E y with the pro pe r ty g ( x ) = z . Now, y = p ( x ) ∈ p ( g − 1 ( S )) and we get h − 1 ( p ′ ( S )) ⊆ p ( g − 1 ( S )).  W e would like to co nstruct a cor resp onding homomor phism ( g , h ) ♮ : A p ′ → A p . F o r a co p en sec tio n S ⊆ E ′ , the inv erse image g − 1 ( S ) is cop en in E b ecause g is cont inuous a nd prop er , and it is a se c tion b ecause g is injectiv e on fibers. Thu s we may define ( g , h ) ♮ ( S ) = g − 1 ( S ). Let us show that g − 1 commutes with the saturatio n op erations σ p and σ p ′ . If S is a cop en section in E ′ , then p ( g − 1 ( S )) = h − 1 ( p ′ ( S )) b y Lemma 4.11. Therefore g − 1 ( σ p ′ ( S )) = p − 1 ( h − 1 ( p ′ ( S )) = p − 1 ( p ( g − 1 ( S )) = σ p ( g − 1 ( S )) , as c laimed. The o pe rations o n A p and A p ′ are defined in terms of basic set- theoretic op erations and the saturatio n maps σ p and σ ′ p . Beca use g − 1 commutes with all o f them, ( g , h ) ♮ is an algebra homomorphism. The following is the counterpart of Pr op osition 4.10 for morphisms. Prop ositi on 4. 12 L et ( g , h ) b e a m orphism b etwe en skew Bo ole an sp ac es p : E → B and p ′ : E ′ → B ′ . Its lattic e r efle ction ( g , h ) ♮ / D : A p ′ / D → A p / D is isomorphi c to t he latt ic e homomorphism h ∗ : ( B ′ ) ∗ → B ∗ define d by h ∗ ( S ) = h − 1 ( S ) . Pr o of. By Pro p o sition 4.10 the vertical maps in the diagra m A p   A p ′ ( g,h ) ♮ o o   B ∗ ( B ′ ) ∗ h ∗ o o 14 are lattice reflectio ns . Beca us e the right-hand vertical arrow is epi it suffices to show that the diag ram commutes, which is just Lemma 4 .11.  4.3 Dualit y for right-ha nded algebras Let us take sto ck of what we hav e do ne s o far, in ter ms of catego ry theory . O n the a lgebraic side w e hav e the ca tegory Sk Alg of skew a lgebras and homomor- phisms, as well as its refle c tive full s ubc ategory SkAlg R on right-handed skew algebras . On the top olo gical side we have the catego ry SkSp of skew Bo olea n spaces. A mor phis m ( g , h ) : p → p ′ betw een s kew Bo olean spaces p : E → B and p ′ : E ′ → B ′ is a commutativ e squa re (6) with prop er contin uous pa rtial maps g and h whose domains of definition ar e o pe n, a nd the top map g is bijective on fiber s. Admitt edly , the mor phisms in SkSp are not v ery nice, but in Section 5 we show that they decomp ose nicely int o pa rtial identities and pullbacks. In Section 4 .1 we defined a functor S : SkAlg op → SkSp which assig ns to each skew a lgebra A a s kew Bo olean space S ( A ) = q A : Sk ( A ) → St ( A ). The functor ta kes a homomor phism f : A → B to the corre- sp onding morphism S ( f ) = ( f ♯ , f ♭ ) : S ( B ) → S ( A ). In Section 4 .2 we defined a functor A : SkSp op → SkAlg R which maps an ´ etale map p : E → B b etw een Bo olea n space s to a r ight- handed skew algebra A ( p ) = A p , and a morphism ( g , h ) : p → p ′ as in (6) to a homomorphism A ( g , h ) = ( g , h ) ♮ : A ( p ′ ) → A ( p ). W e now work tow ards showing tha t S restricted to SkAlg R and A form a duality . F or a skew algebra A define the map φ A : A → A ( S ( A )) by φ A ( a ) = M a . That φ A is an isomorphism o f right-handed skew alg ebras follows from Lemma 4.3 and Pro po sition 4.8. By the lemma φ A preserves ∩ , it obviously preserves 0, and by the propos ition it is a bijectio n which preser ves the righ t-handed skew op erations ∧ a nd ∨ : M a ∧ b = M b ∧ a ∧ b = σ A ( M a ) ∩ M b = M a ∧ M b M a ∨ b = M a ∨ b ∨ a = M a ∪ ( M b \ σ A ( M a )) = M a ∨ M b . Naturality o f φ a mounts to the identit y f − 1 ♯ ( M a ) = M f ( a ) , where f : A → B is a ho momorphism a nd a ∈ A . After unraveling the definition of f ♯ we see that the set on the left-hand side consists o f elements [ f ( a )] P with f ( a ) 6∈ P , which is just the description of the right-hand side. W e have shown that φ is a na tural isomo rphism betw een the identit y and A ◦ S . T o es tablish the equiv a le nce we also need an iso mo rphism ψ p betw een a skew Bo olean spa c e p : E → B a nd q A : Sk ( A p ) → St ( A p ), natur al in p . It consists of 15 t wo homeomo r phism ( ψ p ) ♭ = h and ( ψ p ) ♯ = g for which the follo wing diagra m commutes: E g / / p   Sk ( A p ) q A p   B h / / St ( A p ) (7) Because the top ma p determines the b ottom one, we cons ide r g first. By Pr op o- sition 4.8 the map S 7→ M S is an order isomorphism be t ween A p and the cop en sections of Sk ( A p ). But since A p is the se t of cop en s ections o f E , and the natural par tia l order in A p coincides with the subset r elation in E , the map S 7→ M S maps the basis for E isomo r phically onto the basis for Sk ( A p ). Con- sequently , the top o logies o f E and Sk ( A p ) ar e isomorphic as p osets, to o , and bec ause E and Sk ( A p ) ar e so be r spaces they are homeo morphic. Explicitly , the homeomorphism g : E → Sk ( A p ) induced by the isomor phism S 7→ M S takes a po int y ∈ E to the unique p oint g ( y ) ∈ Sk ( A p ) satisfying, for all cop en se c tions S in E , y ∈ S ⇐ ⇒ g ( y ) ∈ M S . Similarly , the homeomorphism h : B → St ( A p ) is c haracter iz ed by the require- men t, for all co pe n sections S in E , x ∈ p ( S ) ⇐ ⇒ h ( x ) ∈ N S . It is not hard to verify that h ( x ) = {D R | R ∈ A p ∧ x ∈ p ( R ) } and that g ( x ) = [ S ] h ( x ) for any S ∈ A p such that x ∈ p ( S ). W e v erify that (7) commutes by chec king that the corresp o nding square of inv erse image maps do es. On o ne hand, starting w ith a co pe n section N R in St ( A p ), we have p − 1 ( h − 1 ( N R )) = p − 1 ( R ) = S { S | p ( S ) ⊆ R } , where S in the union r anges ov er cop en sectio ns in E . O n the other hand, g − 1 ( q − 1 A p ( N R )) = g − 1 ( S {M S | p ( S ) ⊆ R } ) = S { g − 1 ( M S ) | p ( S ) ⊆ R } = S { S | p ( S ) ⊆ R } , where S again r anges ov er cop en se c tio ns in E . Naturality o f ψ inv olves the comm utativity of a cube which w e prefer no t to dra w b eca use six of its faces c ommute by definition and the tw o r emaining faces are B ( ψ p ) ♭ / / h  St ( A p ) ( g,h ) ♮ ♭  B ′ ( ψ p ′ ) ♭ / / St ( A p ′ ) E ( ψ p ) ♯ / / g  Sk ( A p ) (( g, h ) ♮ ) ♯  E ′ ( ψ p ′ ) ♯ / / Sk ( A p ′ ) 16 where p : E → B and p ′ : E ′ → B ′ are skew Bo olean spaces and ( g , h ) is a morphism fr om p to p ′ . W e chec k commutativit y o f the right-hand square, the other one is similar. Ag ain, we verify that the cor resp onding squar e o f in verse image maps commutes. F or any cop en section M S ′ in the low er-rig ht corner we hav e g − 1 (( ψ p ′ ) − 1 ♯ ( M S ′ )) = g − 1 ( S ′ ) and ( ψ p ) − 1 ♯ ((( g , h ) ♮ ) − 1 ♯ ( M S ′ )) = ( ψ p ) − 1 ♯ ( M ( g,h ) ♮ ( S ′ ) ) = ( g , h ) ♮ ( S ′ ) = g − 1 ( S ′ ) . W e hav e prov ed the following main theo rem. Theorem 4.13 The c ate gory of right-hande d skew Bo ole an algebr as with inter- se ct ions is dual t o the c ate gory of skew Bo ole an sp ac es. Clearly , there is also duality b etw een left -handed skew alg e bras and skew Bo olea n spaces, simply bec ause the categ ories of left-ha nded and the r ight-handed skew algebras ar e isomorphic. 4.4 Dualit y for ske w algebras T o see what is needed for duality in the ca se of a ge neral skew alge br a, consider what happens when we ta ke a skew algebr a A to its skew Bo olean s pace q A : Sk ( A ) → St ( A ), and then the s pace to the right-handed skew algebra A q A . By Prop os itio n 4.8 the ele ments and the natural par tial o rder do not change (up to isomorphism), but the oper ations do. The new ones are expr essed in terms of the or ig inal ones as x ∧ ′ y = y ∧ x ∧ y and x ∨ ′ y = x ∨ y ∨ x. If we wan t to recover ∧ and ∨ fro m ∧ ′ and ∨ ′ we need to break the symmetry that is presen t in ∧ ′ and ∨ ′ by keeping a round enough information a bo ut the original op er a tions of A . Recall that a r ectangular band is a set with an op eration which is idem- po tent , a sso ciative and it sa tisfies the rectangula r identit y . W e can s imilarly define r ectangular bands in a ny ca tegory with finite pr o ducts. F or example, a rectangular band in the categ o ry of ´ etale maps over a g iven base spa ce B is an ´ etale map p : E → B together with a contin uous map ∧ : E × B E → E over B which s a tisfies the requir ed identities fiber -wise. The skew Bo olean space q A : Sk ( A ) → St ( A ) ca rries the structure of a rectangular band whose oper ation f : Sk ( A ) × St ( A ) Sk ( A ) → Sk ( A ) is defined by [ a ] P f [ b ] P = [ a ∧ b ] P . Idempo tency and asso cia tivity of f follow immediately from the co rresp onding prop erties of ∧ and the fact that θ P is a congrue nc e . T o see that the rectangular ident ity is satisfied, let a, b , c 6∈ P . Because ( A, ∧ ) forms a normal band, namely it satisfies the iden tity x ∧ y ∧ z ∧ w = x ∧ z ∧ y ∧ w , it follows tha t a ∧ b ∧ c ≤ a ∧ c and so a ∧ b ∧ c θ P a ∧ c by Lemma 4.2, which is equiv alent to [ a ] P f [ b ] P f [ c ] P = [ a ] P f [ c ] P . 17 W e have to chec k that f is contin uous. Let ([ a ] P , [ b ] P ) b e an y p o int of the domain of f a nd supp ose [ a ∧ b ] P ∈ M c for s ome c ∈ A . W e seek an o p en neighborho o d of ([ a ] P , [ b ] P ) which is mapped in to M c by f . Beca us e basic op en subse ts of Sk ( A ) × St ( A ) Sk ( A ) are of the form { ([ u ] Q , [ v ] Q ) | Q ⊆ A pr ime ideal, u 6∈ Q and v 6∈ Q } , it suffices to find u, v ∈ A such that a θ P u and b θ P v , a nd for all pr ime ideals Q ⊆ A , if u 6∈ Q and v 6∈ Q then u ∧ v θ Q c . W e claim that u = ( a ∧ b ∧ a ) ∩ ( c ∧ a ) v = ( b ∧ a ∧ b ) ∩ ( b ∧ c ) satisfy these conditions. W e note tha t u ∧ v ≤ c b eca us e u ∧ v = (( a ∧ b ∧ a ) ∩ ( c ∧ a )) ∧ (( b ∧ a ∧ b ) ∩ ( b ∧ c )) ≤ ( c ∧ a ) ∧ ( b ∧ c ) ≤ c , where we used the fact that ∧ is compatible with the natural partia l order, which is the case b eca use ( A, ∧ ) is a normal band. Next o bserve tha t a ∧ b ∧ a θ P c ∧ a bec ause [ a ∧ b ] P ∈ M c , hence u = ( a ∧ b ∧ a ) ∩ ( c ∧ a ) θ P ( a ∧ b ∧ a ) ∩ ( a ∧ b ∧ a ) = a ∧ b ∧ a θ P a, where the last step follows fro m Lemma 4.2 and a, b 6∈ P . W e similar ly sho w that v θ P b . If Q ⊆ A is a prime ideal such that u 6∈ Q and v 6∈ Q , then u ∧ v 6∈ Q and since u ∧ v ≤ c we ge t u ∧ v θ Q c , ag ain b y Lemma 4.2. The re c tangular ba nd str uc tur e o n A p is pr e c isely what is needed for duality in the general case. Theorem 4.14 The c ate gory of skew Bo ole an algebr as with int erse ctions is dual to the c ate gory of r e ctangular skew Bo ole an sp ac es. Pr o of. By r e ctangular skew Bo ole a n s pa ce ( p : E → B , f ) we mean a rectangular band in the c a tegory of sur jective ´ etale maps ov er B . Mor e precisely , it is a sk ew Bo olea n spa ce p : E → B toge ther with a (not neces sarily prop er ) contin uous map f over B E × B E # # H H H H H H H H H f / / E           B which makes every fib er E x int o a r ectangular ba nd. Notice that in gener al a rectangular skew Bo olea n space is not a rectang ular band in the ca tegory of skew Boolea n spaces bec ause f need no t b e prop er. A morphism b etw een ( p : E → B , f ) and ( p ′ : E ′ → B ′ , f ′ ) is a morphism of skew Boole a n spac e s ( g , h ) : p → p ′ which commut es with the o p erations on its domain of definition: dom( g ) × B dom( g ) g × g / f   E ′ × B ′ E ′ f ′   dom( g ) g / E ′ 18 Note tha t the comm utativity of the square implies that dom( g ) is closed under f , so dom( g ) ∩ E x is a recta ngular sub-band of E x at every x ∈ B . And since g is bijectiv e o n fiber s, g | x : dom( g ) ∩ E x → E h ( x ) is an isomo rphism of rectangula r bands for every x ∈ dom( h ). W e denote the categ ory of recta ngular skew Bo olean spaces and their morphisms by SkRSp . The duality is witnessed b y a pair of co nt rav a r iant functor s S : SkAlg op → SkR Sp and A : SkRSp op → SkAlg . The functor S maps a skew algebr a A to the re ctangular sk ew Bo olean space S ( A ) = ( p : Sk ( A ) → St ( A ) , f ), as describ ed a b ov e. It takes a morphism f : A → A ′ to the mor phis m of skew Bo olean spaces S ( f ) = ( f ♯ , f ♭ ), which commutes with f b ecause f commutes with ∧ . The functor A maps a rectangular skew Bo o lean space ( p : E → B , f ) to the skew algebra A ( p, f ) whose element s a re the cop en sections of p a nd the op erations ar e defined as follows: 0 = ∅ , S ∧ R = ( S ∩ σ p ( R )) f ( σ p ( S ) ∩ R ) , S ∨ R = ( S − σ p ( R )) ∪ ( R − σ p ( S )) ∪ ( R ∧ S ) , S \ R = S − σ p ( R ) , S ∩ R = S ∩ R, These form a skew algebr a b eca use they are res trictions of the o p e r ations from Theorem 2.1. The functor A maps a morphism ( g , h ) : ( p : E → B , f ) → ( p ′ : E ′ → B ′ , f ′ ) to the homo morphism A ( g , h ) = ( g , h ) ♮ . W e need to verify that A ( g , h ) pr e- serves f . Recall that ( g , h ) ♮ is just g − 1 acting o n cop en sectio ns. In Section 4.2 we chec ked that g − 1 commutes with the saturation op era tions. Because for every x ∈ dom( h ) the map g | x : E x ∩ do m( g ) → E ′ h ( x ) is an is omorphism o f rectangular bands , it is not hard to see that g − 1 commutes with f . T he r efore, g − 1 commutes with all the o pe r ations used to define the o pe r ations on G ( p, f ) and G ( p ′ , f ′ ), so it is a homomor phism of skew algebras. It remains to be chec ked that S ◦ A and A ◦ S a r e naturally is omorphic to ident ity functors. Luckily , we can reuse a great deal o f verification of duality for right-handed algebr a s fro m Section 4 .3. The natural iso morphism φ from the iden tit y to A ◦ S is defined as in the right-handed case: for a skew algebra A set φ A ( a ) = M a . Thus we already know that it is a bijection which preserves intersections and r elative complements, but we still have to chec k that it preserves mee ts and joins. It pr eserves meets bec ause M a ∧ M b = M a ∧ b ∧ a f M b ∧ a ∧ b = M a ∧ b where we us ed Pro p o sition 4.8 in the firs t step and the fact that [ a ∧ b ∧ a ] P f [ b ∧ a ∧ b ] P = [ a ∧ b ] P in the last step. With the help o f Pr op osition 4.8 it is not hard to verify that whenever a and b commute then M a ∨ b = M a ∪ M b = M b ∨ a , so φ A preserves co mm uting joins. B ut since for ar bitrary a and b their join can b e expressed as a commuting join a ∨ b = ( a \ b ) ∨ ( b \ a ) ∨ ( b ∧ a ), and we a lr eady know that φ A preserves \ and ∧ , it fo llows that φ A preserves jo ins. Naturality of φ A is chec ked as in the rig ht -handed case. 19 The natural isomor phism ψ from the identit y to S ◦ A is defined as in the right-handed cas e . Given a r ectangular skew B o olean s pace ( p : E → B , f ), let ψ p, f be the morphism cons is ting of the tw o homeomorphis ms ( ψ p, f ) ♭ = h and ( ψ p, f ) ♯ = g from diagram (7). All that we need to chec k in addition to what was alrea dy check ed for ψ in Section 4 .3 is that g preserves the rectang ular ba nd structure. F or any b ∈ B , x, y ∈ E b and T ∈ A p we hav e g ( x f y ) ∈ M T ⇐ ⇒ x f y ∈ T . On the o ther hand, if g ( x ) = [ S ] h ( b ) and g ( y ) = [ R ] h ( b ) then g ( x ) f g ( y ) ∈ M T ⇐ ⇒ [ S f R ] h ( b ) ∈ M T ⇐ ⇒ S f R θ h ( b ) T a nd b ∈ p ( T ) ⇐ ⇒ x f y ∈ T . W e see that g ( x ) f g ( y ) and g ( x f y ) have the same neighborho o ds , therefor e they ar e equal.  It may b e argued that our duality has not g one all the way fro m algebra to geometry b ecause a rectangular skew Bo olean space still car ries the algebr a ic structure of a r ectangular ba nd. How ever, this is not really an ho ne s t a lgebraic structure, as can b e susp ected fr om the fact that the categor y of non-empty rectangular bands is equiv alent to the categor y of pairs of sets. The equiv alence takes a recta ng ular band ( A, ∧ ) to the pa ir of sets ( A/ R , A/ L ) where A/ R a nd A/ L are the q uotients of A by Green’s re la tions R and L , respec tively . In the other direction, a pair of s ets ( X , Y ) is mapped to the r ectangular band X × Y with the opera tion ( x 1 , y 1 ) ∧ ( x 2 , y 2 ) = ( x 1 , y 2 ). The analogous decomp ositio n of re c tangular s kew Boo le a n spaces yields the following v ar iant of duality for skew alg ebras. Theorem 4.15 The c ate gory of skew Bo ole an algebr as with int erse ctions is dual to the c ate gory of p airs of skew Bo ole an sp ac es with c ommon b ase. Pr o of. A pair of skew Bo olean spa c es with a common base is a diagr am E L p L / / B E R p R o o (8) where p L : E L → B and p R : E R → B are skew Bo o lean spaces. A mo r phism is a commutativ e diagr am E L p L / / g L  B h  E R p R o o g R  E ′ L p ′ L / / B ′ E ′ R p ′ R o o (9) in which the left- a nd r ight-hand square are mor phisms of skew Bo o lean spa ces (in the vertical direction). The diagr ams are comp osed in the obvious w ay and we clear ly get a c a tegory . W e establish the duality by showing that the catego ry of pairs of skew Bo olean spa ces with co mmon base is equiv a lent to the catego ry of rec tangu- lar skew B o olean space s . T he idea is to ha ve equiv a lence functors w ork at the 20 level of fib ers in the same w ay as the equiv alence o f non-e mpt y r ectangular bands and pairs of s ets. T o conv ert a pair of skew Bo olean spaces (8) into a rectangular Bo ole a n space we form the pullback E   / / _  E R p R   E L p L / / B to obtain a skew Bo o le an space p : E → B . Concretely , the fib er E x ov er x ∈ B co ns ists of pair s ( u, v ) ∈ E L × E R such that p L ( u ) = x = p R ( v ), and p ( u, v ) = p L ( u ) = p R ( v ). The r ectangular band op er ation f on p : E → B defined by ( u 1 , v 1 ) f ( u 2 , v 2 ) = ( u 1 , v 2 ) , obviously makes the fib ers into rectangula r bands. A morphis m (9) corresp onds to the mor phism E g / p   E ′ p ′   B h / B ′ where g is the par tial map with domain dom( g L ) × B dom( g R ) defined b y g ( x, y ) = ( g L ( x ) , g R ( y )) . It clear ly preserves f . In the opp osite dir ection we star t with a re c tangular s kew Bo olea n spa ce ( p : E → B , f ) and form a pair of skew Bo o lean spa ces with a common base as follows. First construct the fib er-wise Green’s rela tion L o n E as the e q ualizer L ℓ / / ( ( R R R R R R R R R R R R R R R R R E × B E   id E × B E / / g × B f / / E × B E u u j j j j j j j j j j j j j j j j j B in the top os of ´ etale ma ps with base B . In the ab ov e diagr am g : E × B E → E is the op er a tion as s o ciated with f by x g y = y f x . Still in the topo s, w e form the co equa lizer L π 1 ◦ ℓ / / π 2 ◦ ℓ / / & & M M M M M M M M M M M M M E q L / / p   E L p L x x p p p p p p p p p p p p p B The quotien t E L is Hausdorff b ecause b y Pr op osition 2.2 the map q L is op en, and a pair of p oints in E may always be separated b y clo p en sections. W e now hav e one of the skew Bo olean spa ces p L : E L → B , and there is an analogous construction o f p R : E R → B . On a s ing le fib er E x ov er x ∈ B the functor just per forms the usual decomp osition of the rectangula r band E x int o its left- a nd 21 right-handed factors E x / R x and E x / L x . This is so b ecause by Prop osition 2.2 equalizers and co equalizer s of ´ etale maps are computed fib er -wise. A mo rphism ( g , h ) from ( p : E → B , f ) to ( p ′ : E ′ → B ′ , f ′ ), displayed explicitly a s inclusions o f the domains of definition and total maps, E p   dom( g ) o o   g / / E ′ p ′   B dom( h ) o o h / / B ′ (10) corres p o nds to a mor phis m b etw een pair s of skew Bo o lean spaces as describ ed next. Consider the commutativ e diagra m L ∩ (dom( g ) × B dom( g )) / / / / * * T T T T T T T T T T T T T T T T T T dom( g ) q L / / p   q L (dom( g ))  _   B E L p L o o where the tw o parallel arrows in to dom( g ) are the restrictio ns o f π 1 ◦ l a nd π 2 ◦ l to dom( g ). By Pr op osition 2.2 the map q L is o p en, hence q L (dom( g )) is an op en subspace of E L . Mo reov er, be c ause dom( g ) is a n op en subs pace of E and q L is op en, it is not hard to chec k tha t the top row of the diagram is a co e qualizer. Because g comm utes with f , the map q ′ L ◦ g factors throug h the co e q ualizer, dom( g ) g / / q L   E ′ q ′ L   q L (dom( g )) g L / / E ′ L W e hav e obtained a par tia l map g L : E L ⇀ E ′ L whose do ma in q L (dom( g )) is a n op en subspace of E L . Also q L is prop er b ecaus e g is prop er . W e simila rly obtain the rig ht-handed v ers ion g R : E R ⇀ E ′ R . T his gives us the desired morphism E L p L / / g L  B h  E R p R o o g R  E ′ L p ′ L / / B ′ E ′ R p ′ R o o T o see that the tw o functors just des crib ed form a n equiv ale nce, we use the fact that fib er -wise they cor resp ond to the eq uiv alence b etw een no n-empty re ctan- gular bands and pairs of non-empty se ts . W e o mit the deta ils.  5 V aria tions The mor phisms b etw een skew Bo olea n spaces were determined by our taking al l homomorphisms on the algebraic side of dua lity . In this section we consider 22 several v ar iants in which the ho momorphisms are restricted. W e limit attention to the right-handed ca s e, and ask the kind r e ader who will work out the gener al case to let us know whether there are a ny surpris e s . As a pr eparation we first show ho w morphisms o f skew Boole an spaces decomp o s e into open inclusions and pullbacks. Lemma 5.1 Supp ose p : E → B and p ′ : E ′ → B ar e skew Bo ole an sp ac es and g : E → E ′ is a pr op er c ontinu ous map such that E g / / p   @ @ @ @ @ @ @ E ′ p ′ ~ ~ } } } } } } } B c ommu tes. If g is bije ctive on fib ers then it is a home omorphism. Pr o of. It is obvious that g is a bijection, so we only nee d to chec k that it is a closed map. If K ⊆ E ′ is c o mpact then the res triction g | g − 1 ( K ) : g − 1 ( K ) → K is a clos ed map b ecause it maps from the compact spa c e g − 1 ( K ) to the Hausdorff space E ′ . Therefor e, if F ⊆ E is clo sed then g ( F ) ∩ K = g | g − 1 ( K ) ( F ∩ g − 1 ( K )) is clos ed in K for every compact K ⊆ E ′ . Becaus e E ′ is lo ca lly compact, it is compactly gener ated and we may conclude that g ( F ) is closed.  Lemma 5.2 Supp ose p : E → B and p ′ : E ′ → B ′ ar e skew Bo ole an sp ac es and g : E → E ′ is a pr op er c ontinu ous map. A c ommutat ive squar e E g / / p   E ′ p ′   B h / / B ′ is a pul lb ack if, and only if, g is bi je ctive on fib ers. Pr o of. It is e a sy to chec k that g is bijective on fib er s if the squar e is a pullback. Conv ersely , supp ose g is bijective on fib ers. W e form the pullback of h and p ′ and obtain a factoriz a tion e , as in the diag ram E e / / p   ? ? ? ? ? ? ? ? g ! ! P q / /   _  E ′ p ′   B h / / B ′ The map e is prop er b eca use g is pr op er. Indeed, if K ⊆ P is compact then e − 1 ( K ) is a closed subset of the compac t subset g − 1 ( q ( K )), ther e fore it is co m- pact. F urthermor e, e is bijective on fib ers b ecause g and q a re. By Lemma 5.1 the map e is a homeomorphism, ther efore the outer squa r e is a pullba ck.  23 Consider a mor phism o f B o olean s pa ces, with inclusion o f domains displayed explicitly: E p   dom( g ) / / o o   g / / E ′ p ′   B dom( h ) o o h / / B ′ (11) The left squar e need not b e a mor phism in our category be cause inclusions of op en subsets need not be prop er. If we turn them aro und they be c ome p artial identities with op en do mains of definitions, a nd we do g et a dec o mp o sition E p   / dom( g )   g / / E ′ p ′   B / dom( h ) h / / B ′ (12) in which both squar es are mor phisms b etw een skew Boo lean space s. Now b y Lemma 5.2 the condition that g is bijective on fib ers is equiv alent to the righ t square b eing a pullback. Therefore , every mor phism can b e decomp osed into a partial identit y (with op en domain o f definition) a nd a pullback. What do es the decomp osition (12) co rresp ond to on the alg ebraic side o f duality? In order to answer the question, we need to study a ce rtain kind of idea ls in skew algebra s . A ≤ - ide al of a skew algebr a A is a subset I ⊆ A which is closed under finite joins and the natural partial or de r ≤ . In particular , I is nonempty a s it contains the empty join 0 . A ≤ -idea l may eq uiv alently be describ ed a s a subalg ebra that is closed under the natural pa rtial order becaus e in a skew alg ebra w e alw ays hav e x ∧ y ≤ y ∨ x . The follo wing Lemma giv es an e x plicit des cription of the ≤ -ideal g e ne r ated by a given subset, akin to how se ts generate ideals in rings . Lemma 5.3 The ≤ -ide al h S i ≤ gener ate d by a subset S ⊆ A is forme d as the closur e by finite joins of the downar d closur e of S with r esp e ct t o the natu r al p artial or der: h S i ≤ = { x 1 ∨ · · · ∨ x n | ∀ i ≤ n . ∃ y i ∈ S . x i ≤ y i } . Pr o of. Bec a use a ny ideal that contains S als o c o ntains h S i ≤ we only hav e to chec k that h S i ≤ is a ≤ -idea l. The set h S i ≤ is obviously closed under joins. T o see that it is clo s ed under the natural partia l or der, let x ≤ x 1 ∨ · · · ∨ x n where x i ≤ y i and y i ∈ S . Then x = ( x 1 ∨ · · · ∨ x n ) ∧ x ∧ ( x 1 ∨ · · · ∨ x n ) = ( x 1 ∧ x ∧ x 1 ) ∨ · · · ∨ ( x n ∧ x ∧ x n ) , where we canceled all terms of the form x i ∧ x ∧ x j with i 6 = j , by the usual argument in the s kew lattice theor y that amo unt s to the fact that ( A, ∨ ) is r e gular as a band, i.e., it satisfies the iden tit y a ∨ b ∨ a ∨ c ∨ a = a ∨ b ∨ c ∨ a . Now, x i ∧ x ∧ x i ≤ x i ≤ y i for all i ≤ n and thus x ∈ h S i ≤ .  Note that in the previous lemma the or der of op eratio ns matters. If we first close under joins and then per form the down ward closure we need not get a ≤ -ideal b eca us e the r e s ulting set need not b e closed under joins. 24 W e say that a subset S ⊆ A o f a skew a lgebra A is ≤ -c ofinal when h S i ≤ = A . A ho momorphism is ≤ -cofina l when its image is ≤ -cofinal. If A is commutativ e, S ⊆ A is ≤ -cofina l prec is ely when it is cofinal in the usual sense: for every x ∈ A there is y ∈ S such tha t x ≤ y . Every homomo r phism f : A → A ′ of skew algebra s may b e decomp os ed into a ≤ -co final morphism and an inclusion of a ≤ - ideal A f / / f # # H H H H H H H H H A ′ h im( f ) i ≤ i : : v v v v v v v v v The following tw o prop ositions show that the deco mp o sition is dual to the de- comp osition (12) on the top ologica l side of duality . Prop ositi on 5. 4 Partial identities with op en domains of definition on the top o- lo gic al side ar e dual to inclusions of ≤ -ide als on the alge br aic side . Pr o of. Let p : E → B b e a skew Bo ole a n spac e with ope n subsets U ⊆ E and V ⊆ B such that p ( U ) = V . These determine a morphism of skew Bo o lean spaces E i / p   U p | U   B j / V where i and j are the identit y maps r estricted to U a nd V , resp ectively . It is easy to chec k that the corresp o nding homomorphism f = ( i, j ) ♮ : A p | U → A p is the inclusio n o f the subalg ebra A p | U int o A p . Its image is down ward c losed with r e s p e ct to ≤ b ecause the par tial or der in A p is inclusion of c o p en sections. Conv ersely , let I ⊆ A b e a ≤ -ideal in A and i : I → A the inclusion. The dual of i is the morphism o f Bo olean s pa ces Sk ( A ) i ♯ / q A   Sk ( I ) q I   St ( A ) i ♭ / St ( I ) where i ♯ acts as i ♯ ([ a ] P ) = [ a ] P ∩ I and is defined for any a ∈ I and a pr ime ideal P ⊆ A suc h that a 6∈ P . The domain of i ♯ is ope n be c a use it is the union of those basic cop en sets M a for which a ∈ I . The map i ♯ is op en b ecause it takes M a ⊆ Sk ( A ) to M a ⊆ Sk ( I ). There fore, i ♯ really is (isomor phic to) a partial ident ity with an op en domain of definition. The sa me fact for i ♭ follows easily .  Prop ositi on 5. 5 Pul lb acks on t he t op olo gic al side ar e dual t o ≤ -c ofinal homo- morphisms on t he algebr aic side. 25 Pr o of. Consider a mo rphism of skew Bo o lean spaces that is a pullback, which is equiv alent to it b eing defined everywhere: E _  g / / p   E ′ p ′   B h / / B ′ T o see that the corr esp onding homomorphism f = ( g , h ) ♮ : A p ′ → A p is ≤ - cofinal, let S be a cop en section in E . Be c ause g is everywhere defined and S is c o mpact, there exist finitely ma ny co pe n s ections R 1 , . . . , R n in E ′ such that S is cov ered by the copen sections g − 1 ( R 1 ) , . . . , g − 1 ( R n ). Let T 1 , . . . , T n be defined by T 1 = g − 1 ( R 1 ) ∩ S and T i +1 = ( g − 1 ( R i +1 ) ∩ S ) \ T i for i ≥ 1. Then S = T 1 ∪ . . . ∪ T n = T 1 ∨ . . . ∨ T n where the latter equlity follows b ecause distinct T i ’s hav e disjoint saturations . F urthermore, ea ch T i is contained in g − 1 ( R i ) = f ( R i ) and thus S lies in h im( f ) i ≤ . Next, let f : A → A ′ be a ≤ -co final homomorphism of rig ht -handed skew algebras . W e cla im tha t f ♯ : Sk ( A ′ ) → Sk ( A ) is everywhere defined. T o see this take any [ b ] P ∈ Sk ( A ′ ). Since f is ≤ -cofinal there exist b 1 , . . . , b n ∈ A ′ and a 1 , . . . , a n ∈ A such tha t b = b 1 ∨ . . . ∨ b n and b i ≤ f ( a i ) for all i = 1 , . . . , n . Since b 6∈ P there exists i such that b i 6∈ P . Thus b i θ P f ( a i ) by Lemma 4.2. Let { i 1 , . . . i k } b e the set o f thos e indices i j that satisfy b i j 6∈ P . Then b θ P f ( a i 1 ) ∨ . . . ∨ f ( a i k ) = f ( a i 1 ∨ · · · ∨ a i k ) and thus [ b ] P ∈ do m( f ♯ ).  Let us ca ll a morphis m o f skew Bo olean spaces E g / p   E ′ p ′   B h / B ′ total when bo th g and h are total, and semitotal when h is tota l. A mor- phism is total prec isely when it is a pullback squar e . As direct co nsequences o f Prop os itio ns 5.4 and 5.5 we obtain the dualities stated by the following pair of theorems. Theorem 5.6 The c ate gory of skew Bo ole an sp ac es and p artial identities with op en domains is dual t o the c ate gory of right-hande d skew algebr as and inclusions of ≤ -ide als. Theorem 5.7 The c ate gory of skew Bo ole an sp ac es and total morphisms is dual to the c ate gory of right-hande d skew algebr as and ≤ -c ofinal homomorphisms. The duality fo r the semito ta l morphisms is similar to duality for total mor- phisms, exce pt that we have to replace ≤ -cofinality with  - c o finality: a subset S ⊆ A of a skew alg ebra is  -c ofinal whe n the ideal genera ted by S eq uals A . Such an ideal may b e computed either as the ∨ -clos ur e of  -closur e of S , or as the  -clo sure of ∨ -closur e of S . A homomorphism f : A → A ′ is  -cofinal when its image is  -cofina l. Because the image is closed under finite joins,  -co finality of f amounts to 26 the following condition: for every y ∈ A ′ there is x ∈ A such that y  f ( x ). Cofinal homomorphisms b e tween commutativ e algebras are also known as pr op er homomorphisms, but we av oid this terminolog y b e cause we a lr eady use the term prop er o n the top olog ical side. In a slightly different s etup this kind of duality is considered b y Ga nna Kudryavtsev a [4]. Theorem 5.8 The c ate gory of skew Bo ole an sp ac es and semitotal morphisms is du al to the c ate gory of right-hande d skew algebr as and  -c ofinal homomor- phisms. Pr o of. Assume that h in dia gram (12) is total. F or every S ∈ B ∗ there exist R 1 , . . . , R n ∈ ( B ′ ) ∗ such that S (as a cop en set in B ) is covered by the copen set h − 1 ( R 1 ∪ . . . ∪ R n ). Hence S ≤ h ∗ ( R 1 ∨ . . . ∨ R n ). T o prov e the conv erse , assume that f : A → A ′ is a homomorphism of rig ht - handed skew algebr as that is cofina l with resp ect to  . W e claim that f ♭ : St ( A ′ ) → St ( A ) is ev erywher e defined. T o s ee this tak e any P ∈ St ( A ′ ). Since f is  -cofinal it follows tha t im( f ) is not cont ained in a ny pro p er idea l of A ′ . Hence there exists a ∈ A such that f ( a ) 6∈ P , which implies f − 1 ( P ) 6 = A . Thus f − 1 ( P ) is a pr ime ideal in A a nd so f ♭ is defined a t P , namely f ♭ ( P ) = f − 1 ( P ).  T o get s till more v ariatio ns of dualit y w e co nsider no tions of saturation. W e say that a homomor phis m f : A → A ′ of skew a lgebras is D -s atur ate d when its image is sa turated with resp ect to Green’s rela tion D . Lemma 5.9 A homomorphi sm b etwe en skew algebr as is D -s atu r ate d if, and only if, it map s e ach D -class surje ctively onto a D -class. Pr o of. The “if ” part is obvious. F or the “o nly if ” part, let f : A → A ′ be a D -sa turated homomor phism. Suppose a ∈ A and b ∈ A ′ such that f ( a ) D b . Because f is D -satur ated there is a ′ ∈ A s uch tha t f ( a ′ ) = b . Co nsider a ′′ = ( a ′ ∧ a ∧ a ′ ) ∨ a ∨ ( a ′ ∧ a ∧ a ′ ) . It is obvious that a ′′ D a . Because f ( a ) D b we get b ∧ f ( a ) ∧ b = b and b ∨ f ( a ) ∨ b = b , which implies f ( a ′′ ) = ( b ∧ f ( a ) ∧ b ) ∨ f ( a ) ∨ ( b ∧ f ( a ) ∧ b ) = b ∨ f ( a ) ∨ b = b, as desired.  There is a notion of sa turation of morphisms on the to p o logical side, to o . Say that ( g , h ) : p → p ′ from p : E → B to p ′ : E ′ → B ′ is satura te d when the domain of g is satura ted with resp ect to p , i.e., dom( g ) = p − 1 (dom( h )). This is equiv a le nt to the left squa re in (11) b eing a pullback. In a different c ontext such morphisms were calle d pa r tial pullbacks by Erik Palmgren and Steve Vic kers [8]. The tw o notions of saturation ar e not dua l to each other, so they yield tw o more dualities. 27 Theorem 5.10 The c ate gory of right-hande d skew algebr as and D -satur ate d homomorph isms is dual to the c ate gory of skew Bo ole an sp ac es and those mor- phisms E g / p   E ′ p ′   B h / B ′ that satisfy t he fol lowing lifting pr op erty: for every c op en U ⊆ B ′ and c op en se ct ion S of p ab ove h − 1 ( U ) ther e exists a c op en se ction R of p ′ ab ove U such that S = g − 1 ( R ) . Pr o of. Let f : A → A ′ be a D -satura ted homomorphism. T o show that ( f ♯ , f ♭ ) : A p ′ → A p has the desir ed pr op erty , consider a cop en N a ⊆ St ( A ) and a cop en sectio n M b ∈ Sk ( A ′ ) ab ov e N f ( a ) . Becaus e N f ( a ) = N b , we hav e f ( a ) D b and so b y Lemma 5.9 there exists a ′ ∈ A such that a ′ D a and b = f ( a ′ ). So N a = N a ′ and f − 1 ♯ ( M a ′ ) = M f ( a ′ ) = M b . Conv ersely , supp ose w e hav e a morphism b etw een skew Bo ole a n spaces, as in the statement of the theorem. W e need to show that the corres po nding homomorphism f = ( g , h ) ♮ : A p ′ → A p is D -saturated. Let S ⊆ E ′ be a copen section of p ′ ab ov e V ⊆ B ′ , and let R ⊆ E b e a co p en s e ction of p above h − 1 ( V ), i.e., R D g − 1 ( S ) = f ( S ). By the prop er ty o f o ur mo r phism there ex ists a c op en section S ′ ⊆ E ′ ab ov e V suc h that f ( S ′ ) = g − 1 ( S ′ ) = R , a s required.  Theorem 5.11 The c ate gory of skew Bo ole an sp ac es and satur ate d morphisms is dual to the c ate gory of right-hande d skew algebr as and those homomorphisms f : A → A ′ for which h im ( f ) i ≤ is close d under the n at u r al pr e or der. Pr o of. Consider a saturated mor phism E g / p   E ′ p ′   B h / B ′ and let f = ( g , h ) ♮ : A p ′ → A p be the corr esp onding homomor phism. Let R be a section o f p ′ ab ov e a cop en V ⊆ B ′ and S a c op en section of p a bove a cop en U ⊆ h − 1 ( V ). Because dom( g ) is saturated it contains S . F or every x ∈ S there is a cop en section T of p ′ ab ov e V which passes through g ( x ), hence x is cov ered b y the co p en section g − 1 ( T ). Because S is compact, there are finitely many sections T 1 , . . . , T n of p ′ ab ov e V suc h that each x ∈ S is covered by some g − 1 ( T i ). If we let S i = S ∩ g − 1 ( T i ) then S = S 1 ∨ · · · ∨ S n and S i ⊆ g − 1 ( T i ). W e hav e prov ed that h im( f ) i ≤ is clo s ed under  . Conv ersely , consider a homomo rphism f : A → A ′ such that h im( f ) i ≤ is closed under  . Supp ose P ⊆ A ′ is a prime ideal s uch that f − 1 ( P ) is als o a prime ideal. Giv en any a ′ ∈ A ′ − P , we need to show that f ♯ is defined at [ a ′ ] P . There exists a ∈ A s uch that f ( a ) 6∈ P . B y Lemma 5.12, prov ed b elow, there is b ∈ A ′ − P such that a ′ θ P b a nd b D f ( a ). It follows that b  f ( a ) 28 and thus b ∈ h im( f ) i ≤ by the a s sumption. Hence there exits a 1 , . . . , a n ∈ A and b 1 , . . . , b n ∈ A ′ such tha t b = b 1 ∨ · · · ∨ b n and b i ≤ f ( a i ) fo r all i . Beca use b = b 1 ∨ · · · ∨ b n and b 6∈ P it follows that not all b i can lie in P . Say b 1 , . . . , b j 6∈ P while b j +1 , . . . , b n ∈ P . Then b θ P b 1 ∨ · · · ∨ b j θ P f ( a 1 ∨ · · · ∨ a j ) by Lemma 4.2 and so f ♯ ([ a ′ ] P ) = f ♯ ([ b ] P ) = [ a 1 ∨ · · · ∨ a j ] f ♭ ( P ) is defined.  Lemma 5.12 L et A b e a skew algebr a and P ⊆ A a prime ide al. F or every a, b ∈ A − P ther e is c ∈ A − P s u ch that a θ P c and c D b . Pr o of. If we ta ke c = ( a ∧ b ∧ a ) ∨ b ∨ ( a ∧ b ∧ a ) then c D b obviously holds. Next, a 6∈ P a nd b 6∈ P together imply a ∧ b ∧ a 6∈ P , a nd a ∧ b ∧ a θ P a follows by Lemma 4.2. Finally , a ∧ b ∧ a ≤ c and a ∧ b ∧ a θ P c fo llows, a g ain by Lemma 4.2.  6 Lattice secti ons of sk ew algebras A lattic e se ction o f a skew lattice A is a section ℓ : A/ D → A of the canonical pro jection q : A → A/ D which preserves 0, ∧ and ∨ . W e construct a right- handed skew Bo olean alg ebra without a lattice sectio n. This a nswers negatively the op en question whether every s kew lattice ha s a sectio n. Prop ositi on 6. 1 A right-hande d skew algebr a has a lattic e se ction if, and only if, the c orr esp onding skew Bo ole an sp ac e ha s a glob al se ction. Pr o of. Let A b e a s kew algebra with a lattice section ℓ : A/ D → A . F o r every d ∈ A/ D the ´ e tale map q A : Sk ( A ) → St ( A ) has a lo cal section s d : N d → E which maps N d to M ℓ ( d ) . W e can glue the lo ca l sections into a global one as long as they are compatible. T o see that this is the cas e , tak e any d, e ∈ A/ D and compute by Lemma 4 .8 M ℓ ( d ∧ e ) = M ℓ ( e ∧ d ∧ e ) = M ℓ ( e ) ∧ ℓ ( d ) ∧ ℓ ( e ) = q − 1 A ( N d ) ∩ M ℓ ( e ) = q − 1 A ( N d ∩ N e ) ∩ M ℓ ( e ) , and similarly M ℓ ( e ∧ d ) = M ℓ ( d ∧ e ∧ d ) = q − 1 A ( N d ∩ N e ) ∩ M ℓ ( d ) . Therefore, s d and s e restricted to N d ∩ N e are b oth equal to s d ∧ e . Conv ersely , suppo se p : E → B is a skew Bo olean spa ce with a g lobal sectio n s : B → E . Then the ma p V 7→ s ( V ) is a lattice s ection for A p , since s ( ∅ ) = ∅ , s ( U ) ∧ s ( V ) = p − 1 ( p ( s ( U ))) ∩ s ( V ) = p − 1 ( U ) ∩ s ( V ) = s ( U ) ∩ s ( V ) = s ( U ∩ V ) , and similarly s ( U ) ∨ s ( V ) = s ( U ∪ V ).  The prec eeding construction o f a g lobal sec tio n from a lattice sectio n ℓ : A/ D → A used o nly preserv ation of 0 a nd ∧ by ℓ . Thus w e proved in pass ing that a 29 right-handed skew alg ebra has a lattice section if, and only if, it has a section which pr eserves 0 and ∧ . W e next cons tr uct a skew Bo olean space whic h do es not hav e a glo bal section. Consequently , the corr esp onding s kew algebr a do es not ha ve a lattice section. T o find a counter-example we first lo o k at a sufficient condition for existence o f global sections. Prop ositi on 6. 2 A skew Bo ole an sp ac e has a glob al se ction if the b ase sp ac e is a c ountable un ion of c omp act op en sets. Dual ly, a right-hande d skew algebr a has a lattic e se ct ion if it c ontains a c ofinal c ountable chain for to t he natur al pr e or der. Pr o of. Supp ose the base B of a skew Bo olea n space p : E → B of Bo o lean spaces is cov ered by co unt ably many cop en sets. Beca use a finite union of such sets is again cop en, there is in fact a c o untable chain C 0 ⊆ C 1 ⊆ · · · of copen sets which cover B . Each C i has a lo cal section s i : C i → E . These ca n be used to form a globa l section s : B → E which equals s i on C i − C i − 1 . T o transfer the cons tr uction to the a lgebraic side of duality , observe that a sequence c 0 , c 1 , c 2 , . . . in A is a cofinal chain for the natural preor der if, and only if, the corr esp onding se q uence of cop en sets N c 0 , N c 1 , N c 2 , . . . is a chain that covers St ( A ).  Thu s we m ust lo ok for a counter-example whose base space is fairly large. A simple one is the first unco unt able or dinal ω 1 with the in terv al top ology . Recall that the p o int s of ω 1 are the countable or dinals and that the int erv al topo logy is genera ted by the op en interv als I α,β = { γ ∈ ω 1 | α < γ < β } for α < β < ω 1 . W e also need a total space, for which we take the disjoint s um of interv als E = ` α<ω 1 [0 , α ] , where each interv al [0 , α ] car ries the int erv al top ology . The p o int s of E are pairs ( α, β ) with β ≤ α and the bas ic open sets a re of the for m { α } × I β ,γ for β < γ ≤ α . W e define the ´ e ta le map p : E → ω 1 to b e the second pro jectio n, so that the fiber ab ove β contains p oints ( α, β ) with β ≤ α < ω 1 . A globa l s ection ω 1 → E of p is a map β 7→ ( s ( β ) , β ) where s : ω 1 → ω 1 is a progr essive lo c a lly constant map. By pro gressive we mean that β ≤ s ( β ) for all β < ω 1 . T o see that s is lo cally consta nt , obs e rve that for any β ∈ ω the preimage of the op en subset { s ( β ) } × [0 , s ( β )] ⊆ E under the section is op en, hence β has a n op en neig hborho o d which s maps to { s ( β ) } . The map s preserves suprema of mono tone sequences b ecaus e it is lo cally constant. Ab ov e every β < ω 1 there is a fixed point of s , namely the supremum of the monotone sequence β ≤ s ( β ) ≤ s ( s ( β )) ≤ · · · By using this fact rep ea tedly , we obtain a strictly increasing sequence γ 0 < γ 1 < γ 2 < · · · of fixed p o ints of s . Their supre mum γ ∞ = s up i γ i is a fix e d p oint of s , to o. 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