Right-topological semigroup operations on inclusion hyperspaces

RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPERSP A CES VOLOD YM YR GA VR YLKIV Abstract. W e sho w that for a n y discrete semigroup X th e semi gr oup op- eration can b e extended to a ri gh t-topological semigroup operation on the space G ( X ) of inclusion hyperspaces on X . W e detect some imp ortan t sub- semigroups of G ( X ), study the mi nimal ideal, the (topological) cent er, left cancelable elements of G ( X ), and describ e the structure of the semi gr oups G ( Z n ) for small nu mbers n . Contents Int ro duction 2 1. Inclusion hyperspa ces 3 1.1. General definition and reduction to the compact case 3 1.2. Inclusion hyper spaces in the category of compa c ta 4 1.3. Some imp ortant subspaces of G ( X ) 5 1.4. The inner a lg ebraic structure o f G ( X ) 5 2. Extending alg ebraic op erations to inclusio n hypers paces 6 3. Homomorphisms of semigroups of inclusion h yper spaces 12 4. Subgroup oids of G ( X ) 12 5. Ideals and zer os in G ( X ) 14 6. The center of G ( X ) 18 7. The top ologica l center of G ( X ) 19 8. Left cancela ble elements of G ( X ) 20 9. Right ca ncelable elements of G ( X ) 21 10. The structure of the semig r oups G ( H ) ov er finite groups H 24 11. Ac knowledgmen ts 26 References 27 1991 Mathematics Subject Classific ation. 22A15, 54D35. 1 2 VOLOD YMYR GA VR YLKIV Introduction After the t op ologica l pro of of Hindman theorem [H1] g iven b y Galvin and Glazer (unpublished, see [HS, p.102], [H2]) top ologica l metho ds b ecome a sta ndard to ol in the mode r n combinatorics of num b ers, s ee [HS], [P 1 ]. The crucial p oint is tha t the semigro up o pe r ation ∗ de fined on a n y discre te space S can be extended to a right-topolo gical s emigroup op eration on β S , the S tone- ˇ Cech compa c tification of S . The pro duct of tw o ultr a filters U , V ∈ β S can b e found in tw o steps: firstly for every element a ∈ S of the semig roup we extend the lef t shift L a : S → S , L a : x 7→ a ∗ x , to a contin uous ma p β L a : β S → β S . In such a wa y , for every a ∈ S we define the pro duct a ∗ V = β L a ( V ). Then, extending the function R V : S → β S , R V : a 7→ a ∗ V , to a contin uous map β R V : β S → β S , we define the pro duct U ◦ V = β R V ( U ). This product ca n b e also defined directly: this is an ultrafilter with the ba se S x ∈ U x ∗ V x where U ∈ U and { V x } x ∈ U ⊂ V . Endowed with so-extended op era tion the Stone- ˇ Cech co mpactification β S be c omes a compa c t Hausdorff rig h t-top ologica l semigr oup. Becaus e of the compactness the semigr oup β S has idemp otents, minimal (left) ide a ls, etc., who se ex is tence has many imp orta nt combinatorial co nsequences. The Stone- ˇ Cech co mpa ctification β S c a n be considered as a subset of the dou- ble pow e r -set P ( P ( S )). The pow e r-set P ( X ) of any set X (in particular, X = P ( S )) carries a natur a l co mpact Hausdorff topolog y inherited from the Can tor cube { 0 , 1 } X after identification o f each subset A ⊂ X with its characteris tic func- tion. The pow e r-set P ( X ) is a complete distributive lattice wit h respect to the op erations of union and int ersection. The smallest co mplete s ubla ttice of P ( P ( S )) containing β S coincides with the space G ( S ) o f inclusion hype r spaces, a well-studied ob ject in Categorial T o p o lo gy . By definition, a family A ⊂ P ( S ) of non-empty subsets of S is called a n inclusion hyp ersp ac e if together with each set A ∈ A the family A contains all sup ersets of A in S . In [G1] it is shown that G ( S ) is a compa ct Hausdor ff la ttice with r esp ect to the o p er ations of intersection and union. Our principal o bserv atio n is that the algebra ic op era tion o f the s emigroups S can be extended no t only to β S but also to the complete lattice hull G ( S ) of β S in P ( P ( S )). E ndow ed with so -extended op era tion, the space of inclus ion hype r spaces G ( S ) be comes a co mpact Hausdorff rig ht -top ologica l semigr oup containing β S as a closed subsemigr o up. Besides β S , the semigro up G ( S ) contains man y other impor- tant spaces as clos e d subsemig r oups: the sup erex tension λS of S , the space N k ( S ) of k -linked inclusion hyper spaces, the space Fil( S ) of filter s on S (which contains an isomor phic cop y of the global semigro up Γ( S ) of S ), etc. RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 3 W e sha ll study some pr op erties of the s e migroup op eration on G ( S ) and its int erplay with the la ttice structur e o f G ( S ). W e exp ect that studying the a lg e- braic structure o f G ( S ) will hav e some combinatorial co nsequences that ca nnot b e obtained with help of ultrafilter s, see [BGN] for further developmen t of this sub ject. 1. Inclusion hypersp aces In this s ection we r e call s ome basic information a bo ut inclusion hyper s paces. More detail infor mation can b e found in the paper [G1]. 1.1. General definitio n and reduction to the compact case. F or a topolo g- ical space X by exp( X ) we denote the space o f all non-empty closed subspaces of X endowed with the Vietor is top olog y . By an inclusion hyp ersp ac e we mean a closed subfamily F ⊂ exp( X ) that is monotone in the s e ns e that together with each set A ∈ F the family F contains all c losed subsets B ⊂ X that contain A . By [G1], the closure of eac h monotone family in exp( X ) is an inclusion hyperspac e. Consequently , eac h family B ⊂ exp( X ) generates an inclusion h ype r space cl exp( X ) { A ∈ exp( X ) : ∃ B ∈ B with B ⊂ A } denoted by hB i . 1 In this case B is c alled a b ase of F = hB i . An inclusion hyper space h x i genera ted b y a singleton { x } , x ∈ X , is c alled princip al . If X is discr ete, then each monotone family in exp( X ) is an inclusio n hyperspa ce, see [G1]. Denote b y G ( X ) the space of a ll inclus io n h yper spaces with the top olog y g ener- ated by the subbase U + = {A ∈ G ( X ) : ∃ B ∈ A w ith B ⊂ U } a nd U − = {A ∈ G ( X ) : ∀ B ∈ A B ∩ U 6 = ∅ } , where U is open in X . F or a T 1 -space X the ma p η X : X → G ( X ), η X ( x ) = { F ⊂ cl X : x ∈ F } , is an embedding (see [G1]), so we ca n ide ntify principa l inc lus ion hypers paces with elements o f the s pace X . F or a T 1 -space X the spac e G ( X ) is Hausdor ff if and o nly if the space X is normal, see [G1], [M]. In the latter case the map h : G ( X ) → G ( β X ) , h ( F ) = cl exp( β X ) { cl β X F | F ∈ F } , is a ho meomorphism, so we can identif y the space G ( X ) with the space G ( β X ) of inclusion hypers paces ov er the Stone- ˇ Cech compa c tification β X o f the norma l space X , see [M]. Th us we reduce the study of inclusion h y per spaces o ver normal top ological space s to the compact c a se where this c onstruction is w ell-studied. 1 In [G1] the inclusion hyperspace hB i generated by a base B is denot ed b y ↑B . 4 VOLOD YMYR GA VR YLKIV F or a (discrete) T 1 -space the s pa ce G ( X ) contains a (discr ete and) dens e sub- space G • ( X ) consisting of inclusion hyper spaces with finite supp ort. A n inclusion hyperspace A ∈ G ( X ) is defined to have fin ite su pp ort in X if A = hF i for some finite family F of finite subsets of X . An inclusio n h yp erspace F ∈ G ( X ) on a non-c o mpact space X is called fr e e if for ea ch compact subset K ⊂ X a nd any elemen t F ∈ F ther e is another elemen t E ∈ F s uch that E ⊂ F \ K . By G ◦ ( X ) we shall deno te the subse t of G ( X ) consisting of free inclusio n hypers paces. By [G1], for a nor ma l lo cally c ompact space X the subse t G ◦ ( X ) is closed in G ( X ). In the simplest case o f a countable discrete spa ce X = N free inclusio n hyper spaces (called semifilters) on X = N have bee n introduced a nd int ensively studied in [BZ ]. 1.2. Inc lusion h yp erspaces in the category of compacta. The construction of the space of inclusion h yper spaces is functorial and monadic in the categor y C omp of compact Hausdo rff spaces and their co n tin uous map, see [TZ]. T o complete G to a functor o n C omp observe that each contin uo us map f : X → Y b etw e e n compact Hausdorff space s induces a co n tin uous map Gf : G ( X ) → G ( Y ) defined b y Gf ( A ) = h f ( A ) i = { B ⊂ cl Y : B ⊃ f ( A ) for some A ∈ A} for A ∈ G ( X ). The map Gf is well-defined a nd co n tin uous, and G is a functor in the category C omp of compact Haus dorff spaces and their co nt inuous maps, see [TZ]. By Pr op osition 2.3.2 [TZ], this functor is weakly normal in the sense that it is contin uous, mono morphic, epimorphic a nd pres e rves in tersections, singletons, the empty set and weigh t of infinite compacta. Since the functor G preserves monomor phisms, for each clos e d subspace A of a compact Hausdorff s pace X the inclusion map i : A → X induces a top ological embedding Gi : G ( A ) → G ( X ). So we can iden tify G ( A ) with a subspace of G ( X ). Now fo r each inclus ion hyperspace A ∈ G ( X ) we can consider the suppor t of A supp A = ∩{ A ⊂ cl X : A ∈ G ( A ) } and c o nclude tha t A ∈ G (supp A ) becaus e G pr eserves in tersections, see [TZ, § 2.4]. Next, we cons ider the mona dic pr op erties of the functor G . W e recall that a functor T : C omp → C omp is monadic if it can b e completed to a mo nad T = ( T , η , µ ) whe r e η : Id → T and µ : T 2 → T are na tur al transformatio ns (ca lled the unit and multiplication) such that µ ◦ T ( µ X ) = µ ◦ µ T X : T 3 X → T X and µ ◦ η T X = µ ◦ T ( η X ) = Id T X for each compact Hausdor ff space X , see [TZ]. F or the functor G the unit η : Id → G has been defined ab ove while the m ulti- plication µ = { µ X : G 2 X → G ( X ) } is defined by the formula µ X (Θ) = ∪{∩M | M ∈ Θ } , Θ ∈ G 2 X . RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 5 By Pro po sition 3.2.9 of [TZ], the triple G = ( G, η , µ ) is a monad in C omp . 1.3. Some imp ortan t subspaces of G ( X ) . The s pace G ( X ) o f inclusion h ype r- spaces contains many in teresting subspaces. Let X be a top olog ical space a nd k ≥ 2 be a natural n umber . An inclusion hypers pace A ∈ G ( X ) is defined to be • k - linke d if ∩F 6 = ∅ for a ny s ubfamily F ⊂ A with |F | ≤ k ; • c en t er e d if ∩F 6 = ∅ for any finite subfamily F ⊂ A ; • a filt er if A 1 ∩ A 2 ∈ A for all sets A 1 , A 2 ∈ A ; • an u ltr afilter if A = A ′ for any filter A ′ ∈ G ( X ) containing A ; • maximal k -linke d if A = A ′ for any k -linked inclus ion hyperspa ce A ′ ∈ G ( X ) containing A . By N k ( X ), N <ω ( X ), and Fil( X ) we denote the subse ts of G ( X ) consisting of k -linked, centered, and filter inc lus ion hyper spaces, resp ectively . Also by β ( X ) and λ k ( X ) we denote the s ubsets of G ( X ) consisting of ultrafilters and maximal k-linked inclusio n h yper s paces, resp ectively . The spa ce λ ( X ) = λ 2 ( X ) is called the sup er extension of X . The following diagr am describe s the inclusio n relations be tw een subspaces N k X , N <ω X , Fil( X ), λX and β X o f G ( X ) (an arrow A → B means tha t A is a subset of B ). Fil( X ) → N <ω X → N k X → N 2 X → G ( X ) β X ✻ ✲ λX ✻ F or a nor mal space X all the subspa ces from this diagram are closed in G ( X ), see [G1]. F or a non-compact space X we can also consider the intersections Fil ◦ ( X ) =Fil( X ) ∩ G ◦ ( X ) , N ◦ <ω ( X ) = N <ω ( X ) ∩ G ◦ ( X ) , N ◦ k ( X ) = N k ( X ) ∩ G ◦ ( X ) , λ ◦ k ( X ) = λ k ( X ) ∩ G ◦ ( X ) , and β ◦ ( X ) = β X ∩ G ◦ ( X ) = β X \ X . Elements of those sets will b e called free filters, free centered inc lus ion h yp erspaces, free k -linked inclusion hyper spaces, etc. F or a normal lo cally compact space X the subsets Fil ◦ ( X ), N ◦ <ω ( X ), N ◦ k ( X ), λ ◦ ( X ) = λ ◦ 2 ( X ), and β ◦ ( X ) are closed in G ( X ), see [G1]. In co nt rast, λ ◦ k ( N ) is not closed in G ( N ) for k ≥ 3, see [Iv]. 1.4. The inner algebraic s tructure of G ( X ) . In this subsection w e discuss the algebraic structure of the spa ce of inclusio n hyper spaces G ( X ) over a top olo g ical space X . The space of inclusion hypers pa ces G ( X ) p oss esses t w o binary op erations 6 VOLOD YMYR GA VR YLKIV ∪ , ∩ , a nd one unary opera tion ⊥ : G ( X ) → G ( X ) , ⊥ : F 7→ F ⊥ = { E ⊂ cl X : ∀ F ∈ F E ∩ F 6 = ∅} called the transv ersality ma p. These three opera tions a re contin uous and turn G ( X ) into a s ymmetric lattice, see [G1]. Definition 1. 1. A s ymmetric latt ic e is a complete distr ibutiv e lattice ( L, ∨ , ∧ ) endow ed with a n additional unary o p er ation ⊥ : L → L , ⊥ : x 7→ x ⊥ , that is a n inv olutive an ti-isomorphism in the sense that • x ⊥⊥ = x for a ll x ∈ L ; • ( x ∨ y ) ⊥ = x ⊥ ∧ y ⊥ ; • ( x ∧ y ) ⊥ = x ⊥ ∨ y ⊥ ; The smallest element of the lattice G ( X ) is the inclusion hyper space { X } while the lar g est is exp( X ). F or a discrete s pa ce X the set G ( X ) of all inclusion hyper spaces on X is a subset of the double p ower-set P ( P ( X )) (which is a complete dis tr ibutiv e lattice) and is closed under the op eratio ns of union a nd in tersection (of ar bitr ary families of inclusion h yper s paces). Since each inclusion hyperspace is a union of filters and each filter is an inter- section o f ultrafilter s , we o bta in the following prop osition s howing that the lattice G ( X ) is a rather natural ob ject. Prop ositio n 1 .2. F or a discr ete sp ac e X the lattic e G ( X ) c oincides with the smal l- est c omplete sublattic e of P ( P ( X )) c ontaining al l ult ra filters. 2. Extending algebraic opera tions to inclusion hypersp aces In this sec tio n, given a binary (asso ciative) o p er ation ∗ : X × X → X on a discrete space X we extend this o per ation to a rig ht-top ological (ass o ciative) o per ation on G ( X ). This ca n b e done in t w o steps b y analogy with the extension of the op era tion to the Stone- ˇ Cech co mpactification β X of X . First, for each element a ∈ X c onsider the left shift L a : X → X , L a ( x ) = a ∗ x and extend it to a c ontin uous map ¯ L a : β X → β X b etw een the Stone- ˇ Cech compactifications o f X . Next, a pply to this extension the functor G to obtain the contin uous ma p G ¯ L a : G ( β X ) → G ( β X ). Clearly , fo r every inclusion hyperspace F ∈ G ( β X ) the inclusion h y per space G ¯ L a ( F ) has a base { a ∗ F | F ∈ F } . Thus, we hav e defined the pro duct a ∗ F = G ¯ L a ( F ) of the element a ∈ X and the inclusion hyperspace F . F urther, for eac h inclusion hyperspace F ∈ G ( β X ) = G ( X ) we can consider the map R F : X → G ( β X ) defined by the formula R F ( x ) = x ∗ F for every x ∈ X . Extend the map R F to a contin uous map ¯ R F : β X → G ( β X ) and apply to this RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 7 extension the functor G to obtain a map G ¯ R F : G ( β X ) → G 2 ( β X ). Finally , comp ose the map G ¯ R F with the m ultiplication µX = µ G X : G 2 X → G ( X ) of the monad G = ( G, η , µ ) a nd obtain a map µ X ◦ G ¯ R F : G ( β X ) → G ( β X ). F or an inclusion hype r space U ∈ G ( β X ), the ima ge µ G X ◦ G ¯ R F ( U ) is called the pro duct of the inclusio n hypers pa ces U and F and is deno ted by U ◦ F . It follows fro m con tin uit y of the maps G ¯ R F that the extended bina r y ope ration on G ( X ) is contin uo us with r e s pec t to the first argument with the second ar gument fixed. W e a re going to show that the o pe ration ◦ on G ( X ) nicely agrees with the lattice structure of G ( X ) a nd is asso ciative if so is the op eratio n ∗ . Also we shall establish an ea sy formula U ◦ F = h [ x ∈ U x ∗ F x : U ∈ U , { F x } x ∈ U ⊂ F i for calculating the pro duct U ◦ F of t w o inclusion hypers paces U , F . W e start with necessary definitions. Definition 2 .1. Let ⋆ : G ( X ) × G ( X ) → G ( X ) b e a binar y op era tion on G ( X ). W e shall say that ⋆ r esp e cts the lattice structure of G ( X ) if for an y U , V , W ∈ G ( X ) and a ∈ X (1) ( U ∪ V ) ⋆ W = ( U ⋆ W ) ∪ ( V ⋆ W ); (2) ( U ∩ V ) ⋆ W = ( U ⋆ W ) ∩ ( V ⋆ W ); (3) a ⋆ ( V ∪ W ) = ( a ⋆ V ) ∪ ( a ⋆ W ); (4) a ⋆ ( V ∩ W ) = ( a ⋆ V ) ∩ ( a ⋆ W ). Definition 2.2. W e will say that a binar y ope ration ⋆ : G ( X ) × G ( X ) → G ( X ) is right-topolo gical if • for any U ∈ G ( X ) the rig ht shift R U : G ( X ) → G ( X ), R U : F 7→ F ⋆ U , is contin uous; • for any a ∈ X the left shift L a : G ( X ) → G ( X ), L a : F 7→ a ⋆ F , is contin uous. The following uniqueness theorem will be used to find an equiv ale n t description of the induced oper ation on G ( X ). Theorem 2.3. L et ⋆, ◦ : G ( X ) × G ( X ) → G ( X ) b e two right-top olo gic al binary op er ations t hat r esp e ct the lattic e structu re of G ( X ) . These op er ations c oincid e if and only if they c oincide on the pr o duct X × X ⊂ G ( X ) × G ( X ) . Pr o of. It is clear that if these operatio ns coincide o n G ( X ) × G ( X ), then they coincide on the pro duct X × X identified with a subset of G ( X ) × G ( X ). W e recall that each point x ∈ X is identified with the ultrafilter h x i gener ated by x . Now assume co n versely that x ⋆ y = x ◦ y for any tw o p oints x, y ∈ X ⊂ G ( X ). First we chec k that a ⋆ F = a ◦ F for any a ∈ X and F ∈ G ( X ). Since the left 8 VOLOD YMYR GA VR YLKIV shifts F 7→ a ⋆ F and F 7→ a ◦ F are contin uous, it suffices to establish the equa lit y a ⋆ F = a ◦ F for inclusio n hyperspaces F having finite supp or t in X (b ecause the set G • ( X ) of all such inclusion h yp e rspaces is dense in G ( X ), see [G1]). Any such a hyperspa ce F is g enerated by a finite family of finite s ubsets of X . If F = h F i is g enerated by a sing le finite subset F = { a 1 , . . . , a n } ⊂ X , then F = T n i =1 h a i i is the finite intersection of principa l ultrafilters, and hence h a i ⋆ F = h a i ⋆ n \ i =1 h a i i = n \ i =1 h a i ⋆ h a i i = n \ i =1 h a i ◦ h a i i = h a i ◦ n \ i =1 h a i i = h a i ◦ F . If F = h F 1 , . . . , F n i is generated by finite family of finite sets, then F = S n i =1 h F i i and we can use the preceding case to prov e that h a i ⋆ F = h a i ⋆ n [ i =1 h F i i = n [ i =1 h a i ⋆ h F i i = n [ i =1 h a i ◦ h F i i = h a i ◦ n [ i =1 h F i i = h a i ◦ F . Now fixing a ny inclusio n h yp erspace U ∈ G ( X ) by a similar argument one can prov e the eq uality F ⋆ U = F ◦ U for a ll inclusion hype rspaces F ∈ G • ( X ) having finite s upp or t in X . Finally , using the density o f G • ( X ) in G ( X ) and the contin uit y of right shifts F 7→ F ◦ U and F 7→ F ⋆ U one can establish the equalit y F ⋆ U = F ◦ U for all inclusio n hyperspaces F ∈ G ( X ).  The above theorem will b e applied to sho w that the op eratio n ◦ : G ( X ) × G ( X ) → G ( X ) induced b y the oper ation ∗ : X × X → X co incides w ith the op eration ⋆ : G ( X ) × G ( X ) → G ( X ) defined b y the for mula U ⋆ V = h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V i for U , V ∈ G ( X ). First we establish some proper ties of the opera tion ⋆ . Prop ositio n 2.4. The op er ation ⋆ c ommutes with the tr ansversality op er ation in the sense that ( U ⋆ V ) ⊥ = U ⊥ ⋆ V ⊥ for any U , V ∈ G ( X ) . Pr o of. T o prov e that U ⊥ ⋆ V ⊥ ⊂ ( U ⋆ V ) ⊥ , take any element A ∈ U ⊥ ⋆ V ⊥ . W e should check that A intersects each set B ∈ U ⋆ V . Without loss of generality , the sets A a nd B a re of the basic form: A = [ x ∈ F x ∗ G x for some s ets F ∈ U ⊥ and { G x } x ∈ F ⊂ V ⊥ and B = [ x ∈ U x ∗ V x for some s ets U ∈ U and { V x } x ∈ U ⊂ V . Since U ∈ U a nd F ∈ U ⊥ , the intersection F ∩ U co n tains so me p oint x . F or this p oint x the sets V x ∈ V and G x ∈ V ⊥ are well-defined and their int ersection RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 9 V x ∩ G x contains some point y . Then the in tersection A ∩ B c o nt ains the p oint x ∗ y and hence is not empty , which pr ov es that A ∈ ( U ⋆ V ) ⊥ . T o prove tha t ( U ⋆ V ) ⊥ ⊂ U ⊥ ⋆ V ⊥ , fix a set A ∈ ( U ⋆ V ) ⊥ . W e claim that the set F = { x ∈ X : x − 1 A ∈ V ⊥ } belo ngs to U ⊥ (here x − 1 A = { y ∈ X : x ∗ y ∈ A } ). Assuming conv ersely that F / ∈ U ⊥ , we would find a set U ∈ U with F ∩ U = ∅ . By the definition o f F , for each x ∈ U the s et x − 1 A / ∈ V ⊥ and thus we ca n find a set V x ∈ V with empt y intersection V x ∩ x − 1 A . By the definition o f the pro duct U ⋆ V , the s e t B = S x ∈ U x ∗ V x belo ngs to U ⋆ V and hence intersects the set A . Consequently , x ∗ y ∈ A for s ome x ∈ U and y ∈ V x . The inclus ion x ∗ y ∈ A implies that y ∈ x − 1 A ⊂ X \ V x , which is a contradiction proving that F ∈ U ⊥ . Then the sets A ⊃ S x ∈ F x ∗ x − 1 A b elong to U ⊥ ⋆ V ⊥ .  Prop ositio n 2.5. The e quality ( U ∩ V ) ⋆ W = ( U ⋆ W ) ∩ ( V ⋆ W ) holds for any U , V , W ∈ G ( X ) . Pr o of. It is easy to show that ( U ∩ V ) ⋆ W ⊂ ( U ⋆ W ) ∩ ( V ⋆ W ). T o prove the rev erse inclusion, fix a set F ∈ ( U ⋆ W ) ∩ ( V ⋆ W ). Then F ⊃ [ x ∈ U x ∗ W ′ x and F ⊃ [ y ∈ V y ∗ W ′′ y for some U ∈ U , { W ′ x } x ∈ U ⊂ W , a nd V ∈ V , { W ′′ y } y ∈ V ⊂ W . Since U , V are inclusion hyperspa ces, U ∪ V ∈ U ∩ V . F or eac h z ∈ U ∪ V let W z = W ′ z if z ∈ U and W z = W ′′ z if z / ∈ U . It follows that F ⊃ S z ∈ U ∪ V z ∗ W z and hence F ∈ ( U ∩ V ) ⋆ W .  By a nalogy one can prov e Prop ositio n 2.6. F or any U , V , W ∈ G ( X ) and a ∈ X a ⋆ ( V ∪ W ) = ( a ⋆ V ) ∪ ( a ⋆ W ) and a ⋆ ( V ∩ W ) = ( a ⋆ V ) ∩ ( a ⋆ W ) . Combining Prop ositions 2.4 a nd 2.5 we get Corollary 2.7 . F or any U , V , W ∈ G ( X ) we get ( U ∪ V ) ⋆ W = ( U ⋆ W ) ∪ ( V ⋆ W ) . Pr o of. Indeed, ( U ∪ V ) ⋆ W =  (( U ∪ V ) ⋆ W ) ⊥  ⊥ = (( U ∪ V ) ⊥ ⋆ W ⊥ ) ⊥ = =(( U ⊥ ∩ V ⊥ ) ⋆ W ⊥ ) ⊥ = (( U ⊥ ⋆ W ⊥ ) ∩ ( V ⊥ ⋆ W ⊥ )) ⊥ = =( U ⊥ ⋆ W ⊥ ) ⊥ ∪ ( V ⊥ ⋆ W ⊥ ) ⊥ = ( U ⋆ W ) ∪ ( V ⋆ W ) .  10 VOLOD YMYR GA VR YLKIV Prop ositio n 2.8. The op er ation ⋆ : G ( X ) × G ( X ) → G ( X ) , U ⋆ V = h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V i , r esp e cts the lattic e structur e of G ( X ) and is right-top olo gic al. Pr o of. Pro po sitions 2.5, 2.6 and Corollar y 2.7 imply that the op eration ⋆ r esp ects the lattice str ucture of G ( X ). So it remains to chec k that the op er ation ⋆ is right-topolog ic al. First w e check that fo r any U ∈ G ( X ) the rig ht s hift R U : G ( X ) → G ( X ), R U : F 7→ F ⋆ U , is contin uous. Fix any inclusion hyperspaces F , U ∈ G ( X ) and let W + be a sub-ba s ic neigh- bo rho o d of their pro duct F ⋆ U . Find sets F ∈ F a nd { U x } x ∈ F ⊂ U such that S x ∈ F x ∗ U x ⊂ W . Then F + is a neigh bo r ho o d of F with F + ⋆ U ⊂ W + . Now assume that F ⋆ U ∈ W − for some W ⊂ X . O bserve that for an y inclusion hyperspace V ∈ G ( X ) we get the equiv alenc e s V ∈ W − ⇔ W ∈ V ⊥ ⇔ V ⊥ ∈ W + . Consequently , F ⋆ U ∈ W − is eq uiv alent to F ⊥ ⋆ U ⊥ = ( F ⋆ U ) ⊥ ∈ W + . The preceding case yields a neighborho o d O ( F ⊥ ) such that O ( F ⊥ ) ⋆ U ⊥ ∈ W + . Now the contin uity o f the tra nsversalit y op er a tion implies that O ( F ⊥ ) ⊥ is a neighbor ho o d of F with O ( F ⊥ ) ⊥ ⋆ U ∈ W − . Finally , w e pro ve that for every a ∈ X the left shift L a : G ( X ) → G ( X ), L a : F 7→ a ⋆ F , is con tin uous. Given a sub-basic open set W + ⊂ G ( X ) note that L − 1 a ( W + ) is op en becaus e L − 1 a ( W + ) = ( a − 1 W ) + where a − 1 W = { x ∈ X : a ∗ x ∈ W } . On the other hand, a ⋆ F ∈ W − is equiv alent to a ⋆ F ⊥ = ( a ⋆ F ) ⊥ ∈ ( W − ) ⊥ = W + which implies that the preimage L − 1 a ( W − ) = ( L a ( W + )) ⊥ is also o pe n.  The op era tion ◦ has the same prop er ties. Prop ositio n 2.9. The op er ation ◦ : G ( X ) × G ( X ) → G ( X ) , U ◦ V = µ G X ◦ G ¯ R F ( U ) r esp e cts the lattic e structur e of G ( X ) and is right-top olo gic al. Pr o of. F or any U ∈ G ( X ) the right shift R U = µ G ( X ) ◦ G ¯ R U : G ( X ) → G ( X ), R U : F 7→ F ◦ U is contin uous b eing the comp ositio n of contin uo us ma ps. Next for any a ∈ X and F ∈ G ( X ) w e ha v e L a ( F ) = a ◦ F = µ G X ( h a i ∗ F ) = µ G X ( h a ∗ F i ) = a ∗ F = G ¯ L a ( F ) and the map L a ≡ G ¯ L a is co ntin uous. It is known (and easy to verify) tha t the multiplication µ G ( X ) : G 2 ( X ) → G ( X ) is a lattice homomorphism in the sense that µ G ( X ) ( U ∪ V ) = µ G ( X ) ( U ) ∪ µ G X ( V ) and µ G ( X ) ( U ∩ V ) = µ G ( X ) ( U ) ∩ µ G ( X ) ( V ) RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 11 for any U , V ∈ G ( X ). Then for any U , V , W ∈ G ( X ) and a ∈ X we g et ( U ∪ V ) ◦ W = µ G ( X ) ◦ G ¯ R W ( U ∪ V ) = µ G ( X ) ( G ¯ R W ( U ) ∪ G ¯ R W ( V )) = = µ G ( X ) ◦ G ¯ R W ( U ) ∪ µ G ( X ) ◦ G ¯ R W ( V ) = ( U ◦ W ) ∪ ( U ◦ W ) and similarly ( U ∩ V ) ◦ W = ( U ◦ W ) ∩ ( U ◦ W ) . Note that for any a ∈ X a ◦ W = µ G ( X ) ( G ¯ R W ( h a i ) = h ¯ R W ( { a } ) i = h ¯ R W ( a ) i = a ∗ W . Consequently , a ◦ ( V ∪ W ) = a ∗ ( V ∪ W ) = ( a ∗ V ) ∪ ( a ∗ W ) = ( a ◦ V ) ∪ ( a ◦ W ) and similarly a ◦ ( V ∩ W ) = ( a ◦ V ) ∩ ( a ◦ W ).  Since b oth opera tions ◦ and ⋆ are right-top ological and r e spe ct the la ttice struc- ture of G ( X ) we may a pply Theorem 2 .3 to get Corollary 2. 1 0. F or any binary op er ation ∗ : X × X → X t he op er ations ◦ and ⋆ on G ( X ) c oincide. Conse quently, for any inclusion hyp ersp ac es U , V ∈ G ( X ) t heir pr o duct U ◦ V is the inclusion hyp ersp ac e h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V i =  A ⊂ X : { x ∈ X : x − 1 A ∈ V } ∈ U  . Having the apparent description of the op er ation ◦ we can establish its asso cia- tivit y . Prop ositio n 2.11. If the op er ation ∗ on X is asso ciative, then so is the induc e d op er ation ◦ on G ( X ) . Pr o of. It is necess ary to show that ( U ◦ V ) ◦ W = U ◦ ( V ◦ W ) for a ny inclusion hyperspaces U , V , W . T ake any s ubs et A ∈ ( U ◦ V ) ◦ W and choose a set B ∈ U ◦ V such that A ⊃ S z ∈ B z ∗ W z for some family { W z } z ∈ B ⊂ W . Next, for the set B ∈ U ◦ V choos e a set U ∈ U such that B ⊃ S x ∈ U x ∗ V x for so me fa mily { V x } x ∈ U ⊂ V . It is clear that for each x ∈ U and y ∈ V x the pro duct x ∗ y is in B and hence W x ∗ y is defined. Co nsequently , S y ∈ V x y ∗ W x ∗ y ∈ V ◦ W for a ll x ∈ U a nd hence S x ∈ U x ∗ ( S y ∈ V x y ∗ W x ∗ y ) ∈ U ◦ ( V ◦ W ). Since S x ∈ U S y ∈ V x x ∗ y ∗ W x ∗ y ⊂ A , we get A ∈ U ◦ ( V ◦ W ). This proves the inclusion ( U ◦ V ) ◦ W ⊂ U ◦ ( V ◦ W ). T o prov e the reverse inclusion, fix a set A ∈ U ◦ ( V ◦ W ) a nd c hoo se a set U ∈ U such that A ⊃ S x ∈ U x ∗ B x for some family { B x } x ∈ U ⊂ V ◦ W . Next, for each x ∈ U find a set V x ∈ V such that B x ⊃ S y ∈ V x y ∗ W x,y for some family { W x,y } y ∈ V x ⊂ W . Let Z = S x ∈ U x ∗ V x . F o r each z ∈ Z we can find x ∈ U a nd y ∈ V x such that z = x ∗ y 12 VOLOD YMYR GA VR YLKIV and put W z = W x,y . Then Z ∈ U ◦ V a nd S z ∈ Z z ∗ W z ∈ ( U ◦ V ) ◦ W . T aking into account S z ∈ Z z ∗ W z ⊂ S x ∈ U S y ∈ V x x ∗ y ∗ W x,y ⊂ A , we conclude A ∈ ( U ◦ V ) ◦ W .  3. Homomorphisms of semigroups of inclu sion hypersp aces Let us observe that o ur constr uction o f extension of a binary opera tion for X to G ( X ) works w ell b oth for asso cia tiv e a nd non-asso c ia tive oper a tions. Let us recall that a set S endowed with a binary o per ation ∗ : X × X → X is called a gr oup oid . If the oper ation is a sso ciative, then X is called a semigr oup . In the preceding section we have shown tha t for each gr oup oid (semigroup) X the spa ce G ( X ) is a group oid (semigroup) with respe ct to the extended op eration. A map h : X 1 → X 2 betw een t wo group oids ( X 1 , ∗ 1 ) a nd ( X 2 , ∗ 2 ) is called a homomorph ism if h ( x ∗ 1 y ) = h ( x ) ∗ 2 h ( y ) fo r all x, y ∈ X 1 . Prop ositio n 3.1. F or any homomorphism h : X 1 → X 2 b etwe en gr oup oids ( X 1 , ∗ 1 ) and ( X 2 , ∗ 2 ) the induc e d map Gh : G ( X 1 ) → G ( X 2 ) is a homomorphi sm of the gr oup oids G ( X 1 ) , G ( X 2 ) . Pr o of. Given t w o inclusion hyper s paces U , V ∈ G ( X 1 ) obser ve that Gh ( U ◦ 1 V ) = Gh ( h [ x ∈ U x ∗ 1 V x : U ∈ U , { V x } x ∈ U ⊂ V i ) = = h h ( [ x ∈ U x ∗ 1 V x ) : U ∈ U , { V x } x ∈ U ⊂ V i ) = = h [ x ∈ U h ( x ) ∗ 2 h ( V x ) : U ∈ U , { V x } x ∈ U ⊂ V i = = h [ x ∈ h ( U ) x ∗ 2 h ( V x ) : U ∈ U , { h ( V x ) } x ∈ U ⊂ Gh ( V ) i = = h h ( U ) : U ∈ U i ◦ 2 h h ( V ) : V ∈ V i = Gh ( U ) ◦ 2 Gh ( V ) .  Reformulating Prop osition 2.4 in terms of homomo rphisms, we obtain Prop ositio n 3. 2. F or any gr oup oid X t he tr ansversality map ⊥ : G ( X ) → G ( X ) is a homomo rphism of the gr oup oid G ( X ) . 4. Subgr oupoids of G ( X ) In this section we shall show that for a g roup oid X endowed with the dis c rete top ology all (top ologic a lly) clo sed subspaces o f G ( X ) in tr o duced in Section 1.3 are subgroup oids of G ( X ). A subset A of a group oid ( X , ∗ ) is called a sub gr oup oid of X if A ∗ A ⊂ A , where A ∗ A = { a ∗ b : a, b ∈ A } . RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 13 W e a ssume that ∗ : X × X → X is a binar y op eratio n on a discrete s pace X and ◦ : G ( X ) × G ( X ) → G ( X ) is the extension of ∗ to G ( X ). Applying Prop os ition 3.2 we o btain Prop ositio n 4 .1. If S is a sub gr oup oid of G ( X ) , then S ⊥ is a sub gr oup oid of G ( X ) to o. Our next prop o sitions can b e ea sily derived from Coro llary 2.10. Prop ositio n 4.2. The sets Fil( X ) , N <ω ( X ) and N k ( X ) , k ≥ 2 , ar e su b gr oup oids in G ( X ) . Prop ositio n 4.3. The Stone- ˇ Ce ch extension β X and the sup er extension λX b oth ar e close d sub gr oup oids in G ( X ) . Pr o of. The sup erextension λX is a subgr oup oid o f G ( X ) b eing the intersection λ ( X ) = N 2 ( X ) ∩ ( N 2 ( X )) ⊥ of tw o subg roup oids o f G ( X ). By analo gy , β X = Fil( X ) ∩ λ ( X ) is a subgroup oid of G ( X ).  Remark 4. 4. In contrast to λX fo r k ≥ 3 the subset λ k ( X ) need not b e a sub- group oid o f G ( X ). F o r ex a mple, for the cyclic group Z 5 = { 0 , 1 , 2 , 3 , 4 } the subset λ 3 ( Z 5 ) of G ( Z 5 ) contains a maximal 3-linked system L = h{ 0 , 1 , 2 } , { 0 , 1 , 4 } , { 0 , 2 , 4 } , { 1 , 2 , 4 }i whose squar e L ∗ L = h{ 1 , 2 , 4 , 5 } , { 0 , 2 , 3 , 4 } , { 0 , 1 , 3 , 4 } , { 0 , 1 , 2 , 4 } , { 0 , 1 , 2 , 3 }i is not ma ximal 3-linked. By a direct application of Corollar y 2.10 we can als o prov e Prop ositio n 4. 5. The set G • ( X ) of al l inclusion hyp ersp ac es with finite supp ort is a sub gr oup oid in G ( X ) . Finally we find conditions on the op eration ∗ gua ranteeing that the s ubs e t G ◦ ( X ) of free inclusion h yp erspaces is a subgroup oid of G ( X ). Prop ositio n 4.6. Assu m e that for e ach b ∈ X t her e is a finite s u bset F ⊂ X s uch that for e ach a ∈ X \ F t he set a − 1 b = { x ∈ X : a ∗ x = b } is finite. Then the set G ◦ ( X ) is a close d sub gr oup oid in G ( X ) and c onse quently, Fil ◦ ( X ) , λ ◦ ( X ) , β ◦ ( X ) al l ar e close d sub gr oup oids in G ( X ) . Pr o of. T ake tw o free inclusion h ypers paces A , B ∈ G ( X ) and a subset C ∈ A ◦ B . W e should pr ov e that C \ K ∈ A ◦ B for each compact subset K ⊂ X . Without loss of gene r ality , the set C is o f basic for m: C = S a ∈ A a ∗ B a for some set A ∈ A and some family { B a } a ∈ A ⊂ B . 14 VOLOD YMYR GA VR YLKIV Since X is discrete, the set K is finite. It fo llows from our as s umption that there is a finite set F ⊂ X such that for every a ∈ X \ F the se t a − 1 K = { x ∈ X : a ∗ x ∈ K } is finite. The hyperspac e A , b eing free, contains the set A ′ = A \ F . By the same reason, for each a ∈ A ′ the hyperspa ce B co nt ains the set B ′ a = B a \ a − 1 K . Since C \ K ⊃ S a ∈ A ′ a ∗ B ′ a ∈ A ◦ B , we conclude that C \ K ∈ A ◦ B .  Remark 4.7. If X is a semig roup, then G ( X ) is a s emigroup and all the sub- group oids considered above are closed subsemigroups in G ( X ). So me of them a re well-kno wn in Semigroup Theory . In pa rticular, so is the semigr oup β X of ultra - filter and β ◦ ( X ) = β X \ X of free ultrafilter s. The semigr o up Fil( X ) contains a n isomorphic copy of the global semigr oup of X , which is the hyperspace exp( X ) endow ed with the semigroup o per ation A ∗ B = { a ∗ b : a ∈ A, b ∈ B } . 5. Ideals and zeros in G ( X ) A non-empty subset I o f a gro upo id ( X, ∗ ) is called an ide al (resp. right ide al , left ide al ) if I ∗ X ∪ X ∗ I ⊂ I (res p. I ∗ X ⊂ I , X ∗ I ⊂ I ). An element O of a group oid ( X, ∗ ) is called a zer o (resp. left zer o , right zer o ) in X if { O } is a n ideal (resp. rig ht idea l, left ideal) in X . E ach right o r left zero z ∈ X is an idemp otent in the sense that z ∗ z = z . F or a group oid ( X , ∗ ) righ t zeros in G ( X ) admit a simple description. W e define an inclusion hypers pace A ∈ G ( X ) to b e shift-invariant if for every A ∈ A and x ∈ X the s e ts x ∗ A and x − 1 A = { y ∈ X : x ∗ y ∈ A } b elong to A . Prop ositio n 5. 1. An inclus ion hyp ersp ac e A ∈ G ( X ) is a right zer o in G ( X ) if and only if A is shift-invariant. Pr o of. Assuming that an inclusion hyper space A ∈ G ( X ) is shift-inv ar iant, we shall show that B ◦ A = A for every B ∈ G ( X ). T ake a n y set F ∈ B ◦ A and find a set B ∈ B and a family { A x } x ∈ B ⊂ A such that S x ∈ B x ∗ A x ⊂ F . Since A ∈ G ( X ) is shift-inv ar iant, S x ∈ B x ∗ A x ∈ A and thus F ∈ A . This proves the inclusion B ◦ A ⊂ A . On the other hand, for every F ∈ A and ev ery x ∈ X we get x − 1 F ∈ A and thus F ⊃ S x ∈ X x ∗ x − 1 F ∈ B ◦ A . This shows that A is a right zero of the semig roup G ( X ). Now assume that A is a righ t zero of G ( X ). Observe that for every x ∈ X the equality h x i ◦ A = A implies x ∗ A ∈ A for ev ery A ∈ A . One the o ther hand, the equa lit y { X } ◦ A = A implies that for every A ∈ A there is a family { A x } x ∈ X ⊂ A such that S x ∈ X x ∗ A x ⊂ A . Then for every x ∈ X the set x − 1 A = { z ∈ X : x ∗ z ∈ A } ⊃ A x ∈ A b elongs to A witnessing that A is shift-inv ariant.  RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 15 By ↔ G ( X ) we denote the set of shift-inv ar iant inclus ion hyperspa ces in G ( X ). Prop ositio n 5 .1 implies that A ◦ B = B for every A , B ∈ ↔ G ( X ). This means that ↔ G ( X ) is a re c tangular semigro up. W e recall that a semigr oup ( S, ∗ ) is called r e ctangular (or els e a semigr oup of right zer os ) if x ∗ y = y for all x, y ∈ S . Prop ositio n 5.2 . The set ↔ G ( X ) is close d in G ( X ) , is a r e ctangular su bsemigr oup of t he gr oup oid G ( X ) and is close d c omplete sublattic e of the latt ic e G ( X ) invariant under the tr ansversality map. Mor e over, if ↔ G ( X ) is non- empty, then it is a left ide al that lies in e ach right ide al of G ( X ) . Pr o of. If A ∈ G ( X ) \ ↔ G ( X ), then ther e exists x ∈ X and A ∈ A s uch that x ∗ A / ∈ A or x − 1 A / ∈ A . Then O ( A ) = { A ′ ∈ G ( X ) : A ∈ A ′ and ( x ∗ A / ∈ A ′ or x − 1 A / ∈ A ) } is an op en neighbor ho o d of A missing the s et ↔ G ( X ) and witnes sing that the set ↔ G ( X ) is closed in G ( X ). Since A ◦ B = B for every A , B ∈ ↔ G ( X ), the set ↔ G ( X ) is a rectang ular subsemi- group of the group oid G ( X ). T o show that ↔ G ( X ) is inv ar ia nt under the transversality op eratio n, note that for every A ∈ G ( X ) and Z ∈ ↔ G ( X ) we g et A ◦ Z ⊥ = ( A ⊥ ◦ Z ) ⊥ = Z ⊥ which means that Z ⊥ is a right zero in G ( X ) and thus b elongs to ↔ G ( X ) accor ding to Prop ositio n 5.1. T o show tha t ↔ G ( X ) is a co mplete sublattice of G ( X ) it is ne c essary to chec k that ↔ G ( X ) is clo sed under ar bitrary unions and intersections. It is trivia l to check that arbitrar y unio n of shift-in v ariant inclusion hyperspac es is shift-in v ariant, which means that S α ∈ A Z α ∈ ↔ G ( X ) for any family {Z α } α ∈ A ⊂ ↔ G ( X ). Since ↔ G ( X ) is closed under the transversality op eration we also get \ α ∈ A Z α =  [ α ∈ A Z ⊥ α ) ⊥ ∈ ↔ G ( X ) ⊥ = ↔ G ( X ) . If ↔ G ( X ) is no t empt y , then it is a left ideal in G ( X ) beca use it consists o f right zeros. Now take an y rig h t ideal I in G ( X ) and fix any elemen t R ∈ I . Then for every Z ∈ ↔ G ( X ) w e get Z = R ◦ Z ∈ I which yields ↔ G ( X ) ⊂ I .  Prop ositio n 5.3. If X is a semigr ou p and ↔ G ( X ) is not empty, then ↔ G ( X ) is the minimal ide al of G ( X ) . Pr o of. In light of the preceding prop osition, it suffices to ch eck that ↔ G ( X ) is a right ideal. T ake any inclusion hype rspaces A ∈ ↔ G ( X ) and B ∈ G ( X ) a nd ta ke an y set 16 VOLOD YMYR GA VR YLKIV F ∈ A ◦ B . W e need to show that the sets x ∗ F and x − 1 F b e long to A ◦ B . Without loss of g enerality , F is of the basic form: F = [ a ∈ A a ∗ B a for some set A ∈ A a nd some family { B a } a ∈ A ⊂ B . The a sso ciativity of the semigroup op eratio n on S implies that x ∗ F = [ a ∈ A x ∗ a ∗ B a = [ z ∈ x ∗ A z ∗ B a ( z ) ∈ A ◦ B where a ( z ) ∈ { a ∈ A : x ∗ a = z } for z ∈ x ∗ A . T o see that x − 1 F ∈ A observe that the set A ′ = S z ∈ x − 1 A z ∗ B xz belo ngs to A and each p oint a ′ ∈ A ′ belo ngs to the s et z ∗ B xz for some z ∈ x − 1 A . Then x ∗ a ′ ∈ x ∗ z ∗ B xz ⊂ F and hence A ∋ A ′ ⊂ x − 1 F , which yields the desired inclusion x − 1 F ∈ A .  Now we find conditions on the binary op era tion ∗ : X × X → X guara n teeing that the set ↔ G ( X ) is no t e mpty . By min GX = { X } and max GX = { A ⊂ X : A 6 = ∅} we deno te the minimal and maximal e le men ts of the lattice G ( X ). Prop ositio n 5.4. F or a gr oup oid ( X , ∗ ) the fol lowing c onditions ar e e quivalent: (1) min GX ∈ ↔ G ( X ) ; (2) max GX ∈ ↔ G ( X ) ; (3) fo r e ach a, b ∈ X the e quation a ∗ x = b has a solution x ∈ X . Pr o of. (1) ⇒ (3 ) Assuming that min GX ∈ ↔ G ( X ) and applying Pro po sition 5.1 observe that for every a ∈ X the equation h a i ◦ { X } = { X } implies tha t for ev ery b ∈ X the equation a ∗ x = b has a solution. (3) ⇒ (1) If for every a, b ∈ X the equation a ∗ x = b has a solution, then a ∗ X = X and he nce F ◦ { X } = { X } for all F ∈ G ( X ). This means th at { X } = min G ( X ) is a right zero in G ( X ) and hence belo ngs to ↔ G ( X ) according to Prop ositio n 5.1. (2) ⇒ (3 ) Ass ume that ma x G ( X ) ∈ ↔ G ( X ) and take any p oints a, b ∈ X . Since h a i ◦ max G ( X ) = max G ( X ) ∋ { b } , there is a non-empty s e t X a ∈ max G ( X ) with a ∗ X a ⊂ { b } . Then any x ∈ X a is a solution of a ∗ x = b . (3) ⇒ (2) Assume tha t for every a, b ∈ X the equation a ∗ x = b has a solution. T o show tha t F ◦ max G ( X ) = max G ( X ) it suffices to check that max G ( X ) ⊂ F ◦ max G ( X ). T ake an y set B ∈ ma x G ( X ) a nd an y set F ∈ F . F or ev ery a ∈ F find a p oint x a ∈ X with a ∗ x a ∈ B . Then the sets S a ∈ F a ∗ { x a } ⊂ B b elong to F ◦ max G ( X ), which yields the desir ed inclusion max G ( X ) ⊂ F ◦ max G ( X ).  By analogy we can establish a similar description of zeros and the minimal ideal in the semigr oup G ◦ ( X ) of free inclusio n hyperspaces . RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 17 Prop ositio n 5. 5. Assume that ( X , ∗ ) is an infin ite gr oup oid such that for e ach b ∈ X ther e is a finite su bset F ⊂ X su ch that for e ach a ∈ X \ F the set a − 1 b = { x ∈ X : a ∗ x = b } is finite and not empty. Then (1) G ◦ ( X ) is a close d sub gr oup oid of G ( X ) ; (2) G ◦ ( X ) is a left ide al in G ( X ) pr ovide d if for e ach a, b ∈ X the set a − 1 b is finite; (3) the set ↔ G ◦ ( X ) = ↔ G ( X ) ∩ G ◦ ( X ) of shift-invariant fr e e inclusion hyp ersp ac es is the minimal ide al in G ◦ ( X ) ; (4) the set ↔ G ◦ ( X ) is a r e ctangular subsemigr oup of the gr oup oid G ( X ) and is close d c omplete sublattic e of the lattic e G ( X ) invariant under the tr ansver- sality map. Remark 5 .6. It follows from Prop ositio ns 5.2 and 5.5 that the minimal idea ls of the semigr oups G ( Z ) and G ◦ ( X ) are closed. In co ntrast, the minimal ideals of the semigroups β Z and β ◦ Z = β Z \ Z are not clo sed, see [HS, § 4.4]. Minimal left ideals of the semigr oup β ◦ ( Z ) play an imp ortant role in Co m bina- torics of Numbers, see [HS]. W e b elieve that the same will happe n fo r the semig roup λ ◦ ( Z ). The following propo sition implies that minimal left ideals of λ ◦ ( Z ) co ntain no ultrafilter! Prop ositio n 5.7 . If a gr oup oid X admits a homomorphism h : X → Z 3 such that for every y ∈ Z 3 the pr eimage h − 1 ( y ) is not empty (is infinite) then e ach minimal left ide al I of λ ( X ) (of λ ◦ ( X ) ) is disjoint fr om β ( X ) . Pr o of. It follows that the induced map λh : λ ( X ) → λ ( Z 3 ) is a surjective ho mo- morphism. Consequently , λh ( I ) is a minimal left ideal in λ ( Z 3 ). Now obser ve that λ ( Z 3 ) co nsists of four maximal linked inclusion hyperspac e s. Besides thr ee ultra- filters there is a maximal linked inclusion hyperspace L △ = h{ 0 , 1 } , { 0 , 2 } , { 1 , 2 } i where Z 3 = { 0 , 1 , 2 } . O ne can chec k tha t {L △ } is a zero of the semigr oup λ ( Z 3 ). Consequently , λ ( h )( I ) = {L △ } , which implies that I ∩ β ( X ) = ∅ . Now assume that for ev ery y ∈ Z 3 the preimage h − 1 ( y ) is infinite. W e claim that λh ( λ ◦ ( X )) = λ ( Z 3 ). T ake any maximal linked inclusion hyperspac e L ∈ λ ( Z 3 ). If L is an ultra filter suppo rted by a p o int y ∈ Z 3 , then w e can take any free ultrafilter U on X containing the infinite set h − 1 ( y ) and obse rve that λh ( U ) = L . It r emains to consider the c ase L = L △ . Fix free ultra filters U 0 , U 1 , U 2 on X containing the sets h − 1 (0), h − 1 (1), h − 1 (2), resp ectively . Then L = ( U 0 ∩ U 1 ) ∪ ( U 0 ∩ U 2 ) ∪ ( U 1 ∩ U 2 ) is a free maximal linked inclusion hyperspace whose image λh ( L X ) = L △ . Given any minimal left ideal I ⊂ λ ◦ ( X ) we obtain that the image λh ( I ), b e- ing a minimal left ideal o f λ ( Z 3 ) coincides with {L △ } and is disjoint from β ( Z 3 ). Consequently , I is disjoin t from β ( X ).  18 VOLOD YMYR GA VR YLKIV 6. The center o f G ( X ) In this section we describ e the str ucture of the cent er of the group oid G ( X ) for each (quas i)g roup X . By definitio n, the c enter o f a gro up oid X is the set C = { x ∈ X : ∀ y ∈ X xy = y x } . A gr oup oid X is called a quasigr oup if for ev ery a, b ∈ X the sys tem of equations a ∗ x = b a nd y ∗ a = b has a unique solution ( x, y ) ∈ X × X . It is clea r that each group is a quasigr oup. On the other hand, there a re man y examples of quasig roups, not isomorphic to groups, see [Pf], [CPS]. Theorem 6. 1. L et X b e a quasigr oup. If an inclusion hyp ersp ac e C ∈ G ( X ) c ommutes with the extr emal elements max G ( X ) and min G ( X ) of G ( X ) , then C is a princip al ultr afilter. Pr o of. By Pro p os ition 5.4, the inclusion hyperspaces max G ( X ) and min G ( X ) are right zeros in G ( X ) and thus ma x G ( X ) ◦ C = C ◦ ma x G ( X ) = max G ( X ) a nd min G ( X ) ◦ C = C ◦ min G ( X ) = min G ( X ). It follows that for ev ery b ∈ X we get { b } ∈ max G ( X ) = max G ( X ) ◦ C , which mea ns that a ∗ C ⊂ { b } for s o me C ∈ C and some a ∈ X . Since the equation a ∗ y = b has a unique solution y ∈ X , the set C is a sing leton, say C = { c } . It remains to prov e that C co incides with the principal ultrafilter h c i generated by c . Assuming the co n verse, we would conclude that X \ { c } ∈ C . By our h ypothes is , the equation y ∗ c = c ha s a unique solution y 0 ∈ X . Since the equation y 0 ∗ x = c has a unique solution x = c , y 0 ∗ ( X \ { c } ) ⊂ X \ { c } . Letting C x = { c } for all x ∈ X \ { y 0 } and C x = X \ { c } for x = y 0 , we conclude that X \ { c } ⊃ S x ∈ X x ∗ C x ∈ min G ( X ) ◦ C = C ◦ min G ( X ) = min G ( X ), whic h is not p ossible.  Corollary 6.2. F or any quasigr oup X t he c enter of the gr oup oid G ( X ) c oincides with the c enter of X . Pr o of. If an inclusion hyperspace C belongs to the center of the g roup oid G ( X ), then C is a principal ultrafilter genera ted b y some p o int c ∈ X . Since C commutes with a ll the principal ultrafilters, c commutes with all elements of X and thus c belo ngs to the cen ter of X . Conv er sely , if c ∈ X belo ngs to the center of X , then for ev ery inclusio n h yp er- space F ∈ G ( X ) we get c ◦ F = { c ∗ F : F ∈ F } = { F ∗ c : F ∈ F } = F ◦ c, which means that (the principa l ultrafilter generated b y) c b elong s to the cen ter of the gr o upo id G ( X ).  RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 19 Remark 6. 3. It is in ter e sting to note that for a n y g roup X the center of the semigroup β X a lso c o incides with the cen ter of the g roup X , s ee Theor em 6.5 4 o f [HS]. Problem 6.4. Given a gr oup X describ e the c enters of the subsemigr oups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ 2 of the semigr oup G ( X ) . Is it true that the c en t er of any subsemigr oup S ⊂ G ( X ) with β ( X ) ⊂ S = S ⊥ c oincides with the c enter of X ? Remark 6.5. Let us note that the requirement S = S ⊥ in the preceding question is essential: for a ny nontrivial group X the cen ter of the (non-symmetric ) subsemi- group X ∪ max G ( X ) of G ( X ) con tains max G ( X ) and hence is strinctly larger than the center of the gr oup X . Problem 6. 6. Given an infinite gr oup X desc rib e the c enters of the semigr oups G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , and N ◦ k ( X ) , k ≥ 2 . (By Theo rem 6 .5 4 of [HS], the center of the semig roup of free ultrafilters β ◦ ( X ) is empt y). 7. The topological center o f G ( X ) In this sectio n we describ e the top ologica l ce n ter of G ( X ). By the top olo gic al c enter o f a group o id X endow ed with a top olo gy we under stand the set Λ( X ) consisting of a ll p oints x ∈ X such that the left and right shifts l x : X → X , l x : z 7→ xz , and r x : X → X, r x : z 7→ z x bo th are contin uous. Since all rig ht shifts on G ( X ) are contin uous, the top o logical center of the group oid G ( X ) consists of all inclusio n hyp e rspaces F with c ontin uous left shifts l F . W e re call tha t G • ( X ) stands for the set of inclusion h yp erpsa ces with finite suppo rt. Theorem 7 .1. F or a quasigr oup X the top olo gic al c enter of the gr oup oid G ( X ) c oincides with G • ( X ) . Pr o of. By Prop ositio n 2.8, the top olog ic al center Λ( GX ) of G ( X ) co n tains a ll pr in- cipal ultrafilters a nd is a s ublattice of G ( X ). Consequently , Λ( GX ) c ontains the sublatttice G • ( X ) of G ( X ) generated by X . Next, we show that ea ch inclus ion hyp e rspace F ∈ Λ( GX ) has finite supp ort and hence b elong s to G • ( X ). By Theor e m 9 .1 of [G1], this will follow as so on as we check that both F and F ⊥ hav e bases consisting of finite sets. T ake an y set F ∈ F , choo se any point e ∈ X , and co nsider the inclusion hyper- space U = { U ⊂ X : e ∈ F ∗ U } . Since for every f ∈ F the equatio n f ∗ u = e 20 VOLOD YMYR GA VR YLKIV has a solution in X , we co nclude that { e } ∈ F ◦ U and by the contin uity o f the left shift l F , there is an op en neighbo rho o d O ( U ) of U suc h tha t { e } ∈ F ◦ A for all A ∈ O ( U ). Without loss of generality , the neighborho o d O ( U ) is of basic form O ( U ) = U + 1 ∩ · · · ∩ U + n ∩ V − 1 ∩ · · · ∩ V − m for some sets U 1 , . . . , U n ∈ U and V 1 , . . . , V m ∈ U ⊥ . T ake a ny finite set A ⊂ F − 1 e = { x ∈ X : e ∈ F ∗ x } int ersecting each set U i , i ≤ n , and consider the inclusion hyperspac e A = h A i ⊥ . It is clear that A ⊂ U + 1 ∩ · · · ∩ U + n . Since each set V j , j ≤ m , contains the set F − 1 e ⊃ A , we get also that A ∈ V − 1 ∩ · · · ∩ V − m . Then F ◦ A ∋ { e } and hence there is a set E ∈ F and a family { A x } x ∈ E ⊂ A with S x ∈ E x ∗ A x ⊂ { e } . It follows that the set E ⊂ eA − 1 = { x ∈ X : ∃ a ∈ A with xa = e } is finite. W e claim that E ⊂ F . Indeed, take any p oint x ∈ E and find a po int a ∈ A with x ∗ a = e . Since A ⊂ F − 1 e , there is a p oint y ∈ F with e = y ∗ a . Hence xa = y a and the right cancella tivit y of X yields x = y ∈ F . There fo re, using the co nt inuit y of the left s hift l F , for every F ∈ F w e have found a finite subset E ∈ F with E ⊂ F . This means that F ha s a base of finite sets. The contin uity of the le ft shift l F and Prop osition 2.4 imply the contin uity of the left s hift l F ⊥ . Rep eating the preceding ar gument, w e can prov e that the inclusion hyperspace F ⊥ has a base of finite sets too. Finally , applying Theorem 9.1 of [G1], we co nclude that F ∈ G • ( X ).  Problem 7.2. Given an infinite gr oup G describ e the top olo gic al c enter of the subsemigr oups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ 2 , of the semigr oup G ( X ) . Is it true that the top olo gic al c enter of any s u bsemigr oup S ⊂ G ( X ) c ontaining β ( X ) c oincides with S ∩ G • ( X ) ? (This is true for the subsemigro ups S = G ( X ) (s e e Theorem 7.1) and S = β ( X ), see Theorems 4.24 and 6.54 of [HS]). Problem 7.3. Given an infinite gr oup X describ e the top olo gic al c ent ers of the semigr oups G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , and N ◦ k ( X ) , k ≥ 2 . (It sho uld b e men tioned that the topolo g ical cen ter of the semigroup β ◦ ( X ) o f free ultrafilters is empt y [P 2 ]). 8. Left cancelable elements of G ( X ) An elemen t a of a g roup oid S is called left c anc elable (resp. right c anc elable ) if for a n y po int s x, y ∈ S the eq uation ax = ay (res p. xa = y a ) implies x = y . In this s ection we characterize left cancelable elements of the g roup oid G ( X ) ov er a quasigro up X . Theorem 8.1. L et X b e a quasigr oup. An inclusion hyp ersp ac e F ∈ G ( X ) is left c anc elable in the gr oup oid G ( X ) if and only if F is a princip al ultr afilter. RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 21 Pr o of. Assume tha t F is left cancela ble in G ( X ). Fir st we show that F co n tains some single to n. Assuming the co n verse, ta ke a n y p oint x 0 ∈ X and note that F ∗ ( X \ { x 0 } ) = X for a ny F ∈ F . T o see that this equality holds, take any po int a ∈ X , choose tw o distinct points b, c ∈ F and find solutions x, y ∈ X of the equation b ∗ x = a and c ∗ y = a . Since X is righ t cancellative, x 6 = y . Consequently , one of the p oints x or y is distinct from x 0 . If x 6 = x 0 , then a = b ∗ x ∈ F ∗ ( X \ { x 0 } ). If y 6 = x 0 , then a = c ∗ y ∈ F ∗ ( X \ { x 0 } ). Now fo r the inclusion hyper space U = h X \ { x 0 }i 6 = min G ( X ), we ge t F ◦ U = min G ( X ) = F ◦ min G ( X ), which contradicts the ch oice of F as a left cancelable elemen t of G ( X ). Thu s F co n tains some singleton { c } . W e c la im that F coincides with the principal ultra filter genera ted by c . Assuming the co nv ers e, we would conclude that X \ { c } ∈ F . Let A = h X \ { c }i ⊥ be the inclusion hyperspa ce consisting of subsets that meet X \ { c } . It is clear that A 6 = max G ( X ). W e claim that F ◦ A = max G ( X ) = F ◦ max G ( X ) whic h will contradict the left cancelability of F . Indeed, g iven any singleton { a } ∈ max G ( X ), c o nsider t wo ca ses: if a 6 = c ∗ c , then we can find a unique x ∈ X with c ∗ x = a . Since x 6 = c , { x } ∈ A and hence { a } = c ∗ { x } ∈ F ◦ A . If a = c ∗ c , then for every y ∈ X \ { c } w e can find a y ∈ X with y ∗ a y = a and use the left cancela tivit y of X to conclude that a y 6 = c and hence { a y } ∈ A . Then { a } = S y ∈ X \{ c } y ∗ { a y } ∈ F ◦ A . Therefore F = h c i is a pr inc ipa l ultr afilter, which prov es the “only if ” part o f the theor em. T o prov e the “if ” part, take any principal ultrafilter h x i g enerated by a p oint x ∈ X . W e cla im that tw o inclusio n hyperspac e s F , U ∈ G ( X ) are equal provided h x i ◦ F = h x i ◦ U . Indeed, given any set F ∈ F obse r ve that x ∗ F ∈ h x i ◦ F = h x i ◦ U and hence x ∗ F = x ∗ U for some U ∈ U . The left cancelativity of X implies that F = U ∈ U , which yields F ⊂ U . By the same argument we can a lso chec k that U ⊂ F .  Problem 8. 2. Given an (infin ite) gr oup X describ e left c anc elable elements of the subsemigr oups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ 2 (and G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , N ◦ k ( X ) , for k ≥ 2 ). Remark 8 .3. Theorem 8.1 implies that for a coun table Abelian gr oup X the set of left cancelable elemen ts in G ( X ) coincides with X . On the other hand, the set of (left) cancelable elements of β ( X ) contains an op en dense subset of β ◦ ( X ), see Theorem 8.34 o f [HS]. 9. Right cancelable elements of G ( X ) As we saw in the preceding section, for any qua sigroup X the gr o upo id G ( X ) contains only tr iv ial left c a ncelable elements. F or right c a ncelable elements the situation is m uch more interesting. First note that the rig h t cancelativity o f an 22 VOLOD YMYR GA VR YLKIV inclusion hyperspace F ∈ G ( X ) is equiv alent to the injectivity of the map µ X ◦ G ¯ R F : G ( X ) → G ( X ) considere d at the b egining of Sec tio n 2. W e reca ll that µ X : G 2 ( X ) → G ( X ) is the m ultiplicatio n of the monad G = ( G, µ, η ) while ¯ R F : β X → G ( X ) is the Stone- ˇ Cech extension of the right shift R F : X → G ( X ), R F : x 7→ x ∗ F . The map ¯ R F certainly is not injective if R F is no t an em b edding, which is equiv alent to the disc r eteness of the indexed se t { x ∗ F : x ∈ X } in G ( X ). Therefor e we have obtained the following necessa ry condition for the right cancelability . Prop ositio n 9. 1. L et X b e a gr oup oid. If an inclu ison hyp ersp ac e F ∈ G ( X ) is right c anc elable in G ( X ) , then the indexe d set { x F : x ∈ X } is discr et e in G ( X ) in the sense that e ach p oint x F has a neighb orho o d O ( x F ) c ontaining no other p oints y F with y ∈ X \ { x } . Next we give a sufficien t condition o f the r ight c ancelability . Prop ositio n 9.2. L et X b e a gr oup oid. A n inclusion hyp ersp ac e F ∈ G ( X ) is right c anc elable in G ( X ) pr ovide d t her e is a family of set s { S x } x ∈ X ⊂ F ∩ F ⊥ such that xS x ∩ y S y = ∅ for any distinct x, y ∈ X . Pr o of. Assume that A ◦ F = B ◦ F for tw o inclusion hyper s paces A , B ∈ G ( X ). First w e show that A ⊂ B . T ake any set A ∈ A and observe that the set S a ∈ A aS a belo ngs to A ◦ F = B ◦ F . Co nsequently , there is a set B ∈ B and a family of sets { F b } b ∈ B ⊂ F such that [ b ∈ B bF b ⊂ [ a ∈ A aS a . It follows from S b ∈ F ⊥ that F b ∩ S b is not e mpty for every b ∈ B . Since the sets aS a and bS b are disjoint for different a, b ∈ X , the inclusion [ b ∈ B b ( F b ∩ S b ) ⊂ [ b ∈ B bF b ⊂ [ a ∈ A aS a implies B ⊂ A a nd hence A ∈ B . By a nalogy w e can pro ve that B ⊂ A .  Prop ositio ns 9.1 and 9.2 imply the following characterizatio n of right cancelable ultrafilters in G ( X ) gener alizing a known characterization of rig ht cancelable ele- men ts of the semigroups β X , see [HS , 8.11]. Corollary 9.3. L et X b e a c ountable gr oup oid. F or an ult ra filter U on X the fol lowing c onditions ar e e quivalent: (1) U is right c anc elable in G ( X ) ; (2) U is right c anc elable in β X ; (3) the indexe d set { x U : x ∈ X } is discr ete in β X ; RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 23 (4) ther e is an indexe d family of sets { U x } x ∈ X ⊂ U such that for any distinct x, y ∈ X t he shifts x U x and y U y ar e disjoint. This characterizatio n can b e used to show that for any c o unt able gro up X the semigroup β ◦ ( X ) of free ultrafilters contains a n o pe n dens e subset of right canc e - lable ultrafilters, s e e [HS , 8.10]. It turns out that a similar r esult ca n b e prov ed for the semigr oup G ◦ ( X ). Prop ositio n 9.4. F or any c ountable quasigr oup, t he gr oup oid G ◦ ( X ) c ontains an op en dense subset of right c anc elable fr e e inclusion hyp ersp ac es. Pr o of. Let X = { x n : n ∈ ω } be an injective enumeration of the countable qua s i- group X . Given a free inclusion hyperspa c e F ∈ G ◦ ( X ) and a neigh bo rho o d O ( F ) of F in G ◦ ( X ), we should find a non- empt y op en subs e t in O ( F ). Without lo ss of generality , the neighborho o d O ( F ) is of basic for m: O ( F ) = G ◦ ( X ) ∩ U + 0 ∩ · · · ∩ U + n ∩ U − n +1 ∩ · · · ∩ U − m − 1 for some sets U 1 , . . . , U m − 1 of X . Tho se sets are infinite be c a use F is fre e . W e are going to construct a n infinite set C = { c n : n ∈ ω } ⊂ X that has infinite int ersection with the sets U i , i < m , and such that for any dis tinct x, y ∈ X the int ersection xC ∩ y C is finite. The p oints c k , k ∈ ω , co mp os ing the set C will b e chosen by induction to satisfy the following conditions: • c k ∈ U j where j = k mo d m ; • c k do es not belong to the finite set F k = { z ∈ X : ∃ i , j ≤ k ∃ l < k ( x i z = x j c l ) } . It is clea r that the so-cons tructed s et C = { c k : k ∈ ω } has infinite intersection with each set U i , i < m . Since X is r ight ca ncellative, for an y i < j the set Z i,j = { z ∈ X : x i z = x j z } is finite. No w the choice of the p oints c k for k > j implies that x i C ∩ x j C ⊂ x i ( Z i,j ∪ { c l : l ≤ j } ) is finite. Now let C b e the fr ee inclusion hyperspac e on X g enerated by the sets C and U 0 , . . . , U n . It is clea r that C ∈ O ( F ) and C ∈ C ∩ C ⊥ . Co nsider the o p en neigh- bo rho o d O ( C ) = O ( F ) ∩ C + ∩ ( C + ) ⊥ of C in G ◦ ( X ). W e claim that each inclusion hype r space A ∈ O ( C ) is rig ht cancelable in G ( X ). This will follow f rom Prop osition 9 .2 a s soon as w e construct a family of sets { A i } i ∈ ω ∈ A ∩ A ⊥ such that x i A i ∩ x j A j = ∅ for a n y num b ers i < j . The sets A i , i ∈ ω , can b e defined by the formula A k = C \ F k where F k = { c ∈ C : ∃ i < k with x k c = x i C } 24 VOLOD YMYR GA VR YLKIV is finite by the choice of the set C .  Problem 9.5. Given an (infi n ite) gr oup X describ e righ t c anc elable elements of t he subsemigr oups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ 2 ( λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , N ◦ k ( X ) , for k ≥ 2) . 10. The structure of the semigroups G ( H ) over finite gr oups H In Prop osition 5 .7 w e hav e seen that the str uctural pro per ties o f the finite semi- group λ ( Z 3 ) has non-trivia l implications for the essentially infinite ob ject λ ◦ ( Z ). This obse r v ation is a mo tiv ation for more detail study of spa c e s G ( H ) over finite Abelia n gr o ups H . I n this case the group H acts on G ( H ) by right shifts: s : G ( H ) × H → G ( H ) , s : ( A , h ) 7→ A ◦ h. So w e ca n s pea k ab out the orbit A ◦ H = {A ◦ h : h ∈ H } of an inclus ion h y per space A ∈ G ( H ) and the orbit space G ( H ) /H = {A ◦ H : A ∈ G ( X ) } . By π : G ( H ) → G ( H ) /H we denote the quotient map whic h induces a unique semigroup structure of G ( H ) / H turning π into a semigroup homomorphism. W e s ha ll say that the semigr oup G ( H ) is splittable if there is a semigr o up homo- morphism s : G ( H ) /H → G ( H ) such that π ◦ s is the identit y homomo rphism of G ( H ) /H . Such a homomorphism s will b e called a se ction of π and the semigro up T ( H ) = s ( G ( H ) /H ) will be called a H -t r ansversal semigr oup of G ( H ). It is clear that a H -transversal semigro up T ( H ) has one-po in t in tersection with each or bit of G ( H ). If the semigroup G ( H ) is splittable, then the structure o f G ( H ) ca n b e describ ed as follows. Prop ositio n 10.1. If the semigr oup G ( H ) is splittable and T ( H ) is the tr ansversal semigr oup of G ( H ) , then T ( H ) is isomorph ic to G ( H ) /H and G ( H ) is the quotient semigr oup of the pr o duct T ( H ) × H under t he homomorphism h : T ( H ) × H → G ( H ) , h : ( A , h ) 7→ A ◦ h . It turns out that the semigr oup G ( Z n ) is splittable for n ≤ 3 and no t splittable for n = 5 (the latter follows from the no n-splittability of the semig roup λ ( Z 5 ) established in [BGN]). So b elow w e describ e the structure of the semigro ups G ( Z n ) and their transversal semig r oup T ( Z n ) for n ≤ 3. F or a gr oup X we shall identify the elements x ∈ X with the ultra filters they generate. Also we shall use the notations ∧ a nd ∨ to denote the lattice oper ations ∩ and ∪ on G ( X ), resp ectively . The semig rou p G ( Z 2 ). F o r the cyclic gr oup Z 2 = { e, a } the lattice G ( Z 2 ) contains four inclusion hypers paces: e, a , e ∧ a, e ∨ a , and is shown at the picture: RIGHT-TOPOLOGICAL SEMIGROUP OPERA TIONS ON INCLUSION HYPE RSP ACES 25 r   r e a r e ∨ a ❅ ❅ q e ∧ a The semig r oup G ( Z 2 ) has a unique Z 2 -transversal semigroup T ( Z 2 ) = { e ∧ a, e, e ∨ a } with tw o right zero s: e ∧ a , e ∨ a and o ne unit e . The semig roup G ( Z 3 ) ov er the cyclic g roup Z 3 = { e, a, a − 1 } cont ains 18 ele- men ts: a ∨ e ∨ a − 1 , a ∨ a − 1 , a ∨ e , e ∨ a − 1 , a ∨ ( e ∧ a − 1 ), e ∨ ( a ∧ a − 1 ), a − 1 ∨ ( a ∧ e ), a, e, a − 1 , ( a ∨ e ) ∧ ( a ∨ a − 1 ) ∧ ( e ∨ a − 1 ), a ∧ ( e ∨ a − 1 ), e ∧ ( a ∨ a − 1 ), a − 1 ∧ ( a ∨ e ), a ∧ a − 1 , a ∧ e , e ∧ a − 1 , a ∧ e ∧ a − 1 divided into 8 orbits with resp ect to the action of the group Z 3 . The semigroup G ( Z 3 ) ha s 9 different Z 3 -transversal semigro ups one o f which is drawn a t the picture: T ( Z 3 ) q a ∧ e ∧ a − 1 q a ∧ e q ❅ e ∧ ( a ∨ a − 1 ) q e q ( a ∨ e ) ∧ ( e ∨ a − 1 ) ∧ ( a ∨ a − 1 ) q e ∨ ( a ∧ a − 1 )  q e ∨ a − 1 q a ∨ e ∨ a − 1 The semigr o up G ( Z 3 ) has 3 s hift-in v ariant inclusion h yper spaces which are right zeros: a ∧ e ∧ a − 1 , a ∨ e ∨ a − 1 and ( a ∨ e ) ∧ ( e ∨ a − 1 ) ∧ ( a ∨ a − 1 ). Beside s right zeros G ( Z 3 ) has 3 idemp otents: e , e ∨ ( a ∧ a − 1 ) a nd e ∧ ( a ∨ a − 1 ). The elemen t e is the unit of the semigroup G ( Z 3 ). 26 VOLOD YMYR GA VR YLKIV The complete informa tio n on the structure of the Z 3 -transversal semigro up T ( Z 3 ) (whic h is isomorphic to the quotient s emigroup G ( Z 3 ) / Z 3 ) ca n be der ived from the Cayley ta ble ◦ x − 3 x − 2 x − 1 x 0 x 1 x 2 x 3 x − 3 x − 3 x − 3 x − 3 x 0 x 0 x 0 x 3 x − 2 x − 3 x − 3 x − 2 x 0 x 0 x 1 x 3 x − 1 x − 3 x − 3 x − 1 x 0 x 0 x 2 x 3 x 0 x − 3 x − 3 x 0 x 0 x 0 x 3 x 3 x 1 x − 3 x − 2 x 0 x 0 x 1 x 3 x 3 x 2 x − 3 x − 1 x 0 x 0 x 2 x 3 x 3 x 3 x − 3 x 0 x 0 x 0 x 3 x 3 x 3 of its linea rly order ed subsemigroup T ( Z 3 ) \ { e } having with 7 -elements: x − 3 = e ∧ a ∧ a − 1 , x − 2 = e ∧ a, x − 1 = e ∧ ( a ∨ a − 1 ) , x 0 = ( e ∨ a ) ∧ ( e ∨ a − 1 ) ∧ ( a ∨ a − 1 ) , x 1 = e ∨ ( a ∧ a − 1 ) , x 2 = e ∨ a, x 3 = e ∨ a ∨ a − 1 . 11. 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