Right-topological semigroup operations on inclusion hyperspaces
aa r X i v : . [ m a t h . GN ] F e b RIGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ONINCLUSION HYPERSPACES
VOLODYMYR GAVRYLKIV
Abstract.
We show that for any discrete semigroup X the semigroup op-eration can be extended to a right-topological semigroup operation on thespace G ( X ) of inclusion hyperspaces on X . We detect some important sub-semigroups of G ( X ), study the minimal ideal, the (topological) center, leftcancelable elements of G ( X ), and describe the structure of the semigroups G ( Z n ) for small numbers n . Contents
Introduction 21. Inclusion hyperspaces 31.1. General definition and reduction to the compact case 31.2. Inclusion hyperspaces in the category of compacta 41.3. Some important subspaces of G ( X ) 51.4. The inner algebraic structure of G ( X ) 52. Extending algebraic operations to inclusion hyperspaces 63. Homomorphisms of semigroups of inclusion hyperspaces 124. Subgroupoids of G ( X ) 125. Ideals and zeros in G ( X ) 146. The center of G ( X ) 187. The topological center of G ( X ) 198. Left cancelable elements of G ( X ) 209. Right cancelable elements of G ( X ) 2110. The structure of the semigroups G ( H ) over finite groups H Mathematics Subject Classification.
Introduction
After the topological proof of Hindman theorem [H1] given by Galvin and Glazer(unpublished, see [HS, p.102], [H2]) topological methods become a standard toolin the modern combinatorics of numbers, see [HS], [P ]. The crucial point is thatthe semigroup operation ∗ defined on any discrete space S can be extended to aright-topological semigroup operation on βS , the Stone- ˇCech compactification of S . The product of two ultrafilters U , V ∈ βS can be found in two steps: firstlyfor every element a ∈ S of the semigroup we extend the left shift L a : S → S , L a : x a ∗ x , to a continuous map βL a : βS → βS . In such a way, for every a ∈ S we define the product a ∗ V = βL a ( V ). Then, extending the function R V : S → βS , R V : a a ∗ V , to a continuous map βR V : βS → βS , we definethe product U ◦ V = βR V ( U ). This product can be also defined directly: this is anultrafilter with the base S x ∈ U x ∗ V x where U ∈ U and { V x } x ∈ U ⊂ V . Endowedwith so-extended operation the Stone- ˇCech compactification βS becomes a compactHausdorff right-topological semigroup. Because of the compactness the semigroup βS has idempotents, minimal (left) ideals, etc., whose existence has many importantcombinatorial consequences.The Stone- ˇCech compactification βS can be considered as a subset of the dou-ble power-set P ( P ( S )). The power-set P ( X ) of any set X (in particular, X = P ( S )) carries a natural compact Hausdorff topology inherited from the Cantorcube { , } X after identification of each subset A ⊂ X with its characteristic func-tion. The power-set P ( X ) is a complete distributive lattice with respect to theoperations of union and intersection.The smallest complete sublattice of P ( P ( S )) containing βS coincides with thespace G ( S ) of inclusion hyperspaces, a well-studied object in Categorial Topology.By definition, a family A ⊂ P ( S ) of non-empty subsets of S is called an inclusionhyperspace if together with each set A ∈ A the family A contains all supersets of A in S . In [G1] it is shown that G ( S ) is a compact Hausdorff lattice with respectto the operations of intersection and union.Our principal observation is that the algebraic operation of the semigroups S can be extended not only to βS but also to the complete lattice hull G ( S ) of βS in P ( P ( S )). Endowed with so-extended operation, the space of inclusion hyperspaces G ( S ) becomes a compact Hausdorff right-topological semigroup containing βS as aclosed subsemigroup. Besides βS , the semigroup G ( S ) contains many other impor-tant spaces as closed subsemigroups: the superextension λS of S , the space N k ( S )of k -linked inclusion hyperspaces, the space Fil( S ) of filters on S (which containsan isomorphic copy of the global semigroup Γ( S ) of S ), etc. IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 3
We shall study some properties of the semigroup operation on G ( S ) and itsinterplay with the lattice structure of G ( S ). We expect that studying the alge-braic structure of G ( S ) will have some combinatorial consequences that cannot beobtained with help of ultrafilters, see [BGN] for further development of this subject.1. Inclusion hyperspaces
In this section we recall some basic information about inclusion hyperspaces.More detail information can be found in the paper [G1].1.1.
General definition and reduction to the compact case.
For a topolog-ical space X by exp( X ) we denote the space of all non-empty closed subspacesof X endowed with the Vietoris topology. By an inclusion hyperspace we meana closed subfamily F ⊂ exp( X ) that is monotone in the sense that together witheach set A ∈ F the family F contains all closed subsets B ⊂ X that contain A .By [G1], the closure of each monotone family in exp( X ) is an inclusion hyperspace.Consequently, each family B ⊂ exp( X ) generates an inclusion hyperspacecl exp( X ) { A ∈ exp( X ) : ∃ B ∈ B with B ⊂ A } denoted by hBi . In this case B is called a base of F = hBi . An inclusion hyperspace h x i generated by a singleton { x } , x ∈ X , is called principal .If X is discrete, then each monotone family in exp( X ) is an inclusion hyperspace,see [G1].Denote by G ( X ) the space of all inclusion hyperspaces with the topology gener-ated by the subbase U + = {A ∈ G ( X ) : ∃ B ∈ A with B ⊂ U } and U − = {A ∈ G ( X ) : ∀ B ∈ A B ∩ U = ∅} , where U is open in X .For a T -space X the map ηX : X → G ( X ), ηX ( x ) = { F ⊂ cl X : x ∈ F } , isan embedding (see [G1]), so we can identify principal inclusion hyperspaces withelements of the space X .For a T -space X the space G ( X ) is Hausdorff if and only if the space X isnormal, see [G1], [M]. In the latter case the map h : G ( X ) → G ( βX ) , h ( F ) = cl exp( βX ) { cl βX F | F ∈ F} , is a homeomorphism, so we can identify the space G ( X ) with the space G ( βX )of inclusion hyperspaces over the Stone- ˇCech compactification βX of the normalspace X , see [M]. Thus we reduce the study of inclusion hyperspaces over normaltopological spaces to the compact case where this construction is well-studied. In [G1] the inclusion hyperspace hBi generated by a base B is denoted by ↑B . VOLODYMYR GAVRYLKIV
For a (discrete) T -space the space G ( X ) contains a (discrete and) dense sub-space G • ( X ) consisting of inclusion hyperspaces with finite support. An inclusionhyperspace A ∈ G ( X ) is defined to have finite support in X if A = hFi for somefinite family F of finite subsets of X .An inclusion hyperspace F ∈ G ( X ) on a non-compact space X is called free iffor each compact subset K ⊂ X and any element F ∈ F there is another element E ∈ F such that E ⊂ F \ K . By G ◦ ( X ) we shall denote the subset of G ( X )consisting of free inclusion hyperspaces. By [G1], for a normal locally compactspace X the subset G ◦ ( X ) is closed in G ( X ). In the simplest case of a countablediscrete space X = N free inclusion hyperspaces (called semifilters) on X = N havebeen introduced and intensively studied in [BZ].1.2. Inclusion hyperspaces in the category of compacta.
The constructionof the space of inclusion hyperspaces is functorial and monadic in the category C omp of compact Hausdorff spaces and their continuous map, see [TZ]. To complete G toa functor on C omp observe that each continuous map f : X → Y between compactHausdorff spaces induces a continuous map Gf : G ( X ) → G ( Y ) defined by 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 and continuous, and G is a functorin the category Comp of compact Hausdorff spaces and their continuous maps, see[TZ]. By Proposition 2.3.2 [TZ], this functor is weakly normal in the sense thatit is continuous, monomorphic, epimorphic and preserves intersections, singletons,the empty set and weight of infinite compacta.Since the functor G preserves monomorphisms, for each closed subspace A of acompact Hausdorff space X the inclusion map i : A → X induces a topologicalembedding Gi : G ( A ) → G ( X ). So we can identify G ( A ) with a subspace of G ( X ).Now for each inclusion hyperspace A ∈ G ( X ) we can consider the support of A supp A = ∩{ A ⊂ cl X : A ∈ G ( A ) } and conclude that A ∈ G (supp A ) because G preserves intersections, see [TZ, § G . We recall that afunctor T : C omp → C omp is monadic if it can be completed to a monad T =( T, η, µ ) where η : Id → T and µ : T → T are natural transformations (calledthe unit and multiplication) such that µ ◦ T ( µ X ) = µ ◦ µ T X : T X → T X and µ ◦ η T X = µ ◦ T ( η X ) = Id T X for each compact Hausdorff space X , see [TZ].For the functor G the unit η : Id → G has been defined above while the multi-plication µ = { µ X : G X → G ( X ) } is defined by the formula µ X (Θ) = ∪{∩M | M ∈ Θ } , Θ ∈ G X. IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 5
By Proposition 3.2.9 of [TZ], the triple G = ( G, η, µ ) is a monad in C omp .1.3. Some important subspaces of G ( X ) . The space G ( X ) of inclusion hyper-spaces contains many interesting subspaces. Let X be a topological space and k ≥ A ∈ G ( X ) is defined to be • k -linked if ∩F 6 = ∅ for any subfamily F ⊂ A with |F| ≤ k ; • centered if ∩F 6 = ∅ for any finite subfamily F ⊂ A ; • a filter if A ∩ A ∈ A for all sets A , A ∈ A ; • an ultrafilter if A = A ′ for any filter A ′ ∈ G ( X ) containing A ; • maximal k -linked if A = A ′ for any k -linked inclusion hyperspace A ′ ∈ G ( X ) containing A .By N k ( X ), N <ω ( X ), and Fil( X ) we denote the subsets of G ( X ) consisting of k -linked, centered, and filter inclusion hyperspaces, respectively. Also by β ( X )and λ k ( X ) we denote the subsets of G ( X ) consisting of ultrafilters and maximalk-linked inclusion hyperspaces, respectively. The space λ ( X ) = λ ( X ) is called thesuperextension of X .The following diagram describes the inclusion relations between subspaces N k X , N <ω X , Fil( X ), λX and βX of G ( X ) (an arrow A → B means that A is a subsetof B ). Fil( X ) → N <ω X → N k X → N X → G ( X ) βX ✻ ✲ λX ✻ For a normal space X all the subspaces from this diagram are closed in G ( X ),see [G1].For a non-compact space X we can also consider the intersectionsFil ◦ ( 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 ) = βX ∩ G ◦ ( X ) = βX \ X. Elements of those sets will be called free filters, free centered inclusion hyperspaces,free k -linked inclusion hyperspaces, etc. For a normal locally compact space X thesubsets Fil ◦ ( X ), N ◦ <ω ( X ), N ◦ k ( X ), λ ◦ ( X ) = λ ◦ ( X ), and β ◦ ( X ) are closed in G ( X ),see [G1]. In contrast, λ ◦ k ( N ) is not closed in G ( N ) for k ≥
3, see [Iv].1.4.
The inner algebraic structure of G ( X ) . In this subsection we discuss thealgebraic structure of the space of inclusion hyperspaces G ( X ) over a topologicalspace X . The space of inclusion hyperspaces G ( X ) possesses two binary operations VOLODYMYR GAVRYLKIV ∪ , ∩ , and one unary operation ⊥ : G ( X ) → G ( X ) , ⊥ : F 7→ F ⊥ = { E ⊂ cl X : ∀ F ∈ F E ∩ F = ∅} called the transversality map. These three operations are continuous and turn G ( X ) into a symmetric lattice, see [G1]. Definition 1.1. A symmetric lattice is a complete distributive lattice ( L, ∨ , ∧ )endowed with an additional unary operation ⊥ : L → L , ⊥ : x x ⊥ , that is aninvolutive anti-isomorphism in the sense that • x ⊥⊥ = x for all x ∈ L ; • ( x ∨ y ) ⊥ = x ⊥ ∧ y ⊥ ; • ( x ∧ y ) ⊥ = x ⊥ ∨ y ⊥ ;The smallest element of the lattice G ( X ) is the inclusion hyperspace { X } whilethe largest is exp( X ).For a discrete space X the set G ( X ) of all inclusion hyperspaces on X is asubset of the double power-set P ( P ( X )) (which is a complete distributive lattice)and is closed under the operations of union and intersection (of arbitrary familiesof inclusion hyperspaces).Since each inclusion hyperspace is a union of filters and each filter is an inter-section of ultrafilters, we obtain the following proposition showing that the lattice G ( X ) is a rather natural object. Proposition 1.2.
For a discrete space X the lattice G ( X ) coincides with the small-est complete sublattice of P ( P ( X )) containing all ultrafilters. Extending algebraic operations to inclusion hyperspaces
In this section, given a binary (associative) operation ∗ : X × X → X on a discretespace X we extend this operation to a right-topological (associative) operation on G ( X ). This can be done in two steps by analogy with the extension of the operationto the Stone- ˇCech compactification βX of X .First, for each element a ∈ X consider the left shift L a : X → X , L a ( x ) = a ∗ x and extend it to a continuous map ¯ L a : βX → βX between the Stone- ˇCechcompactifications of X . Next, apply to this extension the functor G to obtain thecontinuous map G ¯ L a : G ( βX ) → G ( βX ). Clearly, for every inclusion hyperspace F ∈ G ( βX ) the inclusion hyperspace G ¯ L a ( F ) has a base { a ∗ F | F ∈ F} . Thus, wehave defined the product a ∗ F = G ¯ L a ( F ) of the element a ∈ X and the inclusionhyperspace F .Further, for each inclusion hyperspace F ∈ G ( βX ) = G ( X ) we can consider themap R F : X → G ( βX ) defined by the formula R F ( x ) = x ∗ F for every x ∈ X .Extend the map R F to a continuous map ¯ R F : βX → G ( βX ) and apply to this IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 7 extension the functor G to obtain a map G ¯ R F : G ( βX ) → G ( βX ). Finally,compose the map G ¯ R F with the multiplication µX = µ G X : G X → G ( X ) of themonad G = ( G, η, µ ) and obtain a map µ X ◦ G ¯ R F : G ( βX ) → G ( βX ). For aninclusion hyperspace U ∈ G ( βX ), the image µ G X ◦ G ¯ R F ( U ) is called the productof the inclusion hyperspaces U and F and is denoted by U ◦ F .It follows from continuity of the maps G ¯ R F that the extended binary operationon G ( X ) is continuous with respect to the first argument with the second argumentfixed. We are going to show that the operation ◦ on G ( X ) nicely agrees with thelattice structure of G ( X ) and is associative if so is the operation ∗ . Also we shallestablish an easy formula U ◦ F = h [ x ∈ U x ∗ F x : U ∈ U , { F x } x ∈ U ⊂ Fi for calculating the product U ◦ F of two inclusion hyperspaces U , F . We start withnecessary definitions. Definition 2.1.
Let ⋆ : G ( X ) × G ( X ) → G ( X ) be a binary operation on G ( X ).We shall say that ⋆ respects the lattice structure of G ( X ) if for any 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.
We will say that a binary operation ⋆ : G ( X ) × G ( X ) → G ( X ) isright-topological if • for any U ∈ G ( X ) the right shift R U : G ( X ) → G ( X ), R U : F 7→ F ⋆ U , iscontinuous; • for any a ∈ X the left shift L a : G ( X ) → G ( X ), L a : F 7→ a ⋆ F , iscontinuous.The following uniqueness theorem will be used to find an equivalent descriptionof the induced operation on G ( X ). Theorem 2.3.
Let ⋆, ◦ : G ( X ) × G ( X ) → G ( X ) be two right-topological binaryoperations that respect the lattice structure of G ( X ) . These operations coincide ifand only if they coincide on the product X × X ⊂ G ( X ) × G ( X ) .Proof. It is clear that if these operations coincide on G ( X ) × G ( X ), then theycoincide on the product X × X identified with a subset of G ( X ) × G ( X ). We recallthat each point x ∈ X is identified with the ultrafilter h x i generated by x .Now assume conversely that x ⋆ y = x ◦ y for any two points x, y ∈ X ⊂ G ( X ).First we check that a ⋆ F = a ◦ F for any a ∈ X and F ∈ G ( X ). Since the left VOLODYMYR GAVRYLKIV shifts
F 7→ a ⋆ F and F 7→ a ◦ F are continuous, it suffices to establish the equality a ⋆ F = a ◦ F for inclusion hyperspaces F having finite support in X (because theset G • ( X ) of all such inclusion hyperspaces is dense in G ( X ), see [G1]). Any sucha hyperspace F is generated by a finite family of finite subsets of X .If F = h F i is generated by a single finite subset F = { a , . . . , a n } ⊂ X , then F = T ni =1 h a i i is the finite intersection of principal 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 , . . . , F n i is generated by finite family of finite sets, then F = S ni =1 h F i i and we can use the preceding case to prove 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 any inclusion hyperspace
U ∈ G ( X ) by a similar argument one canprove the equality F ⋆ U = F ◦ U for all inclusion hyperspaces
F ∈ G • ( X ) havingfinite support in X . Finally, using the density of G • ( X ) in G ( X ) and the continuityof right shifts F 7→ F ◦U and
F 7→ F ⋆ U one can establish the equality F ⋆ U = F ◦U for all inclusion hyperspaces
F ∈ G ( X ). (cid:3) The above theorem will be applied to show that the operation ◦ : G ( X ) × G ( X ) → G ( X ) induced by the operation ∗ : X × X → X coincides with the operation ⋆ : G ( X ) × G ( X ) → G ( X ) defined by the formula U ⋆ V = h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ Vi for U , V ∈ G ( X ).First we establish some properties of the operation ⋆ . Proposition 2.4.
The operation ⋆ commutes with the transversality operation inthe sense that ( U ⋆ V ) ⊥ = U ⊥ ⋆ V ⊥ for any U , V ∈ G ( X ) .Proof. To prove that U ⊥ ⋆ V ⊥ ⊂ ( U ⋆ V ) ⊥ , take any element A ∈ U ⊥ ⋆ V ⊥ . Weshould check that A intersects each set B ∈ U ⋆ V . Without loss of generality, thesets A and B are of the basic form: A = [ x ∈ F x ∗ G x for some sets F ∈ U ⊥ and { G x } x ∈ F ⊂ V ⊥ and B = [ x ∈ U x ∗ V x for some sets U ∈ U and { V x } x ∈ U ⊂ V . Since U ∈ U and F ∈ U ⊥ , the intersection F ∩ U contains some point x . Forthis point x the sets V x ∈ V and G x ∈ V ⊥ are well-defined and their intersection IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 9 V x ∩ G x contains some point y . Then the intersection A ∩ B contains the point x ∗ y and hence is not empty, which proves that A ∈ ( U ⋆ V ) ⊥ .To prove that ( U ⋆ V ) ⊥ ⊂ U ⊥ ⋆ V ⊥ , fix a set A ∈ ( U ⋆ V ) ⊥ . We claim that the set F = { x ∈ X : x − A ∈ V ⊥ } belongs to U ⊥ (here x − A = { y ∈ X : x ∗ y ∈ A } ). Assuming conversely that F / ∈ U ⊥ , we would find a set U ∈ U with F ∩ U = ∅ . By the definition of F ,for each x ∈ U the set x − A / ∈ V ⊥ and thus we can find a set V x ∈ V withempty intersection V x ∩ x − A . By the definition of the product U ⋆ V , the set B = S x ∈ U x ∗ V x belongs to U ⋆ V and hence intersects the set A . Consequently, x ∗ y ∈ A for some x ∈ U and y ∈ V x . The inclusion x ∗ y ∈ A implies that y ∈ x − A ⊂ X \ V x , which is a contradiction proving that F ∈ U ⊥ . Then the sets A ⊃ S x ∈ F x ∗ x − A belong to U ⊥ ⋆ V ⊥ . (cid:3) Proposition 2.5.
The equality ( U ∩ V ) ⋆ W = ( U ⋆ W ) ∩ ( V ⋆ W ) holds for any U , V , W ∈ G ( X ) .Proof. It is easy to show that (
U ∩ V ) ⋆ W ⊂ ( U ⋆ W ) ∩ ( V ⋆ W ).To prove the reverse 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 , and V ∈ V , { W ′′ y } y ∈ V ⊂ W . Since U , V areinclusion hyperspaces, U ∪ V ∈ U ∩ V . For each 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 . (cid:3) By analogy one can prove
Proposition 2.6.
For any U , V , W ∈ G ( X ) and a ∈ Xa ⋆ ( V ∪ W ) = ( a ⋆ V ) ∪ ( a ⋆ W ) and a ⋆ ( V ∩ W ) = ( a ⋆ V ) ∩ ( a ⋆ W ) . Combining Propositions 2.4 and 2.5 we get
Corollary 2.7.
For any U , V , W ∈ G ( X ) we get ( U ∪ V ) ⋆ W = ( U ⋆ W ) ∪ ( V ⋆ W ) . Proof.
Indeed,(
U ∪ V ) ⋆ W = (cid:0) (( U ∪ V ) ⋆ W ) ⊥ (cid:1) ⊥ = (( U ∪ V ) ⊥ ⋆ W ⊥ ) ⊥ ==(( U ⊥ ∩ V ⊥ ) ⋆ W ⊥ ) ⊥ = (( U ⊥ ⋆ W ⊥ ) ∩ ( V ⊥ ⋆ W ⊥ )) ⊥ ==( U ⊥ ⋆ W ⊥ ) ⊥ ∪ ( V ⊥ ⋆ W ⊥ ) ⊥ = ( U ⋆ W ) ∪ ( V ⋆ W ) . (cid:3) Proposition 2.8.
The operation ⋆ : G ( X ) × G ( X ) → G ( X ) , U ⋆ V = h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ Vi , respects the lattice structure of G ( X ) and is right-topological.Proof. Propositions 2.5, 2.6 and Corollary 2.7 imply that the operation ⋆ respectsthe lattice structure of G ( X ).So it remains to check that the operation ⋆ is right-topological. First we checkthat for any U ∈ G ( X ) the right shift R U : G ( X ) → G ( X ), R U : F 7→ F ⋆ U , iscontinuous.Fix any inclusion hyperspaces F , U ∈ G ( X ) and let W + be a sub-basic neigh-borhood of their product F ⋆ U . Find sets F ∈ F and { U x } x ∈ F ⊂ U such that S x ∈ F x ∗ U x ⊂ W . Then F + is a neighborhood of F with F + ⋆ U ⊂ W + .Now assume that F ⋆ U ∈ W − for some W ⊂ X . Observe that for any inclusionhyperspace V ∈ G ( X ) we get the equivalences V ∈ W − ⇔ W ∈ V ⊥ ⇔ V ⊥ ∈ W + .Consequently, F ⋆ U ∈ W − is equivalent to F ⊥ ⋆ U ⊥ = ( F ⋆ U ) ⊥ ∈ W + . Thepreceding case yields a neighborhood O ( F ⊥ ) such that O ( F ⊥ ) ⋆ U ⊥ ∈ W + . Now thecontinuity of the transversality operation implies that O ( F ⊥ ) ⊥ is a neighborhoodof F with O ( F ⊥ ) ⊥ ⋆ U ∈ W − .Finally, we prove that for every a ∈ X the left shift L a : G ( X ) → G ( X ), L a : F 7→ a ⋆ F , is continuous. Given a sub-basic open set W + ⊂ G ( X ) note that L − a ( W + ) is open because L − a ( W + ) = ( a − W ) + where a − W = { x ∈ X : a ∗ x ∈ W } . On the other hand, a ⋆ F ∈ W − is equivalent to a ⋆ F ⊥ = ( a ⋆ F ) ⊥ ∈ ( W − ) ⊥ = W + which implies that the preimage L − a ( W − ) = ( L a ( W + )) ⊥ is also open. (cid:3) The operation ◦ has the same properties. Proposition 2.9.
The operation ◦ : G ( X ) × G ( X ) → G ( X ) , U ◦V = µ G X ◦ G ¯ R F ( U ) respects the lattice structure of G ( X ) and is right-topological.Proof. For 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 continuous being the composition of continuous maps. Next forany a ∈ X and F ∈ G ( X ) we have L a ( F ) = a ◦F = µ G X ( h a i∗F ) = µ G X ( h a ∗Fi ) = a ∗ F = G ¯ L a ( F ) and the map L a ≡ G ¯ L a is continuous.It is known (and easy to verify) that the multiplication µ G ( X ) : G ( 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 ) IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 11 for any U , V ∈ G ( X ). Then for any U , V , W ∈ G ( X ) and a ∈ X we get( 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 ∈ Xa ◦ 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 ). (cid:3) Since both operations ◦ and ⋆ are right-topological and respect the lattice struc-ture of G ( X ) we may apply Theorem 2.3 to get Corollary 2.10.
For any binary operation ∗ : X × X → X the operations ◦ and ⋆ on G ( X ) coincide. Consequently, for any inclusion hyperspaces U , V ∈ G ( X ) theirproduct U ◦ V is the inclusion hyperspace h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ Vi = (cid:8) A ⊂ X : { x ∈ X : x − A ∈ V} ∈ U (cid:9) . Having the apparent description of the operation ◦ we can establish its associa-tivity. Proposition 2.11.
If the operation ∗ on X is associative, then so is the inducedoperation ◦ on G ( X ) .Proof. It is necessary to show that (
U ◦ V ) ◦ W = U ◦ ( V ◦ W ) for any inclusionhyperspaces U , V , W .Take any subset 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 choosea set U ∈ U such that B ⊃ S x ∈ U x ∗ V x for some family { V x } x ∈ U ⊂ V . It is clearthat for each x ∈ U and y ∈ V x the product x ∗ y is in B and hence W x ∗ y isdefined. Consequently, S y ∈ V x y ∗ W x ∗ y ∈ V ◦ W for all x ∈ U and 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 ).To prove the reverse inclusion, fix a set A ∈ U ◦ ( V ◦ W ) and choose 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 . For each z ∈ Z we can find x ∈ U and y ∈ V x such that z = x ∗ y and put W z = W x,y . Then Z ∈ U ◦ V and S z ∈ Z z ∗ W z ∈ ( U ◦ V ) ◦ W . Taking intoaccount 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 . (cid:3) Homomorphisms of semigroups of inclusion hyperspaces
Let us observe that our construction of extension of a binary operation for X to G ( X ) works well both for associative and non-associative operations. Let us recallthat a set S endowed with a binary operation ∗ : X × X → X is called a groupoid . Ifthe operation is associative, then X is called a semigroup . In the preceding sectionwe have shown that for each groupoid (semigroup) X the space G ( X ) is a groupoid(semigroup) with respect to the extended operation.A map h : X → X between two groupoids ( X , ∗ ) and ( X , ∗ ) is called a homomorphism if h ( x ∗ y ) = h ( x ) ∗ h ( y ) for all x, y ∈ X . Proposition 3.1.
For any homomorphism h : X → X between groupoids ( X , ∗ ) and ( X , ∗ ) the induced map Gh : G ( X ) → G ( X ) is a homomorphism of thegroupoids G ( X ) , G ( X ) .Proof. Given two inclusion hyperspaces U , V ∈ G ( X ) observe that Gh ( U ◦ V ) = Gh ( h [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ Vi ) == h h ( [ x ∈ U x ∗ V x ) : U ∈ U , { V x } x ∈ U ⊂ Vi ) == h [ x ∈ U h ( x ) ∗ h ( V x ) : U ∈ U , { V x } x ∈ U ⊂ Vi == h [ x ∈ h ( U ) x ∗ h ( V x ) : U ∈ U , { h ( V x ) } x ∈ U ⊂ Gh ( V ) i == h h ( U ) : U ∈ Ui ◦ h h ( V ) : V ∈ Vi = Gh ( U ) ◦ Gh ( V ) . (cid:3) Reformulating Proposition 2.4 in terms of homomorphisms, we obtain
Proposition 3.2.
For any groupoid X the transversality map ⊥ : G ( X ) → G ( X ) is a homomorphism of the groupoid G ( X ) . Subgroupoids of G ( X )In this section we shall show that for a groupoid X endowed with the discretetopology all (topologically) closed subspaces of G ( X ) introduced in Section 1.3 aresubgroupoids of G ( X ). A subset A of a groupoid ( X, ∗ ) is called a subgroupoid of X if A ∗ A ⊂ A , where A ∗ A = { a ∗ b : a, b ∈ A } . IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 13
We assume that ∗ : X × X → X is a binary operation on a discrete space X and ◦ : G ( X ) × G ( X ) → G ( X ) is the extension of ∗ to G ( X ). Applying Proposition 3.2we obtain Proposition 4.1. If S is a subgroupoid of G ( X ) , then S ⊥ is a subgroupoid of G ( X ) too. Our next propositions can be easily derived from Corollary 2.10.
Proposition 4.2.
The sets
Fil( X ) , N <ω ( X ) and N k ( X ) , k ≥ , are subgroupoidsin G ( X ) . Proposition 4.3.
The Stone- ˇCech extension βX and the superextension λX bothare closed subgroupoids in G ( X ) .Proof. The superextension λX is a subgroupoid of G ( X ) being the intersection λ ( X ) = N ( X ) ∩ ( N ( X )) ⊥ of two subgroupoids of G ( X ). By analogy, βX =Fil( X ) ∩ λ ( X ) is a subgroupoid of G ( X ). (cid:3) Remark 4.4.
In contrast to λX for k ≥ λ k ( X ) need not be a sub-groupoid of G ( X ). For example, for the cyclic group Z = { , , , , } the subset λ ( Z ) of G ( Z ) contains a maximal 3-linked system L = h{ , , } , { , , } , { , , } , { , , }i whose square L ∗ L = h{ , , , } , { , , , } , { , , , } , { , , , } , { , , , }i is not maximal 3-linked.By a direct application of Corollary 2.10 we can also prove Proposition 4.5.
The set G • ( X ) of all inclusion hyperspaces with finite supportis a subgroupoid in G ( X ) . Finally we find conditions on the operation ∗ guaranteeing that the subset G ◦ ( X )of free inclusion hyperspaces is a subgroupoid of G ( X ). Proposition 4.6.
Assume that for each b ∈ X there is a finite subset F ⊂ X suchthat for each a ∈ X \ F the set a − b = { x ∈ X : a ∗ x = b } is finite. Then the set G ◦ ( X ) is a closed subgroupoid in G ( X ) and consequently, Fil ◦ ( X ) , λ ◦ ( X ) , β ◦ ( X ) all are closed subgroupoids in G ( X ) .Proof. Take two free inclusion hyperspaces A , B ∈ G ( X ) and a subset C ∈ A ◦ B .We should prove that C \ K ∈ A ◦ B for each compact subset K ⊂ X . Without lossof generality, the set C is of basic form: C = S a ∈ A a ∗ B a for some set A ∈ A andsome family { B a } a ∈ A ⊂ B . Since X is discrete, the set K is finite. It follows from our assumption that thereis a finite set F ⊂ X such that for every a ∈ X \ F the set a − K = { x ∈ X : a ∗ x ∈ K } is finite. The hyperspace A , being free, contains the set A ′ = A \ F . By thesame reason, for each a ∈ A ′ the hyperspace B contains the set B ′ a = B a \ a − K .Since C \ K ⊃ S a ∈ A ′ a ∗ B ′ a ∈ A ◦ B , we conclude that C \ K ∈ A ◦ B . (cid:3) Remark 4.7. If X is a semigroup, then G ( X ) is a semigroup and all the sub-groupoids considered above are closed subsemigroups in G ( X ). Some of them arewell-known in Semigroup Theory. In particular, so is the semigroup βX of ultra-filter and β ◦ ( X ) = βX \ X of free ultrafilters. The semigroup Fil( X ) contains anisomorphic copy of the global semigroup of X , which is the hyperspace exp( X )endowed with the semigroup operation A ∗ B = { a ∗ b : a ∈ A, b ∈ B } .5. Ideals and zeros in G ( X )A non-empty subset I of a groupoid ( X, ∗ ) is called an ideal (resp. right ideal , left ideal ) if I ∗ X ∪ X ∗ I ⊂ I (resp. I ∗ X ⊂ I , X ∗ I ⊂ I ). An element O of agroupoid ( X, ∗ ) is called a zero (resp. left zero , right zero ) in X if { O } is an ideal(resp. right ideal, left ideal) in X . Each right or left zero z ∈ X is an idempotent in the sense that z ∗ z = z .For a groupoid ( X, ∗ ) right zeros in G ( X ) admit a simple description. We definean inclusion hyperspace A ∈ G ( X ) to be shift-invariant if for every A ∈ A and x ∈ X the sets x ∗ A and x − A = { y ∈ X : x ∗ y ∈ A } belong to A . Proposition 5.1.
An inclusion hyperspace
A ∈ G ( X ) is a right zero in G ( X ) ifand only if A is shift-invariant.Proof. Assuming that an inclusion hyperspace
A ∈ G ( X ) is shift-invariant, weshall show that B ◦ A = A for every B ∈ G ( X ). Take any set F ∈ B ◦ A andfind 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-invariant, S x ∈ B x ∗ A x ∈ A and thus F ∈ A . This proves theinclusion B ◦ A ⊂ A . On the other hand, for every F ∈ A and every x ∈ X we get x − F ∈ A and thus F ⊃ S x ∈ X x ∗ x − F ∈ B ◦ A . This shows that A is a right zeroof the semigroup G ( X ).Now assume that A is a right zero of G ( X ). Observe that for every x ∈ X theequality h x i ◦ A = A implies x ∗ A ∈ A for every A ∈ A .One the other hand, the equality { 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 − A = { z ∈ X : x ∗ z ∈ A } ⊃ A x ∈ A belongs to A witnessing that A isshift-invariant. (cid:3) IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 15 By ↔ G ( X ) we denote the set of shift-invariant inclusion hyperspaces in G ( X ).Proposition 5.1 implies that A ◦ B = B for every A , B ∈ ↔ G ( X ). This means that ↔ G ( X ) is a rectangular semigroup.We recall that a semigroup ( S, ∗ ) is called rectangular (or else a semigroup ofright zeros ) if x ∗ y = y for all x, y ∈ S . Proposition 5.2.
The set ↔ G ( X ) is closed in G ( X ) , is a rectangular subsemigroupof the groupoid G ( X ) and is closed complete sublattice of the lattice G ( X ) invariantunder the transversality map. Moreover, if ↔ G ( X ) is non-empty, then it is a left idealthat lies in each right ideal of G ( X ) .Proof. If A ∈ G ( X ) \ ↔ G ( X ), then there exists x ∈ X and A ∈ A such that x ∗ A / ∈ A or x − A / ∈ A . Then O ( A ) = {A ′ ∈ G ( X ) : A ∈ A ′ and ( x ∗ A / ∈ A ′ or x − A / ∈ A ) } is an open neighborhood of A missing the set ↔ G ( X ) and witnessing 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 rectangular subsemi-group of the groupoid G ( X ).To show that ↔ G ( X ) is invariant under the transversality operation, note thatfor every A ∈ G ( X ) and Z ∈ ↔ G ( X ) we get A ◦ Z ⊥ = ( A ⊥ ◦ Z ) ⊥ = Z ⊥ whichmeans that Z ⊥ is a right zero in G ( X ) and thus belongs to ↔ G ( X ) according toProposition 5.1.To show that ↔ G ( X ) is a complete sublattice of G ( X ) it is necessary to checkthat ↔ G ( X ) is closed under arbitrary unions and intersections. It is trivial to checkthat arbitrary union of shift-invariant inclusion hyperspaces is shift-invariant, whichmeans that S α ∈ A Z α ∈ ↔ G ( X ) for any family {Z α } α ∈ A ⊂ ↔ G ( X ). Since ↔ G ( X ) isclosed under the transversality operation we also get \ α ∈ A Z α = (cid:0) [ α ∈ A Z ⊥ α ) ⊥ ∈ ↔ G ( X ) ⊥ = ↔ G ( X ) . If ↔ G ( X ) is not empty, then it is a left ideal in G ( X ) because it consists of rightzeros. Now take any right ideal I in G ( X ) and fix any element R ∈ I . Then forevery Z ∈ ↔ G ( X ) we get Z = R ◦ Z ∈ I which yields ↔ G ( X ) ⊂ I . (cid:3) Proposition 5.3. If X is a semigroup and ↔ G ( X ) is not empty, then ↔ G ( X ) is theminimal ideal of G ( X ) .Proof. In light of the preceding proposition, it suffices to check that ↔ G ( X ) is a rightideal. Take any inclusion hyperspaces A ∈ ↔ G ( X ) and B ∈ G ( X ) and take any set F ∈ A ◦ B . We need to show that the sets x ∗ F and x − F belong to A ◦ B . Withoutloss of generality, F is of the basic form: F = [ a ∈ A a ∗ B a for some set A ∈ A and some family { B a } a ∈ A ⊂ B . The associativity of thesemigroup operation 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 . To see that x − F ∈ A observethat the set A ′ = S z ∈ x − A z ∗ B xz belongs to A and each point a ′ ∈ A ′ belongsto the set z ∗ B xz for some z ∈ x − A . Then x ∗ a ′ ∈ x ∗ z ∗ B xz ⊂ F and hence A ∋ A ′ ⊂ x − F , which yields the desired inclusion x − F ∈ A . (cid:3) Now we find conditions on the binary operation ∗ : X × X → X guaranteeing thatthe set ↔ G ( X ) is not empty. By min GX = { X } and max GX = { A ⊂ X : A = ∅} we denote the minimal and maximal elements of the lattice G ( X ). Proposition 5.4.
For a groupoid ( X, ∗ ) the following conditions are equivalent: (1) min GX ∈ ↔ G ( X ) ; (2) max GX ∈ ↔ G ( X ) ; (3) for each a, b ∈ X the equation a ∗ x = b has a solution x ∈ X .Proof. (1) ⇒ (3) Assuming that min GX ∈ ↔ G ( X ) and applying Proposition 5.1observe that for every a ∈ X the equation h a i ◦ { X } = { X } implies that for every 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 hence F ◦ { X } = { X } for all F ∈ G ( X ). This means that { X } = min G ( X ) is a right zero in G ( X ) and hence belongs to ↔ G ( X ) according toProposition 5.1.(2) ⇒ (3) Assume that max G ( X ) ∈ ↔ G ( X ) and take any points a, b ∈ X . Since h a i ◦ max G ( X ) = max G ( X ) ∋ { b } , there is a non-empty set 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 that for every a, b ∈ X the equation a ∗ x = b has a solution.To show that F ◦ max G ( X ) = max G ( X ) it suffices to check that max G ( X ) ⊂F ◦ max G ( X ). Take any set B ∈ max G ( X ) and any set F ∈ F . For every a ∈ F find a point x a ∈ X with a ∗ x a ∈ B . Then the sets S a ∈ F a ∗ { x a } ⊂ B belong to F ◦ max G ( X ), which yields the desired inclusion max G ( X ) ⊂ F ◦ max G ( X ). (cid:3) By analogy we can establish a similar description of zeros and the minimal idealin the semigroup G ◦ ( X ) of free inclusion hyperspaces. IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 17
Proposition 5.5.
Assume that ( X, ∗ ) is an infinite groupoid such that for each b ∈ X there is a finite subset F ⊂ X such that for each a ∈ X \ F the set a − b = { x ∈ X : a ∗ x = b } is finite and not empty. Then (1) G ◦ ( X ) is a closed subgroupoid of G ( X ) ; (2) G ◦ ( X ) is a left ideal in G ( X ) provided if for each a, b ∈ X the set a − b isfinite; (3) the set ↔ G ◦ ( X ) = ↔ G ( X ) ∩ G ◦ ( X ) of shift-invariant free inclusion hyperspacesis the minimal ideal in G ◦ ( X ) ; (4) the set ↔ G ◦ ( X ) is a rectangular subsemigroup of the groupoid G ( X ) and isclosed complete sublattice of the lattice G ( X ) invariant under the transver-sality map. Remark 5.6.
It follows from Propositions 5.2 and 5.5 that the minimal ideals ofthe semigroups G ( Z ) and G ◦ ( X ) are closed. In contrast, the minimal ideals of thesemigroups β Z and β ◦ Z = β Z \ Z are not closed, see [HS, § β ◦ ( Z ) play an important role in Combina-torics of Numbers, see [HS]. We believe that the same will happen for the semigroup λ ◦ ( Z ). The following proposition implies that minimal left ideals of λ ◦ ( Z ) containno ultrafilter! Proposition 5.7.
If a groupoid X admits a homomorphism h : X → Z such thatfor every y ∈ Z the preimage h − ( y ) is not empty (is infinite) then each minimalleft ideal I of λ ( X ) (of λ ◦ ( X ) ) is disjoint from β ( X ) .Proof. It follows that the induced map λh : λ ( X ) → λ ( Z ) is a surjective homo-morphism. Consequently, λh ( I ) is a minimal left ideal in λ ( Z ). Now observe that λ ( Z ) consists of four maximal linked inclusion hyperspaces. Besides three ultra-filters there is a maximal linked inclusion hyperspace L △ = h{ , } , { , } , { , }i where Z = { , , } . One can check that {L △ } is a zero of the semigroup λ ( Z ).Consequently, λ ( h )( I ) = {L △ } , which implies that I ∩ β ( X ) = ∅ .Now assume that for every y ∈ Z the preimage h − ( y ) is infinite. We claim that λh ( λ ◦ ( X )) = λ ( Z ). Take any maximal linked inclusion hyperspace L ∈ λ ( Z ). If L is an ultrafilter supported by a point y ∈ Z , then we can take any free ultrafilter U on X containing the infinite set h − ( y ) and observe that λh ( U ) = L . It remainsto consider the case L = L △ . Fix free ultrafilters U , U , U on X containing thesets h − (0), h − (1), h − (2), respectively. Then L = ( U ∩ U ) ∪ ( U ∩ U ) ∪ ( U ∩ U )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 ), be-ing a minimal left ideal of λ ( Z ) coincides with {L △ } and is disjoint from β ( Z ).Consequently, I is disjoint from β ( X ). (cid:3) The center of G ( X )In this section we describe the structure of the center of the groupoid G ( X ) foreach (quasi)group X . By definition, the center of a groupoid X is the set C = { x ∈ X : ∀ y ∈ X xy = yx } . A groupoid X is called a quasigroup if for every a, b ∈ X the system of equations a ∗ x = b and y ∗ a = b has a unique solution ( x, y ) ∈ X × X . It is clear that eachgroup is a quasigroup. On the other hand, there are many examples of quasigroups,not isomorphic to groups, see [Pf], [CPS]. Theorem 6.1.
Let X be a quasigroup. If an inclusion hyperspace C ∈ G ( X ) commutes with the extremal elements max G ( X ) and min G ( X ) of G ( X ) , then C isa principal ultrafilter.Proof. By Proposition 5.4, the inclusion hyperspaces max G ( X ) and min G ( X ) areright zeros in G ( X ) and thus max G ( X ) ◦ C = C ◦ max G ( X ) = max G ( X ) andmin G ( X ) ◦ C = C ◦ min G ( X ) = min G ( X ). It follows that for every b ∈ X we get { b } ∈ max G ( X ) = max G ( X ) ◦C , which means that a ∗ C ⊂ { b } for some C ∈ C andsome a ∈ X . Since the equation a ∗ y = b has a unique solution y ∈ X , the set C is a singleton, say C = { c } . It remains to prove that C coincides with the principalultrafilter h c i generated by c . Assuming the converse, we would conclude that X \ { c } ∈ C . By our hypothesis, the equation y ∗ c = c has a unique solution y ∈ X .Since the equation y ∗ x = c has a unique solution x = c , y ∗ ( X \ { c } ) ⊂ X \ { c } .Letting C x = { c } for all x ∈ X \ { y } and C x = X \ { c } for x = y , we concludethat X \ { c } ⊃ S x ∈ X x ∗ C x ∈ min G ( X ) ◦ C = C ◦ min G ( X ) = min G ( X ), which isnot possible. (cid:3) Corollary 6.2.
For any quasigroup X the center of the groupoid G ( X ) coincideswith the center of X .Proof. If an inclusion hyperspace C belongs to the center of the groupoid G ( X ),then C is a principal ultrafilter generated by some point c ∈ X . Since C commuteswith all the principal ultrafilters, c commutes with all elements of X and thus c belongs to the center of X .Conversely, if c ∈ X belongs to the center of X , then for every inclusion hyper-space F ∈ G ( X ) we get c ◦ F = { c ∗ F : F ∈ F} = { F ∗ c : F ∈ F} = F ◦ c, which means that (the principal ultrafilter generated by) c belongs to the center ofthe groupoid G ( X ). (cid:3) IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 19
Remark 6.3.
It is interesting to note that for any group X the center of thesemigroup βX also coincides with the center of the group X , see Theorem 6.54 of[HS]. Problem 6.4.
Given a group X describe the centers of the subsemigroups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ of the semigroup G ( X ) . Is it true that the centerof any subsemigroup S ⊂ G ( X ) with β ( X ) ⊂ S = S ⊥ coincides with the center of X ? Remark 6.5.
Let us note that the requirement S = S ⊥ in the preceding questionis essential: for any nontrivial group X the center of the (non-symmetric) subsemi-group X ∪ max G ( X ) of G ( X ) contains max G ( X ) and hence is strinctly larger thanthe center of the group X . Problem 6.6.
Given an infinite group X describe the centers of the semigroups G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , and N ◦ k ( X ) , k ≥ . (By Theorem 6.54 of [HS],the center of the semigroup of free ultrafilters β ◦ ( X ) is empty).7. The topological center of G ( X )In this section we describe the topological center of G ( X ). By the topologicalcenter of a groupoid X endowed with a topology we understand the set Λ( X )consisting of all points x ∈ X such that the left and right shifts l x : X → X, l x : z xz, and r x : X → X, r x : z zx both are continuous.Since all right shifts on G ( X ) are continuous, the topological center of thegroupoid G ( X ) consists of all inclusion hyperspaces F with continuous left shifts l F .We recall that G • ( X ) stands for the set of inclusion hyperpsaces with finitesupport. Theorem 7.1.
For a quasigroup X the topological center of the groupoid G ( X ) coincides with G • ( X ) .Proof. By Proposition 2.8, the topological center Λ( GX ) of G ( X ) contains all prin-cipal ultrafilters and is a sublattice of G ( X ). Consequently, Λ( GX ) contains thesublatttice G • ( X ) of G ( X ) generated by X .Next, we show that each inclusion hyperspace F ∈ Λ( GX ) has finite supportand hence belongs to G • ( X ). By Theorem 9.1 of [G1], this will follow as soon aswe check that both F and F ⊥ have bases consisting of finite sets.Take any set F ∈ F , choose any point e ∈ X , and consider the inclusion hyper-space U = { U ⊂ X : e ∈ F ∗ U } . Since for every f ∈ F the equation f ∗ u = e has a solution in X , we conclude that { e } ∈ F ◦ U and by the continuity of the leftshift l F , there is an open neighborhood O ( U ) of U such that { e } ∈ F ◦ A for all A ∈ O ( U ). Without loss of generality, the neighborhood O ( U ) is of basic form O ( U ) = U +1 ∩ · · · ∩ U + n ∩ V − ∩ · · · ∩ V − m for some sets U , . . . , U n ∈ U and V , . . . , V m ∈ U ⊥ . Take any finite set A ⊂ F − e = { x ∈ X : e ∈ F ∗ x } intersecting each set U i , i ≤ n , and consider theinclusion hyperspace A = h A i ⊥ . It is clear that A ⊂ U +1 ∩ · · · ∩ U + n . Since eachset V j , j ≤ m , contains the set F − e ⊃ A , we get also that A ∈ V − ∩ · · · ∩ 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 − = { x ∈ X : ∃ a ∈ A with xa = e } is finite. We claim that E ⊂ F . Indeed, take any point x ∈ E and find apoint a ∈ A with x ∗ a = e . Since A ⊂ F − e , there is a point y ∈ F with e = y ∗ a .Hence xa = ya and the right cancellativity of X yields x = y ∈ F . Therefore, usingthe continuity of the left shift l F , for every F ∈ F we have found a finite subset E ∈ F with E ⊂ F . This means that F has a base of finite sets.The continuity of the left shift l F and Proposition 2.4 imply the continuity of theleft shift l F ⊥ . Repeating the preceding argument, we can prove that the inclusionhyperspace F ⊥ has a base of finite sets too. Finally, applying Theorem 9.1 of [G1],we conclude that F ∈ G • ( X ). (cid:3) Problem 7.2.
Given an infinite group G describe the topological center of thesubsemigroups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ , of the semigroup G ( X ) . Isit true that the topological center of any subsemigroup S ⊂ G ( X ) containing β ( X ) coincides with S ∩ G • ( X ) ? (This is true for the subsemigroups S = G ( X ) (seeTheorem 7.1) and S = β ( X ), see Theorems 4.24 and 6.54 of [HS]). Problem 7.3.
Given an infinite group X describe the topological centers of thesemigroups G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , and N ◦ k ( X ) , k ≥ . (It should bementioned that the topological center of the semigroup β ◦ ( X ) of free ultrafilters isempty [P ]). 8. Left cancelable elements of G ( X )An element a of a groupoid S is called left cancelable (resp. right cancelable ) iffor any points x, y ∈ S the equation ax = ay (resp. xa = ya ) implies x = y . Inthis section we characterize left cancelable elements of the groupoid G ( X ) over aquasigroup X . Theorem 8.1.
Let X be a quasigroup. An inclusion hyperspace F ∈ G ( X ) is leftcancelable in the groupoid G ( X ) if and only if F is a principal ultrafilter. IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 21
Proof.
Assume that F is left cancelable in G ( X ). First we show that F containssome singleton. Assuming the converse, take any point x ∈ X and note that F ∗ ( X \ { x } ) = X for any F ∈ F . To see that this equality holds, take anypoint a ∈ X , choose two distinct points b, c ∈ F and find solutions x, y ∈ X of theequation b ∗ x = a and c ∗ y = a . Since X is right cancellative, x = y . Consequently,one of the points x or y is distinct from x . If x = x , then a = b ∗ x ∈ F ∗ ( X \{ x } ).If y = x , then a = c ∗ y ∈ F ∗ ( X \ { x } ). Now for the inclusion hyperspace U = h X \ { x }i 6 = min G ( X ), we get F ◦ U = min G ( X ) = F ◦ min G ( X ), whichcontradicts the choice of F as a left cancelable element of G ( X ).Thus F contains some singleton { c } . We claim that F coincides with theprincipal ultrafilter generated by c . Assuming the converse, we would concludethat X \ { c } ∈ F . Let A = h X \ { c }i ⊥ be the inclusion hyperspace consistingof subsets that meet X \ { c } . It is clear that A 6 = max G ( X ). We claim that F ◦ A = max G ( X ) = F ◦ max G ( X ) which will contradict the left cancelability of F . Indeed, given any singleton { a } ∈ max G ( X ), consider two cases: if a = c ∗ c ,then we can find a unique x ∈ X with c ∗ x = a . Since x = c , { x } ∈ A and hence { a } = c ∗ { x } ∈ F ◦ A . If a = c ∗ c , then for every y ∈ X \ { c } we can find a y ∈ X with y ∗ a y = a and use the left cancelativity of X to conclude that a y = c andhence { a y } ∈ A . Then { a } = S y ∈ X \{ c } y ∗ { a y } ∈ F ◦ A .Therefore F = h c i is a principal ultrafilter, which proves the “only if” part ofthe theorem. To prove the “if” part, take any principal ultrafilter h x i generatedby a point x ∈ X . We claim that two inclusion hyperspaces F , U ∈ G ( X ) areequal provided h x i ◦ F = h x i ◦ U . Indeed, given any set F ∈ F observe that x ∗ F ∈ h x i ◦ F = h x i ◦ U and hence x ∗ F = x ∗ U for some U ∈ U . The leftcancelativity of X implies that F = U ∈ U , which yields F ⊂ U . By the sameargument we can also check that
U ⊂ F . (cid:3) Problem 8.2.
Given an (infinite) group X describe left cancelable elements ofthe subsemigroups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ (and G ◦ ( X ) , λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , N ◦ k ( X ) , for k ≥ ). Remark 8.3.
Theorem 8.1 implies that for a countable Abelian group X the setof left cancelable elements in G ( X ) coincides with X . On the other hand, the setof (left) cancelable elements of β ( X ) contains an open dense subset of β ◦ ( X ), seeTheorem 8.34 of [HS].9. Right cancelable elements of G ( X )As we saw in the preceding section, for any quasigroup X the groupoid G ( X )contains only trivial left cancelable elements. For right cancelable elements thesituation is much more interesting. First note that the right cancelativity of an inclusion hyperspace F ∈ G ( X ) is equivalent to the injectivity of the map µ X ◦ G ¯ R F : G ( X ) → G ( X ) considered at the begining of Section 2. We recall that µ X : G ( X ) → G ( X ) is the multiplication of the monad G = ( G, µ, η ) while¯ R F : βX → G ( X ) is the Stone- ˇCech extension of the right shift R F : X → G ( X ), R F : x x ∗ F . The map ¯ R F certainly is not injective if R F is not an embedding,which is equivalent to the discreteness of the indexed set { x ∗ F : x ∈ X } in G ( X ). Therefore we have obtained the following necessary condition for the rightcancelability. Proposition 9.1.
Let X be a groupoid. If an incluison hyperspace F ∈ G ( X ) isright cancelable in G ( X ) , then the indexed set { x F : x ∈ X } is discrete in G ( X ) inthe sense that each point x F has a neighborhood O ( x F ) containing no other points y F with y ∈ X \ { x } . Next we give a sufficient condition of the right cancelability.
Proposition 9.2.
Let X be a groupoid. An inclusion hyperspace F ∈ G ( X ) isright cancelable in G ( X ) provided there is a family of sets { S x } x ∈ X ⊂ F ∩ F ⊥ suchthat xS x ∩ yS y = ∅ for any distinct x, y ∈ X .Proof. Assume that
A ◦ F = B ◦ F for two inclusion hyperspaces A , B ∈ G ( X ).First we show that A ⊂ B . Take any set A ∈ A and observe that the set S a ∈ A aS a belongs to A ◦ F = B ◦ F . Consequently, 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 empty 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 and hence A ∈ B .By analogy we can prove that B ⊂ A . (cid:3) Propositions 9.1 and 9.2 imply the following characterization of right cancelableultrafilters in G ( X ) generalizing a known characterization of right cancelable ele-ments of the semigroups βX , see [HS, 8.11]. Corollary 9.3.
Let X be a countable groupoid. For an ultrafilter U on X thefollowing conditions are equivalent: (1) U is right cancelable in G ( X ) ; (2) U is right cancelable in βX ; (3) the indexed set { x U : x ∈ X } is discrete in βX ; IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 23 (4) there is an indexed family of sets { U x } x ∈ X ⊂ U such that for any distinct x, y ∈ X the shifts x U x and y U y are disjoint. This characterization can be used to show that for any countable group X thesemigroup β ◦ ( X ) of free ultrafilters contains an open dense subset of right cance-lable ultrafilters, see [HS, 8.10]. It turns out that a similar result can be proved forthe semigroup G ◦ ( X ). Proposition 9.4.
For any countable quasigroup, the groupoid G ◦ ( X ) contains anopen dense subset of right cancelable free inclusion hyperspaces.Proof. Let X = { x n : n ∈ ω } be an injective enumeration of the countable quasi-group X . Given a free inclusion hyperspace F ∈ G ◦ ( X ) and a neighborhood O ( F )of F in G ◦ ( X ), we should find a non-empty open subset in O ( F ). Without loss ofgenerality, the neighborhood O ( F ) is of basic form: O ( F ) = G ◦ ( X ) ∩ U +0 ∩ · · · ∩ U + n ∩ U − n +1 ∩ · · · ∩ U − m − for some sets U , . . . , U m − of X . Those sets are infinite because F is free. Weare going to construct an infinite set C = { c n : n ∈ ω } ⊂ X that has infiniteintersection with the sets U i , i < m , and such that for any distinct x, y ∈ X theintersection xC ∩ yC is finite. The points c k , k ∈ ω , composing the set C will bechosen by induction to satisfy the following conditions: • c k ∈ U j where j = k mod m ; • c k does not belong to the finite set F k = { z ∈ X : ∃ i, j ≤ k ∃ l < k ( x i z = x j c l ) } . It is clear that the so-constructed set C = { c k : k ∈ ω } has infinite intersectionwith each set U i , i < m . Since X is right cancellative, for any i < j the set Z i,j = { z ∈ X : x i z = x j z } is finite. Now the choice of the points 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 be the free inclusion hyperspace on X generated by the sets C and U , . . . , U n . It is clear that C ∈ O ( F ) and C ∈ C ∩ C ⊥ . Consider the open neigh-borhood O ( C ) = O ( F ) ∩ C + ∩ ( C + ) ⊥ of C in G ◦ ( X ).We claim that each inclusion hyperspace A ∈ O ( C ) is right cancelable in G ( X ).This will follow from Proposition 9.2 as soon as we construct a family of sets { A i } i ∈ ω ∈ A ∩ A ⊥ such that x i A i ∩ x j A j = ∅ for any numbers i < j . The sets A i , i ∈ ω , can be defined by the formula A k = C \ F k where F k = { c ∈ C : ∃ i < k with x k c = x i C } is finite by the choice of the set C . (cid:3) Problem 9.5.
Given an (infinite) group X describe right cancelable elements of thesubsemigroups λ ( X ) , Fil( X ) , N <ω ( X ) , N k ( X ) , k ≥ λ ◦ ( X ) , Fil ◦ ( X ) , N ◦ <ω ( X ) , N ◦ k ( X ) , for k ≥ . The structure of the semigroups G ( H ) over finite groups H In Proposition 5.7 we have seen that the structural properties of the finite semi-group λ ( Z ) has non-trivial implications for the essentially infinite object λ ◦ ( Z ).This observation is a motivation for more detail study of spaces G ( H ) over finiteAbelian groups H . In 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 we can speak about the orbit A◦ H = {A◦ h : h ∈ H } of an inclusion hyperspace 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 which induces a unique semigroup structureof G ( H ) /H turning π into a semigroup homomorphism.We shall say that the semigroup G ( H ) is splittable if there is a semigroup homo-morphism s : G ( H ) /H → G ( H ) such that π ◦ s is the identity homomorphism of G ( H ) /H . Such a homomorphism s will be called a section of π and the semigroup T ( H ) = s ( G ( H ) /H ) will be called a H -transversal semigroup of G ( H ). It is clearthat a H -transversal semigroup T ( H ) has one-point intersection with each orbit of G ( H ).If the semigroup G ( H ) is splittable, then the structure of G ( H ) can be describedas follows. Proposition 10.1.
If the semigroup G ( H ) is splittable and T ( H ) is the transversalsemigroup of G ( H ) , then T ( H ) is isomorphic to G ( H ) /H and G ( H ) is the quotientsemigroup of the product T ( H ) × H under the homomorphism h : T ( H ) × H → G ( H ) , h : ( A , h )
7→ A ◦ h . It turns out that the semigroup G ( Z n ) is splittable for n ≤ n = 5 (the latter follows from the non-splittability of the semigroup λ ( Z )established in [BGN]). So below we describe the structure of the semigroups G ( Z n )and their transversal semigroup T ( Z n ) for n ≤ X we shall identify the elements x ∈ X with the ultrafilters theygenerate. Also we shall use the notations ∧ and ∨ to denote the lattice operations ∩ and ∪ on G ( X ), respectively. The semigroup G ( Z ). For the cyclic group Z = { e, a } the lattice G ( Z )contains four inclusion hyperspaces: e, a, e ∧ a, e ∨ a , and is shown at the picture: IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 25 r (cid:0)(cid:0) r e a r e ∨ a ❅❅ q e ∧ a The semigroup G ( Z ) has a unique Z -transversal semigroup T ( Z ) = { e ∧ a, e, e ∨ a } with two right zeros: e ∧ a , e ∨ a and one unit e . The semigroup G ( Z ) over the cyclic group Z = { e, a, a − } contains 18 ele-ments: a ∨ e ∨ a − , a ∨ a − , a ∨ e , e ∨ a − , a ∨ ( e ∧ a − ), e ∨ ( a ∧ a − ), a − ∨ ( a ∧ e ), a, e, a − ,( a ∨ e ) ∧ ( a ∨ a − ) ∧ ( e ∨ a − ), a ∧ ( e ∨ a − ), e ∧ ( a ∨ a − ), a − ∧ ( a ∨ e ), a ∧ a − , a ∧ e , e ∧ a − , a ∧ e ∧ a − divided into 8 orbits with respect to the action of the group Z .The semigroup G ( Z ) has 9 different Z -transversal semigroups one of which isdrawn at the picture: T ( Z ) q a ∧ e ∧ a − q a ∧ e q ❅ e ∧ ( a ∨ a − ) q e q ( a ∨ e ) ∧ ( e ∨ a − ) ∧ ( a ∨ a − ) q e ∨ ( a ∧ a − ) (cid:0) q e ∨ a − q a ∨ e ∨ a − The semigroup G ( Z ) has 3 shift-invariant inclusion hyperspaces which are rightzeros: a ∧ e ∧ a − , a ∨ e ∨ a − and ( a ∨ e ) ∧ ( e ∨ a − ) ∧ ( a ∨ a − ). Besides right zeros G ( Z ) has 3 idempotents: e , e ∨ ( a ∧ a − ) and e ∧ ( a ∨ a − ). The element e is theunit of the semigroup G ( Z ). The complete information on the structure of the Z -transversal semigroup T ( Z )(which is isomorphic to the quotient semigroup G ( Z ) / Z ) can be derived from theCayley table ◦ x − x − x − x x x x x − x − x − x − x x x x x − x − x − x − x x x x x − x − x − x − x x x x x x − x − x x x x x x x − x − x x x x x x x − x − x x x x x x x − x x x x x x of its linearly ordered subsemigroup T ( Z ) \ { e } having with 7-elements: x − = e ∧ a ∧ a − ,x − = e ∧ a,x − = e ∧ ( a ∨ a − ) ,x = ( e ∨ a ) ∧ ( e ∨ a − ) ∧ ( a ∨ a − ) ,x = e ∨ ( a ∧ a − ) ,x = e ∨ a,x = e ∨ a ∨ a − . Acknowledgments
The author express his sincere thanks to Taras Banakh and Oleg Nykyforchyn forhelp during preparation of the paper and also to the referee for inspiring criticism.
IGHT-TOPOLOGICAL SEMIGROUP OPERATIONS ON INCLUSION HYPERSPACES 27
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