Algebra in superextension of groups, II: cancelativity and centers
aa r X i v : . [ m a t h . GN ] F e b ALGEBRA IN SUPEREXTENSION OF GROUPS, II:CANCELATIVITY AND CENTERS
TARAS BANAKH AND VOLODYMYR GAVRYLKIV
Abstract.
Given a countable group X we study the algebraic structure of itssuperextension λ ( X ). This is a right-topological semigroup consisting of allmaximal linked systems on X endowed with the operation A ◦ B = { C ⊂ X : { x ∈ X : x − C ∈ B} ∈ A} that extends the group operation of X . We show that the subsemigroup λ ◦ ( X )of free maximal linked systems contains an open dense subset of right cance-lable elements. Also we prove that the topological center of λ ( X ) coincideswith the subsemigroup λ • ( X ) of all maximal linked systems with finite sup-port. This result is applied to show that the algebraic center of λ ( X ) coincideswith the algebraic center of X provided X is countably infinite. On the otherhand, for finite groups X of order 3 ≤ | X | ≤ λ ( X ) isstrictly larger than the algebraic center of X . Introduction
After the topological proof (see [HS, p.102], [H2]) of Hindman theorem [H1],topological methods become a standard tool in the modern combinatorics of num-bers, see [HS], [P]. The crucial point is that any semigroup operation ∗ defined onany discrete space X can be extended to a right-topological semigroup operationon β ( X ), the Stone- ˇCech compactification of X . The extension of the operationfrom X to β ( X ) can be defined by the simple formula:(1) U ∗ V = n [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V o , where U , V are ultrafilters on X . Endowed with the so-extended operation, theStone- ˇCech compactification β ( X ) becomes a compact right-topological semigroup.The algebraic properties of this semigroup (for example, the existence of idem-potents or minimal left ideals) have important consequences in combinatorics ofnumbers, see [HS], [P].The Stone- ˇCech compactification β ( X ) of X is the subspace of the double power-set P ( P ( X )), which is a complete lattice with respect to the operations of unionand intersection. In [G ] it was observed that the semigroup operation extends notonly to β ( X ) but also to the complete sublattice G ( X ) of P ( P ( X )) generated by β ( X ). This complete sublattice consists of all inclusion hyperspaces over X .By definition, a family F of non-empty subsets of a discrete space X is calledan inclusion hyperspace if F is monotone in the sense that a subset A ⊂ X belongsto F provided A contains some set B ∈ F . On the set G ( X ) there is an impor-tant transversality operation assigning to each inclusion hyperspace F ∈ G ( X ) the Mathematics Subject Classification. inclusion hyperspace F ⊥ = { A ⊂ X : ∀ F ∈ F ( A ∩ F = ∅ ) } . This operation is involutive in the sense that ( F ⊥ ) ⊥ = F .It is known that the family G ( X ) of inclusion hyperspaces on X is closed inthe double power-set P ( P ( X )) = { , } P ( X ) endowed with the natural producttopology.The extension of a binary operation ∗ from X to G ( X ) can be defined in the sameway as for ultrafilters, i.e., by the formula (1) applied to any two inclusion hyper-spaces U , V ∈ G ( X ). In [G ] it was shown that for an associative binary operation ∗ on X the space G ( X ) endowed with the extended operation becomes a com-pact right-topological semigroup. Besides the Stone- ˇCech extension, the semigroup G ( X ) contains many important spaces as closed subsemigroups. In particular, thespace λ ( X ) = {F ∈ G ( X ) : F = F ⊥ } of maximal linked systems on X is a closed subsemigroup of G ( X ). The space λ ( X )is well-known in General and Categorial Topology as the superextension of X , see[vM], [TZ]. Endowed with the extended binary operation, the superextension λ ( X )of a semigroup X is a supercompact right-topological semigroup containing β ( X )as a subsemigroup.The thorough study of algebraic properties of the superextensions of groups wasstarted in [BGN] where we described right and left zeros in λ ( X ) and detected allgroups X with commutative superextension λ ( X ) (those are groups of cardinality | X | ≤ λ ( X ) for allfinite groups X of cardinality | X | ≤
5. In [BG ] we shall describe the structure ofminimal left ideals of the superextensions of groups. In this paper we concentrate atcancellativity and centers (topological and algebraic) in the superextensions λ ( X )of groups X . Since λ ( X ) is an intermediate subsemigroup between β ( X ) and G ( X )the obtained results for λ ( X ) in a sense are intermediate between those for β ( X )and G ( X ).In section 2 we describe cancelable elements of λ ( X ). In particular, we showthat for a finite group X all left or right cancelable elements of λ ( X ) are principalultrafilters. On the other hand, if a group X is countable, then the set of rightcancelable elements has open dense intersection with the subsemigroup λ ◦ ( X ) ⊂ λ ( X ) of free maximal linked systems, see Theorem 2.4. This resembles the situationwith the semigroup β ( X ) \ X which contains a dense open subset of right cancelableelements (see [HS, 8.10]), and also with the semigroup G ( X ) whose right cancelableelements form a subset having open dense intersection with the set G ◦ ( X ) of freeinclusion hyperspaces, see [G ].The section 3 is devoted to describing the topological center of λ ( X ). By defi-nition, the topological center of a right-topological semigroup S is the set Λ( S ) ofall elements a ∈ S such that the left shift l a : S → S , l a : x a ∗ x , is continuous.By [HS] for every group X the topological center of the semigroup β ( X ) coincideswith X . On the other hand, the topological center of the semigroup G ( X ) coincideswith the subspace G • ( X ) of G ( X ) consisting of inclusion hyperspaces with finitesupport, see [G , 7.1]. A similar results holds also for the semigroup λ ( X ): for anyat most countable group X the topological center of λ ( X ) coincides with λ • ( X ),see Theorem 3.4. LGEBRA IN SUPEREXTENSION OF GROUPS, II: CANCELATIVITY AND CENTERS 3
The final section 4 is devoted to describing the algebraic center of λ ( X ). Werecall that the algebraic center of a semigroup S consists of all elements s ∈ S thatcommute with all other elements of S . In Theorem 4.2 we shall prove that for anycountable infinite group X the algebraic center of λ ( X ) coincides with the algebraiccenter of X . It is interesting to note that for any group X the algebraic centers ofthe semigroups β ( X ) and G ( X ) also coincide with the center of the group X , see[HS, 6.54] and [G , 6.2]. In contrast, for finite groups X of cardinality 3 ≤ | X | ≤ λ ( X ) is strictly larger than the algebraic center of X , seeRemark 4.4. 1. Inclusion hyperspaces and superextensions
In this section we recall the necessary definitions and facts.A family L of subsets of a set X is called a linked system on X if A ∩ B = ∅ for all A, B ∈ L and L is closed under taking supersets. Such a linked system L is maximal linked if L coincides with any linked system L ′ on X that contains L . Each (ultra)filter on X is a (maximal) linked system. By λ ( X ) we denote thefamily of all maximal linked systems on X . Since each ultrafilter on X is a maximallinked system, λ ( X ) contains the Stone- ˇCech extension β ( X ) of X . It is easy to seethat each maximal linked system on X is an inclusion hyperspace on X and hence λ ( X ) ⊂ G ( X ). Moreover, it can be shown that λ ( X ) = {A ∈ G ( X ) : A = A ⊥ } ,see [G ].By [G ] the subspace λ ( X ) is closed in the space G ( X ) endowed with the topologygenerated by the sub-base consisting of the sets U + = {A ∈ G ( X ) : U ∈ A} and U − = {A ∈ G ( X ) : U ∈ A ⊥ } where U runs over subsets of X . By [G ] and [vM] the spaces G ( X ) and λ ( X )are supercompact in the sense that any their cover by the sub-basic sets containsa two-element subcover. Observe that U + ∩ λ ( X ) = U − ∩ λ ( X ) and hence thetopology on λ ( X ) is generated by the sub-basis consisting of the sets U ± = {A ∈ λ ( X ) : U ∈ A} , U ⊂ X. We say that an inclusion hyperspace
A ∈ G ( X ) • has finite support if there is a finite family F ⊂ A of finite subsets of X such that each set A ∈ A contains a set F ∈ F ; • is free if for each A ∈ A and each finite subset F ⊂ X the complement A \ F belongs to A .By G • ( X ) we denote the subspace of G ( X ) consisting of inclusion hyperspaceswith finite support and G ◦ ( X ) stands for the subset of free inclusion hyperspaceson X . Those two sets induce the subsets λ • ( X ) = G • ( X ) ∩ λ ( X ) and λ ◦ ( X ) = G ◦ ( X ) ∩ λ ( X )in the superextension λ ( X ) of X . By [G ], λ • ( X ) is an open dense subset of λ ( X )while λ ◦ ( X ) is closed and nowhere dense in λ ( X ).Given any semigroup operation ∗ : X × X → X on a set X we can extend thisoperation to G ( X ) letting U ∗ V = n [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V o TARAS BANAKH AND VOLODYMYR GAVRYLKIV for inclusion hyperspaces U , V ∈ G ( X ). Equivalently, the product U ∗ V can bedefined as(2)
U ∗ V = { A ⊂ X : { x ∈ X : x − A ∈ V} ∈ U} where x − A = { z ∈ X : x ∗ z ∈ A } . By [G ] the so-extended operation turns G ( X )into a right-topological semigroup. The structure of this semigroup was studied indetails in [G ]. In this paper we shall concentrate at the study of the algebraicstructure of the semigroup λ ( X ) for a group X .The formula (2) implies that the product U ∗ V of two maximal linked systems U and V is a principal ultrafilter if and only if both U and V are principal ultrafilters.So we get the following Proposition 1.1.
For any group X the set λ ( X ) \ X is a two-sided ideal in λ ( X ) . Cancelable elements of λ ( X )In this section, given a group X we shall detect cancelable elements of λ ( X ).We recall that an element x of a semigroup S is right (resp. left ) cancelable iffor every a, b ∈ X the equation x ∗ a = b (resp. a ∗ x = b ) has at most one solution x ∈ S . This is equivalent to saying that the right (resp. left) shift r a : S → S , r a : x x ∗ a , (resp. l a : S → S , l a : x a ∗ x ) is injective. Proposition 2.1.
Let G be a finite group. If C ∈ λ ( G ) is left or right cancelable,then C is a principal ultrafilter.Proof. Assume that some maximal linked system a ∈ λ ( G ) \ G is left cancelable.This means that the left shift l a : λ ( G ) → λ ( G ), l a : x a ◦ x , is injective. ByProposition 1.1, the set λ ( G ) \ G is an ideal in λ ( G ). Consequently, l a ( λ ( G )) = a ∗ λ ( G ) ⊂ λ ( G ) \ G . Since λ ( G ) is finite, l a cannot be injective. (cid:3) Thus the semigroups λ ( X ) can have non-trivial cancelable elements only forinfinite groups X . According to [HS, 8.11] an ultrafilter U ∈ β ( X ) is right cancelableif and only if the orbit { x U : x ∈ X } is discrete in β ( X ) if and only if for every x ∈ X there is a set U x ∈ U such that the indexed family { x ∗ U x : x ∈ X } isdisjoint.This characterization admits a partial generalization to the semigroup G ( X ).According to [G ] if an inclusion hyperspace A ∈ G ( X ) is right cancelable in G ( X ),then its orbit { x ∗A : x ∈ X } is discrete in G ( X ). On the other hand, A is cancelableprovided for every x ∈ X there is a set A x ∈ A ∩ A ⊥ such that the indexed family { x ∗ A x : x ∈ X } is disjoint. The latter means that x ∗ A x ∩ y ∗ A y = ∅ for anydistinct points x, y ∈ X . This result on right cancelable elements in G ( X ) will helpus to prove a similar result on the right cancelable elements in the semigroup λ ( X ). Theorem 2.2.
Let X be a group and L ∈ λ ( X ) be a maximal linked system on X . (1) If L is right cancelable in λ ( X ) , then the orbit { x L : x ∈ X } is discrete in λ ( X ) and x L 6 = y L for all x, y ∈ X . (2) L is right cancelable in λ ( X ) provided for every x ∈ X there is a set S x ∈ L such that the family { x ∗ S x : x ∈ X } is disjoint.Proof.
1. First note that the right cancelativity of a maximal linked system
L ∈ λ ( X ) is equivalent to the injectivity of the map µ X ◦ λ ¯ R L : λ ( X ) → λ ( X ), see[G ]. We recall that µ X : λ ( X ) → λ ( X ) is the multiplication of the monad LGEBRA IN SUPEREXTENSION OF GROUPS, II: CANCELATIVITY AND CENTERS 5 λ = ( λ, µ, η ) while ¯ R L : β ( X ) → λ ( X ) is the Stone- ˇCech extension of the rightshift R L : X → λ ( X ), R L : x x ∗ L . The map ¯ R L certainly is not injective if R L is not an embedding, which is equivalent to the discreteness of the indexed set { x ∗ L : x ∈ X } in λ ( X ).2. Assume that { S x } x ∈ X ⊂ L is a family such that { x ∗ S x : x ∈ X } is disjoint.To prove that L is right cancelable, take two maximal linked systems A , B ∈ λ ( X )with A ◦ L = B ◦ L . It is sufficient to show that
A ⊂ B . Take any set A ∈ A andobserve that the set S a ∈ A aS a belongs to A ◦ L = B ◦ L . Consequently, there is aset B ∈ B and a family of sets { L b } b ∈ B ⊂ L such that [ b ∈ B bL b ⊂ [ a ∈ A aS a . It follows from S b ∈ L that L 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 ( L b ∩ S b ) ⊂ [ b ∈ B bL b ⊂ [ a ∈ A aS a implies B ⊂ A and hence A ∈ B . (cid:3) It is interesting to remark that the first item gives a necessary but not sufficientcondition of the right cancelability in λ ( X ) (in contrast to the situation in β ( X )). Example 2.3.
By [BGN, 6.3], the superextension λ ( C ) of the 4-element cyclicgroup C is isomorphic to the direct product C × C , where C = C ∪ { e } isthe 2-element cyclic group with attached external unit e (the latter means that ex = xe = x for all x ∈ C ). Consequently, each element of the ideal λ ( C ) \ C is not cancelative but has the discrete 4-element orbit { x L : x ∈ C } . In factall the (left or right) cancelable elements of λ ( C ) are principal ultrafilters, seeProposition 2.1.According to [HS, 8.10], for each infinite group the semigroup β ( X ) containsmany right cancelable elements. In fact, the set of right cancelable elements containsan open dense subset of β ( X ) \ X . A similar result holds also for the semigroup G ( X ) over a countable group X : the set of right cancelable elements of G ( X )contains an open dense subset of the subsemigroup G ◦ ( X ). Theorem 2.2 will helpus to prove a similar result for the semigroup λ ( X ). Theorem 2.4.
For each counatable group X the subsemigroup λ ◦ ( X ) of free max-imal linked systems contains an open dense subset consisting of right cancelableelements in the semigroup λ ( X ) .Proof. Let X = { x n : n ∈ ω } be an injective enumeration of the countable group X . Given a free maximal linked system L ∈ λ ◦ ( X ) and a neighborhood O ( L ) of L in λ ◦ ( X ), we should find a non-empty open subset of right cancelable elements in O ( L ). Without loss of generality, the neighborhood O ( L ) is of basic form: O ( L ) = λ ◦ ( X ) ∩ U ± ∩ · · · ∩ U ± n − for some sets U , . . . , U n − of X . Those sets are infinite because L is free. Weare going to construct an infinite set C = { c n : n ∈ ω } ⊂ X that has infiniteintersection with the sets U i , i < n , 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: TARAS BANAKH AND VOLODYMYR GAVRYLKIV • c k ∈ U j where j = k mod n ; • 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 < n . The choice of the points c k for k > j implies that x i C ∩ x j C ⊂ { x i c m : m ≤ j } ) is finite.Now let C be a free maximal linked system on X enlarging the linked systemgenerated by the sets C and U , . . . , U n − . It is clear that C ∈ O ( L ). Consider theopen neighborhood O ( C ) = O ( L ) ∩ C ± of C in λ ◦ ( X ).We claim that each maximal linked system A ∈ O ( C ) is right cancelable in λ ( X ). This will follow from Proposition 2.2 as soon as we construct a family of sets { A i } i ∈ ω ∈ A such that x i A i ∩ x j A j = ∅ for any numbers i < j . Observe that thesets A i = C \ { x − i x k c m : k < i, m ≤ i } , i ∈ ω, have the required property. (cid:3) By [HS, 8.34], the semigroup β ( Z ) contains many free ultrafilters that are leftcancelable in β ( Z ). On the other hand, by [G , 8.1], the only left cancelable elementsof the semigroup G ( Z ) are principal ultrafilters. Problem 2.5.
Is there a maximal linked system
U ∈ λ ( Z ) \ Z which is left cancelablein the semigroup λ ( Z ) ? The topological center of λ ( X )In this section we describe the topological center of the superextension λ ( X ) of agroup X . By definition, the topological center of a right-topological semigroup S isthe set Λ( S ) of all elements a ∈ S such that the left shift l a : S → S , l a : x a ∗ x ,is continuous.By [HS, 4.24, 6.54] for every group X the topological center of the semigroup β ( X ) coincides with X . On the other hand, the topological center of the semigroup G ( X ) coincides with G • ( X ), see [G , 7.1]. A similar results holds also for thesemigroup λ ( X ): the topological center of λ ( X ) coincides with λ • ( X ) (at least forcountable groups X ).To prove this result we shall use so-called detecting ultrafilters. Definition 3.1.
A free ultrafilter D on a group X is called detecting if there is anindexed family of sets { D x : x ∈ X } ⊂ D such that for any A ⊂ X (1) the set U A = S x ∈ A xD x has the property: U A ∪ yU A = X for all y ∈ X ;(2) for every D ∈ D the set { x ∈ X : xD ⊂ U A } is finite and lies in A . Lemma 3.2.
On each countable group X there is a detecting ultrafilter.Proof. Let X = { x n : n ∈ ω } be an injective enumeration of the group X such that x is the neutral element of X . For every n ∈ ω let F n = { x i , x − i : i ≤ n } . Let a = x and inductively, for every n ∈ ω choose an element a n ∈ X so that a n / ∈ F − n F n A Let us show that for any distinct numbers n, m the intersection x n A ≥ n ∩ x m A ≥ m is empty. Otherwise there would exist two numbers i ≥ n and j ≥ m such that x n a i = x m a j . It follows from x n = x m that i = j . We lose no generality assumingthat j > i . Then x n a i = x m a j implies that a j = x − m x n a i ∈ F − j F j A Let X be a group admitting a detecting ultrafilter D . For a maximallinked system A ∈ λ ( X ) the following conditions are equivalent: (1) the left shift L A : G ( X ) → G ( X ) , L A : F 7→ A ◦ F , is continuous; (2) the left shift l A : λ ( X ) → λ ( X ) , l A : L 7→ A ◦ L , is continuous; (3) the left shift l A : λ ( X ) → λ ( X ) is continuous at the detecting ultrafilter D ; (4) A ∈ λ • ( X ) .Proof. The implications (1) ⇒ (2) ⇒ (3) are trivial while (4) ⇒ (1) follows fromTheorem 7.1 [G ] asserting that the topological center of the semigroup G ( X )coincides with G • ( X ). To prove that (3) ⇒ (4), assume that the left shift l A : λ ( X ) → λ ( X ) is continuous at the detecting ultrafilter D .We need to show that A ∈ λ • ( G ). By Theorem 8.1 of [G ], it suffices to checkthat each set A ∈ A contains a finite set F ∈ A . TARAS BANAKH AND VOLODYMYR GAVRYLKIV Since D is a detecting ultrafilter, there is a family of sets { D x : x ∈ X } ⊂ D such that for every D ∈ D the set { x ∈ X : xD ⊂ S a ∈ A aD A } is finite and lies in A . Consider the set U A = S x ∈ A xD x belonging to the product A◦D . The continuityof the left shift l A : λ ( X ) → λ ( X ) at D yields us a set D ∈ D , such that l A ( D ± ) ⊂ U ± A . This means that U A ∈ A ◦ L for any maximal linked system L ∈ λ ( X ) thatcontains D .The choice of D and { D x } x ∈ X guarantees that S = { x ∈ X : xD ⊂ U A } is a finite subset lying in A . We claim that there is a maximal linked system˜ L ∈ λ ( X ) such that D ∈ ˜ L and x − U A / ∈ ˜ L for all x / ∈ S . Such a system ˜ L can beconstructed as an enlargement of the linked system L = { D, X \ x − U A : x ∈ X \ S } . The latter system is linked because of the definition of S = { x ∈ X : D ⊂ x − U A } and the property (1) of the family ( D x ) x ∈ X from Definition 3.1.Take any maximal linked system ˜ L containing L and observe that D ∈ L and { x ∈ X : x − U A ∈ ˜ L} = { x ∈ X : x − U A ∈ L} = S ⊂ A. Taking into account that D ∈ L , we conclude that A ◦ L = l A ( L ) ∈ U ± A and hencethe set S = { x ∈ X : x − U A ∈ L} ∈ A . This set S is the required finite subset of A belonging to A . (cid:3) Combining Theorem 3.3 with Lemma 3.2 we obtain the main result of this sec-tion. Corollary 3.4. For any countable group X the topological center of the semigroup λ ( X ) coincides with λ • ( X ) . Question 3.5. Is Theorem 3.4 true for a group X of arbitrary cardinality? The algebraic center of λ ( X )This section is devoted to studying the algebraic center of λ ( X ). We recall thatthe algebraic center of a semigroup S consists of all elements s ∈ S that commutewith all other elements of S . Such elements s are called central in S . Lemma 4.1. Let X be a group with the neutral element e . A maximal linkedsystem A ∈ λ ( X ) is not central in λ ( X ) provided there are sets S, T ⊂ X such that (1) | T | = 3 ; (2) for each A ∈ A we get A ∩ S ∈ A and | A ∩ S | ≥ ; (3) there is a finite set B ∈ A such that BS − ∩ T − T ⊂ { e } .Proof. We claim that A does not commute with the maximal linked system T = { A ⊂ X : | A ∩ T | ≥ } . By (3), the maximal linked system A contains a finite set B ∈ A such that BS − ∩ T T − ⊂ { e } . By (2), we can assume that B ⊂ S and B isminimal in the sense that each B ′ ⊂ B with B ′ ∈ A is equal to B . By (2), | B | ≥ { T b } b ∈ B of 2-element subsets of T such that S b ∈ B T b = T . Sucha choice is possible because | B | ≥ S b ∈ B bT b belongs to A ◦ T = T ◦ A and hence we can find a subset D ∈ T and a family { A d } d ∈ D ⊂ A with S d ∈ D dA d ⊂ S b ∈ B bT b . By (2), we can LGEBRA IN SUPEREXTENSION OF GROUPS, II: CANCELATIVITY AND CENTERS 9 assume that each A d ⊂ S . Replacing D by a smaller set, if necessary, we canassume that D ⊂ T and | D | = 2. We claim that A d = B for all d ∈ D and T b = D for all b ∈ B .Indeed, take any d ∈ D and any a ∈ A d . Since da ∈ S x ∈ D xA x ⊂ S b ∈ B bT b , thereare b ∈ B and t ∈ T b with da = bt . Then T − T ∋ t − d = ba − ∈ BA − d ⊂ BS − .Taking into account that T − T ∩ BS − ⊂ { e } , we conclude that t − d = ba − is the neutral element of X . Consequently, a = b ∈ B and d = t ∈ T b . Since a ∈ A d was arbitrary, we get A ∋ A d ⊂ B . The minimality of B ∈ A implies that A d = B . It follows from d = t ∈ T b for d ∈ D that D ⊂ T b . Since | D | = | T b | = 2,we get D = T b for every b ∈ B = A d . Consequently, D = S b ∈ B T b = T whichcontradictions (1). (cid:3) By [HS, 6.54], for every group X the algebraic center of the semigroups β ( X )coincides with the center of the group X . Consequently, the semigroup β ( X ) \ X contains no central elements. A similar result holds also for the semigroup λ ( X ). Theorem 4.2. For any countable infinite group X the algebraic center of λ ( X ) coincides with the algebraic center of X .Proof. It is clear that all central elements of X are central in λ ( X ). Now assumethat a maximal linked system C ∈ λ ( X ) is a central element of the semigroup λ ( X ).Observe that the left shift l C : λ ( X ) → λ ( X ), l C : X 7→ C ◦ X is continuous becauseit coincides with the right shift r C : λ ( X ) → λ ( X ), r C : X 7→ X ◦ C . Consequently, C belongs to the topological center of λ ( X ). Applying Theorem 3.4, we concludethat C ∈ λ • ( X ). We claim that A is a principal ultrafilter.Assuming the converse, consider the family C of minimal finite subsets in C .Since C ∈ λ • ( X ), the family C is finite and hence has finite union S = ∪C . Takeany set B ∈ C and observe that | B | ≥ C is not a principal ultrafilter).Since the group X is infinite, we can choose a 3-element subset T ⊂ X such that T − T ∩ BS − ⊂ { e } . Now we see that the maximal linked system C satisfies theconditions of Lemma 4.1 and hence is not central in λ ( X ), which is a contradiction. (cid:3) We do not know if Theorem 4.2 is true for any infinite group X . Question 4.3. Let X be an infinite group. Does the algebraic center of λ ( X ) coincides with the algebraic center of X ? Remark 4.4. Theorem 4.2 certainly is not true for finite groups. According to[BGN, § X of cardinality 3 ≤ | X | ≤ λ ( X ) containsa central element, which is not a principal ultrafilter. Problem 4.5. Characterize (finite) abelian groups X whose superextensions λ ( X ) have central elements distinct from principal ultrafilters. Have all such groups X cardinality | X | ≤ ? It is interesting to remark that the semigroup λ ( X ) contains many non-principalmaximal linked systems that commute with all ultrafilters. Proposition 4.6. Let X be a group and Y, Z ⊂ X be non-empty subsets suchthat yz = zy for all y ∈ Y , z ∈ Z . Then for any L ∈ λ • ( Y ) ⊂ λ • ( X ) and U ∈ β ( Z ) ⊂ β ( X ) we get L ◦ U = U ◦ L . Proof. It is sufficient to prove that L ◦ U ⊂ U ◦ L . Let S x ∈ L x ∗ U x ∈ L ◦ U .Without loss of generality we may assume that L = { x , . . . , x n } is finite, L ⊂ Y and U x i ⊂ Z . Denote V = U x ∩ . . . ∩ U x n ∈ U . Then [ x ∈ L x ∗ U x = [ x ∈ L U x ∗ x ⊃ V ∗ L ∈ U ◦ L . It follows that S x ∈ L x ∗ U x ∈ U ◦ L and the proof is complete. (cid:3) References [BGN] T. Banakh, V. Gavrylkiv, O. Nykyforchyn. Algebra in superextensions of groups, I: zerosand commutativity , preprint (arXiv:0802.1853).[BG ] T. Banakh, V. Gavrylkiv. Algebra in superextension of groups, III: the minimal ideal ,preprint.[G ] V. Gavrylkiv. The spaces of inclusion hyperspaces over noncompact spaces // Matem. Studii. :1 (2007), 92–110.[G ] V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces // Matem.Studii (to appear).[H1] N. Hindman, Finite sums from sequences within cells of partition of N // J. Combin. TheorySer. A (1974), 1–11.[H2] N. Hindman, Ultrafilters and combinatorial number theory // Lecture Notes in Math. (1979), 49–184.[HS] N. Hindman, D. Strauss, Algebra in the Stone- ˇCech compactification, de Gruyter, Berlin,New York, 1998.[TZ] A. Teleiko, M. Zarichnyi. Categorical Topology of Compact Hausdofff Spaces, VNTL, Lviv,1999.[P] I. Protasov. Combinatorics of Numbers, VNTL, Lviv, 1997.[vM] J. van Mill, Supercompactness and Wallman spaces, Math Centre Tracts. . Amsterdam:Math. Centrum., 1977. Ivan Franko National University of Lviv, Ukraine andAkademia ´Swie¸tokrzyska, Kielce, Poland E-mail address : [email protected] Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine E-mail address ::