Algebra in superextension of groups, II: cancelativity and centers
Given a countable group $X$ we study the algebraic structure of its superextension $\lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\mathcal A\circ\mathcal B=\{C\subset X…
Authors: Taras Banakh, Volodymyr Gavrylkiv
ALGEBRA IN SUPEREXTENSION OF GROUPS, I I: CANCELA TIVITY AND CENTER S T A R AS BANAKH AND VOLODYMYR GA VR YLKIV Abstract. Giv en a coun table group X we stu dy the algebraic structure of its superextension λ ( X ). This is a righ t-topological semigroup consisting of all maximal li nk ed systems on X endow ed with the operation A ◦ B = { C ⊂ X : { x ∈ X : x − 1 C ∈ B } ∈ A} that extends the group op eration of X . W e sho w that the subsemigroup λ ◦ ( X ) of free maximal li nked systems cont ains an op en dense subset of right cance- lable element s. Also w e pro v e that the topological center of λ ( X ) coincides with the subsemigroup λ • ( X ) of all maximal link ed systems with finite sup- port. This result is applied to s ho w that the algebraic center of λ ( X ) coincides with the algebraic cen ter of X pr o vided X is countab ly infinite. On the other hand, f or finite groups X of order 3 ≤ | X | ≤ 5 the algebraic cen ter of λ ( X ) is strictly larger than the algebraic cen ter of X . Introduction After the top ological proo f (see [HS, p.102], [H2]) of Hindman theorem [H 1], top ological metho ds beco me a standard to o l in the mo dern combin atoric s of num - ber s, se e [HS], [P]. The crucial p oint is that any semigroup op er ation ∗ defined o n any discrete space X can be extended to a r ight-topolog ical semigr oup o p e ration on β ( X ), the Stone- ˇ Cech co mpactification of X . The extension of the o p e r ation from X to β ( X ) ca n b e defined by the simple for m ula: (1) U ∗ V = n [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V o , where U , V a re ultra filters on X . Endow ed with the so-extended op eratio n, the Stone- ˇ Cech compa c tifica tion β ( X ) beco mes a co mpact r ight-topolog ical semigroup. The algebra ic prop e rties of this semigr oup (for example, the ex istence of idem- po tent s o r minimal left ideals) ha ve imp ortant cons equences in com binatorics of nu mbers, see [HS], [P]. The Stone- ˇ Cech compa c tifica tion β ( X ) of X is the subspace of the double power- set P ( P ( X )), which is a co mplete lattice with resp ect to the o per ations of union and intersection. In [G 2 ] it was o bserved that the semigr oup op er a tion extends not only to β ( X ) but also to the complete sublattice G ( X ) of P ( P ( X )) genera ted by β ( X ). This complete sublattice consists of all inclusion h yp erspaces o ver X . By definition, a family F of non-empty s ubs e ts of a dis c rete space X is called an inclusion hyp ersp ac e if F is monoto ne in the sense that a subset A ⊂ X be longs to F provided A contains so me set B ∈ F . On the set G ( X ) there is an imp or - tant trans versality op eration assigning to ea ch inclusion hypers pa ce F ∈ G ( X ) the 1991 Mathematics Subje c t Classific ation. 20M99, 54B20. 1 2 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV inclusion hyper space F ⊥ = { A ⊂ X : ∀ F ∈ F ( A ∩ F 6 = ∅ ) } . This op er ation is in volutiv e in the sense that ( F ⊥ ) ⊥ = F . It is k nown that the fa mily G ( X ) of inclus ion h y p er spaces on X is closed in the double p ow er-set P ( P ( X )) = { 0 , 1 } P ( X ) endow e d with the natural pro duct top ology . The e xtension of a binary o p eration ∗ from X to G ( X ) can b e defined in the same wa y as for ultrafilters, i.e., by the form ula (1) applied to an y t wo inclusion hyper- spaces U , V ∈ G ( X ). In [G 2 ] it was shown that for a n as s o ciative binary op er a tion ∗ on X the space G ( X ) endo wed with the extended op eration b ecomes a com- pact rig ht -top olog ical semigr oup. Besides the Stone- ˇ Cech ex tension, the semigro up G ( X ) con tains ma ny imp orta nt spaces as closed subsemigroups. In particular , the space λ ( X ) = {F ∈ G ( X ) : F = F ⊥ } of maximal linked systems o n X is a closed subsemigroup of G ( X ). The s pace λ ( X ) is w ell-known in Genera l and Categorial T opo logy as the sup er extension of X , se e [vM], [TZ]. Endow ed with the extended binary op er a tion, the sup ere x tension λ ( X ) of a se migroup X is a s uper compact rig ht-topologic a l semigroup containing β ( X ) as a subsemigroup. The thorough s tudy of algebra ic prop erties o f the sup erextensio ns of gro ups w as started in [BGN] where we describ ed right and left zer o s in λ ( X ) and detected all groups X with commutativ e superextensio n λ ( X ) (those ar e gro ups of ca rdinality | X | ≤ 4 ). In [B GN] we also describ ed the structure of the semig r oups λ ( X ) for all finite groups X of cardinality | X | ≤ 5. In [BG 3 ] w e sha ll describe the structure of minimal left idea ls of the s uper extensions of g r oups. In this pa per we co ncentrate at cancellativity and centers (topolog ical and algebraic) in the super extensions λ ( X ) of groups X . Since λ ( X ) is an in termediate subsemig r oup betw een β ( X ) and G ( X ) the obta ined results for λ ( X ) in a s ense a re intermediate b etw een those for β ( X ) and G ( X ). In section 2 we describ e cancelable elemen ts of λ ( X ). In particular , w e s how that for a finite group X all left or r ight cancela ble elements of λ ( X ) are principal ultrafilters. On the o ther hand, if a group X is co unt able, then the set of rig ht cancelable ele ments has o pe n dense intersection with the subsemigro up λ ◦ ( X ) ⊂ λ ( X ) of free maximal linked systems, see Theor em 2.4. This resembles the situation with the semigroup β ( X ) \ X which contains a dense op en subs et of r ight ca nce lable elements (see [HS, 8.10 ]), and a lso with the semigr oup G ( X ) who se r ight cance lable elements form a subset having op en dense intersection with the s et G ◦ ( X ) of free inclusion hyper spaces, see [G 2 ]. The se c tio n 3 is devoted to descr ibing the top olo gical center of λ ( X ). By defi- nition, the top olo gic al c ent er of a rig h t-top olog ical semig roup S is the set Λ( S ) of all elements a ∈ S such that the left shift l a : S → S , l a : x 7→ a ∗ x , is co ntin uous. By [HS ] for every group X the topo logical ce nter of the semigroup β ( X ) coinc ide s with X . On the o ther hand, the top olog ical center of the semigr oup G ( X ) coincides with the subspa ce G • ( X ) o f G ( X ) co nsisting of inclus io n hyperspa c e s with finite suppo rt, see [G 2 , 7.1 ]. A s imilar results holds also for the semigroup λ ( X ): for any at most co un table gr oup X the top olo gical center of λ ( X ) coincides with λ • ( X ), see Theo rem 3 .4. ALGEBRA IN SUPEREXTENSION OF GROUPS, II: CANCELA TIVITY AND CENTERS 3 The final section 4 is dev oted to describing the alg ebraic ce n ter of λ ( X ). W e recall that the algebr aic c ent er of a semigroup S consists of all elements s ∈ S that commute with all other elements of S . In Theorem 4.2 we shall pro ve that for any countable infinite gr oup X the algebraic center o f λ ( X ) coincides with the algebraic center of X . It is in ter esting to note that for an y gr oup X the algebra ic centers of the semigro ups β ( X ) and G ( X ) also coincide with the center of the group X , see [HS, 6.54 ] a nd [G 2 , 6.2 ]. In contrast, for finite gr oups X of ca rdinality 3 ≤ | X | ≤ 5 the alg ebraic center o f λ ( X ) is strictly la rger than the alg ebraic center of X , se e Remark 4.4. 1. Inclusion hypersp aces and superextensions In this section w e reca ll the necessary definitions and facts. A family L of subsets of a set X is called a linke d system on X if A ∩ B 6 = ∅ for all A, B ∈ L and L is closed under taking super s ets. Such a linked sys tem L is maximal linke d if L co incides w ith any linked sy s tem L ′ on X that co n tains L . Each (ultra )filter on X is a (maximal) linked system. By λ ( X ) we denote the family o f all maximal linked systems on X . Since each ultrafilter on X is a maximal linked system, λ ( X ) cont ains the Stone- ˇ Cech extension β ( X ) of X . It is ea sy to see that each maximal linked system on X is a n inclusion hyp e rspace on X and hence λ ( X ) ⊂ G ( X ). Moreov er, it can be s hown that λ ( X ) = {A ∈ G ( X ) : A = A ⊥ } , see [G 1 ]. By [G 1 ] the subspace λ ( X ) is closed in the space G ( X ) endowed with the to po logy generated by the sub-base co nsisting o f the sets U + = { A ∈ G ( X ) : U ∈ A} and U − = {A ∈ G ( X ) : U ∈ A ⊥ } where U runs o ver subsets of X . By [G 1 ] a nd [vM] the spaces G ( X ) a nd λ ( X ) are sup ercompa c t in the sense that a n y their cov er by the sub-basic sets contains a tw o-element subcover. Obser ve that U + ∩ λ ( X ) = U − ∩ λ ( X ) and hence the top ology o n λ ( X ) is generated by the sub-basis co nsisting o f the sets U ± = {A ∈ λ ( X ) : U ∈ A} , U ⊂ X . W e say that an inclusion hypers pace A ∈ G ( X ) • has finite supp ort if ther e is a finite family F ⊂ A of finite s ubsets of X such that ea ch set A ∈ A con tains a set F ∈ F ; • is fr e e if for each A ∈ A and ea ch finite subset F ⊂ X the co mplemen t A \ F belo ngs to A . By G • ( X ) w e denote the subspace of G ( X ) consisting of inclusion hyperspace s with finite supp or t and G ◦ ( X ) stands for the subs et of free inclusion hypers paces on X . Those tw o sets induce the s ubsets λ • ( X ) = G • ( X ) ∩ λ ( X ) and λ ◦ ( X ) = G ◦ ( X ) ∩ λ ( X ) in the s uper extension λ ( X ) o f X . By [G 1 ], λ • ( X ) is an open dense subset o f λ ( X ) while λ ◦ ( X ) is clo sed a nd no where dense in λ ( X ). Given any semigroup op e ration ∗ : X × X → X on a set X we can extend this op eration to G ( X ) letting U ∗ V = n [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V o 4 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV for inclusion hyp e rspaces U , V ∈ G ( X ). Equiv alently , the pro duct U ∗ V can be defined a s (2) U ∗ V = { A ⊂ X : { x ∈ X : x − 1 A ∈ V } ∈ U } where x − 1 A = { z ∈ X : x ∗ z ∈ A } . By [G 2 ] the s o-extended op eration turns G ( X ) int o a right-topolo gical semigro up. The structur e of this semigroup was studied in details in [G 2 ]. In this paper w e shall concen trate at the study o f the algebraic structure of the s e migroup λ ( X ) for a group X . The formula (2) implies that the pro duct U ∗ V of tw o maximal linked systems U and V is a principal ultr afilter if a nd o nly if bo th U a nd V are principal ultr a filters. So we get the following Prop ositio n 1.1 . F or any gr oup X t he set λ ( X ) \ X is a t wo-side d ide al in λ ( X ) . 2. Cancelable elements of λ ( X ) In this section, giv en a group X we shall detect cancela ble elemen ts o f λ ( X ). W e recall that an element x of a semigroup S is right (r esp. left ) c anc elable if for every a, b ∈ X the eq uation x ∗ a = b (resp. a ∗ x = b ) ha s at most o ne solutio n x ∈ S . This is equiv alent to s aying that the right (resp. left) shift r a : S → S , r a : x 7→ x ∗ a , (resp. l a : S → S , l a : x 7→ a ∗ x ) is injective. Prop ositio n 2.1. L et G b e a fin ite gr oup. I f C ∈ λ ( G ) is left or right c anc elable, then C is a princip al ultr afilter. Pr o of. Assume that some maxima l linked system a ∈ λ ( G ) \ G is left cancela ble. This means that the left shift l a : λ ( G ) → λ ( G ), l a : x 7→ a ◦ x , is injectiv e. By Prop ositio n 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 b e injective. Thu s the s e mig roups λ ( X ) can hav e non-trivial ca ncelable elements only for infinite gro ups X . Acco rding to [HS, 8.11] an ultrafilter U ∈ β ( X ) is r ight cancelable if and o nly if the or bit { x U : x ∈ X } is dis crete in β ( X ) if and o nly if for every x ∈ X there is a s et U x ∈ U such that the indexe d family { x ∗ U x : x ∈ X } is disjoint. This c haracter ization admits a partial g eneralizatio n to the semigr oup G ( X ). According to [G 2 ] if an inclusion hypers pa ce A ∈ G ( X ) is right ca ncelable in G ( X ), then its orbit { x ∗ A : x ∈ X } is discrete in G ( X ). O n the other hand, A is cancelable provided 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 any distinct p oints x, y ∈ X . This result on right cancelable elemen ts in G ( X ) will help us to prov e a similar result on the right cancelable elements in the semigr oup λ ( X ). Theorem 2.2. L et X b e a gr oup and L ∈ λ ( X ) b e a maximal linke d system on X . (1) If L is right c anc elable in λ ( X ) , then the orbit { x L : x ∈ X } is discr ete in λ ( X ) and x L 6 = y L for al l x, y ∈ X . (2) L is right c anc elable in λ ( X ) pr ovide d for every x ∈ X t her e is a set S x ∈ L such that the family { x ∗ S x : x ∈ X } is disjoi nt. Pr o of. 1. First note that the rig ht cancelativity of a maximal linked sys tem L ∈ λ ( X ) is eq uiv alent to the injectivit y of the map µ X ◦ λ ¯ R L : λ ( X ) → λ ( X ), see [G 2 ]. W e recall that µ X : λ 2 ( X ) → λ ( X ) is the multiplication of the monad ALGEBRA IN SUPEREXTENSION OF GROUPS, II: CANCELA TIVITY AND CENTERS 5 λ = ( λ, µ, η ) while ¯ R L : β ( X ) → λ ( X ) is the Stone- ˇ Cech extens ion of the r ight shift R L : X → λ ( X ), R L : x 7→ x ∗ L . The map ¯ R L certainly is not injective if R L is not an em bedding , which is equiv alent to the discreteness o f the indexed se t { 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 disjoin t. T o prove tha t L is right c a ncelable, take tw o maximal linked sys tems A , B ∈ λ ( X ) with A ◦ L = B ◦ L . It is sufficien t to show that A ⊂ B . T ake any set A ∈ A and observe that the set S a ∈ A aS a belo ngs to A ◦ L = B ◦ L . Consequently , there is a set B ∈ B and a family of sets { L b } b ∈ B ⊂ L such that [ b ∈ B bL b ⊂ [ a ∈ A aS a . It fo llows from S b ∈ L that L b ∩ S b is not empt y for every b ∈ B . Since the sets aS a and bS b are disjoin t for differen t 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 . It is interesting to r emark that the first item gives a necessary but not sufficient condition o f the right c a ncelability in λ ( X ) (in contrast to the situation in β ( X )). Example 2 .3. By [BGN, 6.3], the sup erextens io n λ ( C 4 ) of the 4- e le men t cyclic group C 4 is isomo rphic to the direct pr o duct C 4 × C 1 2 , where C 1 2 = C 2 ∪ { e } is the 2-element cyclic gro up with attached external unit e (the latter means that ex = xe = x for all x ∈ C 1 2 ). Consequent ly , each element o f the ideal λ ( C 4 ) \ C 4 is not cancelative but has the discrete 4-element o rbit { x L : x ∈ C 4 } . In fact all the (left or rig ht ) cancelable elements of λ ( C 4 ) are principal ultrafilter s, see Prop ositio n 2.1. According to [HS, 8.10], for each infinite group the semigroup β ( X ) contains many right c a ncelable elements. In fact, the set o f r ight cancela ble elemen ts contains an o p e n dense subse t of β ( X ) \ X . A similar r esult holds also for the semigro up G ( X ) o ver a coun table g roup X : the set of right cancelable elements of G ( X ) contains an open dense subset of the subsemigr o up G ◦ ( X ). Theorem 2 .2 will help us to prov e a s imilar r esult for the semig roup λ ( X ). Theorem 2.4. F or e ach c ounatable gr oup X t he su bsemigr oup λ ◦ ( X ) of fr e e max- imal linke d systems c ontains an op en dense subset c onsisting of right c anc elable elements in the semigr oup λ ( X ) . Pr o of. Let X = { x n : n ∈ ω } be a n injective enum eration of the countable gro up X . Given a free maximal linked system L ∈ λ ◦ ( X ) and a neighbo rho o d O ( L ) of L in λ ◦ ( X ), w e s hould find a non-empty o pe n subset of right ca ncelable elemen ts in O ( L ). Without loss o f generality , the neigh b orho o d O ( L ) is of basic form: O ( L ) = λ ◦ ( X ) ∩ U ± 0 ∩ · · · ∩ U ± n − 1 for so me s e ts U 1 , . . . , U n − 1 of X . Those sets are infinite b ecause L is free. W e are going to co nstruct an infinite set C = { c n : n ∈ ω } ⊂ X that has infinite int ersection with the sets U i , i < n , and such that for any distinct x, y ∈ X the int ersection xC ∩ y C is finite. The p oints c k , k ∈ ω , co mpo sing the se t C will b e chosen by induction to satisfy the follo wing c o nditions: 6 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV • c k ∈ U j where j = k mo d n ; • c k do es not b elong 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 intersection with each set U i , i < n . The choice of the p oints c k for k > j implies that x i C ∩ x j C ⊂ { x i c m : m ≤ j } ) is finite. Now let C b e a fre e maximal link ed system on X enlarging the linked sys tem generated by the sets C and U 0 , . . . , U n − 1 . It is clear that C ∈ O ( L ). Consider the op en neighbo rho o d O ( C ) = O ( L ) ∩ C ± of C in λ ◦ ( X ). W e claim that each maximal link ed system A ∈ O ( C ) is right cancelable in λ ( X ). This will follow from Pro po sition 2.2 as so on as w e construct a family of se ts { A i } i ∈ ω ∈ A such that x i A i ∩ x j A j = ∅ for a ny num b ers i < j . Observe that the sets A i = C \ { x − 1 i x k c m : k < i , m ≤ i } , i ∈ ω , hav e the r e quired pr op erty . By [HS, 8.3 4], the semigroup β ( Z ) contains many free ultr a filters that a re left cancelable in β ( Z ). On the other hand, by [G 2 , 8 .1], the only left cancelable elements of the semigroup G ( Z ) are principal ultrafilters . Problem 2. 5. Is ther e a maximal linke d s yst em U ∈ λ ( Z ) \ Z which is left c anc elable in the semigr oup λ ( Z ) ? 3. The topological center o f λ ( X ) In this section we describ e the top olo gical center o f the supe r extension λ ( X ) of a group X . By definition, the t op olo gic al c enter of a right-topo logical se migroup S is the s et Λ( S ) of a ll elemen ts a ∈ S such that the left shift l a : S → S , l a : x 7→ a ∗ x , is contin uous. By [HS, 4.24, 6.54 ] for ev ery g roup X the topo logical cen ter of the semig roup β ( X ) coincides with X . On the other hand, the top olog ical center of the semigr oup G ( X ) co incides with G • ( X ), see [G 2 , 7.1]. A similar results holds also for the semigroup λ ( X ): the top olog ic al center of λ ( X ) co incides with λ • ( X ) (at lea st for countable groups X ). T o prov e this result w e sha ll use so-called de tec ting ultrafilters. Definition 3 .1. A free ultra filter D on a group X is called dete cting if there is an indexed family of sets { D x : x ∈ X } ⊂ D such that for any A ⊂ X (1) the s et U A = S x ∈ A xD x has the prop erty: U A ∪ y U A 6 = 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 e ach c ount able gr oup X ther e is a dete cting ultr afilter. Pr o of. Let X = { x n : n ∈ ω } b e an injective enumeration of the gro up X such that x 0 is the neutral elemen t of X . F or every n ∈ ω let F n = { x i , x − 1 i : i ≤ n } . Let a 0 = x 0 and inductively , for every n ∈ ω choose a n elemen t a n ∈ X so that a n / ∈ F − 1 n F n A i . Then x n a i = x m a j implies that a j = x − 1 m x n a i ∈ F − 1 j F j A m . W e claim that a k / ∈ U A ∪ y U A . Otherwis e, a k ∈ x n D n ∪ x − 1 m x n D n for s o me n ∈ A . It follo ws that a k = x n a i or a k = x − 1 m x n a i for some even i ≥ n . If k > i , then b oth the equa lities a re forbidden by the choice of a k / ∈ F − 1 k F k A
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