Algebra in the superextensions of groups: minimal left ideals

ALGEBRA IN THE SUPEREXTENSIONS OF GR OUPS, I I I: MINIMAL LEFT IDEALS OF λ ( Z ) T.BANAKH, V.GA VR YLK IV Abstract. W e pr o v e that the minimal left ideals of the sup erextension λ ( Z ) of the discrete group Z of in tegers are metrizable t opological s emigroups, topo- logically isomorphic to minimal l eft ideals of the sup erexte nsion λ ( Z 2 ) of the compact group Z 2 of inte ger 2-adic n umbers. The super ext ension λ ( X ) of a discrete group X is the compact Hausdorff right-to p ological semigroup consisting of maximal l ink ed systems on X and endo w ed with the semigroup oper at ion A ∗ B = { A ⊂ X : { x ∈ X : x − 1 A ∈ B} ∈ A} . Introduction After the topolog ic a l pro of (see [HS, p.102], [H2]) of Hindman theo rem [H1], top ological metho ds beco me a standard to ol in the moder n combinatorics o f num - ber s, see [HS], [P]. The cr ucial po in t is that any se mig roup o peratio n ∗ defined on any discrete space X ca n b e extended to a right-topo logical se mig roup o peratio n on β ( X ), the Stone- ˇ Cech compactification o f X . The extension of the op e ration from X to β ( X ) can b e 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 . Endow ed with the so-extended op eration, the Stone- ˇ Cech co mpactification β ( X ) b ecomes a compac t right-topolo g ical s emigroup. The alg ebraic prop erties of this semigro up (for example, the existence of idem- po ten ts or minimal left ideals ) hav e impor tan t cons e quences in combinatorics of nu mbers, se e [HS], [P]. The Stone- ˇ Cech co mpactification β ( X ) of X is the subspa ce of the double p o wer- set P ( P ( X )), which is a complete la ttice with resp ect to the op erations of union and intersection. In [G 2 ] it was o bserv ed that the semigroup o peration e xtends not only to β ( X ) but als o to the complete s ubla ttice G ( X ) of P ( P ( X )), generated by β ( X ). This complete sublattice co nsists of all inclus ion hyperspa ces over X . By definition, a fa mily F of non-empty subsets of a discrete spa c e X is called an inclusion hyp ersp ac e if F is monotone in the sense tha t a subset A ⊂ X b elongs to F provided A contains some se t B ∈ F . Besides the op erations o f union and int ersection, the set G ( X ) possess es an importa n t transversality op eration assigning to each inclusion hyper space F ∈ G ( X ) the inclusion h ype rspace F ⊥ = { A ⊂ X : ∀ F ∈ F ( A ∩ F 6 = ∅ ) } . This op eration is inv olutive in the sense that ( F ⊥ ) ⊥ = F . 1991 M athema tics Subje ct Classific ation. 20M12, 54B99, 54H10. 1 2 T.BANAKH, V.GA VR YLKIV It is known that the family G ( X ) o f inclus ion hyperspa ces on X is closed in the double pow er-set P ( P ( X )) = { 0 , 1 } P ( X ) endow ed with the na tural pro duct top ology . The induce d top ology on G ( X ) can b e descr ibed dir ectly: it is generated by the sub-base consisting of the sets U + = { F ∈ G ( X ) : U ∈ F } and U − = { F ∈ G ( X ) : U ∈ F ⊥ } where U runs over subsets of X . Endow ed with this topo logy , G ( X ) b ecomes a Hausdorff sup ercompact space. The latter mea ns that each cov er of G ( X ) by the sub-basic sets has a 2-element sub co ver. The extension o f a binary o pera tio n ∗ from X to G ( X ) can be defined in the same wa y as for ultra filter s, i.e., b y the formula (1 ) applied to an y tw o inclusion hyper - spaces U , V ∈ G ( X ). In [G 2 ] it was shown that for an a s socia tive binary o peratio n ∗ on X the spa ce G ( X ) endow ed with the extended operatio n b ecomes a com- pact right-topo logical semigroup. Besides the Stone- ˇ Cech extension, the semigroup G ( X ) contains many imp ortant s paces as closed subsemigroups. In pa rticular, the space λ ( X ) = {F ∈ G ( X ) : F = F ⊥ } of maximal linked systems on X is a closed subsemigro up of G ( X ). The space λ ( X ) is well-kno wn in Gener al and Ca teg orial T o pology as the sup er ext ension of X , see [vM], [TZ]. Endow ed with the ex tended binary op eration, the sup erextension λ ( X ) of a semigroup X is a s up ercompact right-topolo gical s e migroup containing β ( X ) as a subsemigroup. The thorough study of alg ebraic prop erties o f the superextensions λ ( X ) of gro ups X w as s ta rted in [BGN] and cont inued in [BG 2 ]. In this pape r we concentrate a t describing the minimal (left) ideals of λ ( X ). Understanding the s tructure of minimal left ideals of the semigr oup β ( X ) had impo rtan t c om bina to rial conse q uences. F or example, prop erties of ultra filters fro m a minimal left ideal of β ( X ) were explo ited in the top ological pro of o f the class ical V an der W aerden Theorem [HS, 14.3 ] due to F usten b erg a nd Katznels on [FK]. Minimal left ideals of the semigro up β ( Z ) play a lso a n importa n t role in T op ological Dynamics, see [BB], [BF], [HS, Ch.19 ]. W e believe that studying the structure of minimal (left) ideals of the semigr oups λ ( X ) a lso will hav e some combinatorial or dynamical co nsequences. The main result of this pap er is Theo rem 5.1 asser ting that the minimal left ideals of the semigr o up λ ( Z ) a re compact metrizable top ological semigro ups topo - logically isomor phic to minimal left ideals of the sup erextension λ ( Z 2 ) of the (co m- pact metrizable) gro up Z 2 of integer 2-adic num b ers. 1. Right-topological semigroups In this section we r e c all some infor mation fro m [HS] related to r igh t-top ological semigroups. By definitio n, a right-topo logical semig r oup is a top ological space S endow ed with a semigr oup op eration ∗ : S × S → S such tha t for e v er y a ∈ S the right shift r a : S → S , r a : x 7→ x ∗ a , is contin uous. If the se mig roup o peratio n ∗ : S × S → S is co ntin uous, then ( S, ∗ ) is a t op olo gic al semigr oup . A non-empt y subset I of a semigroup S is called a left (resp. right ) ide al if S I ⊂ I (resp. I S ⊂ I ). If I is b oth a left and right ideal in S , then I is calle d a n ide al in S . Observe that for every x ∈ S the set S x = { sx : s ∈ S } (resp. xS = { xs : s ∈ S } ) is a le ft (resp. right) idea l in S . Suc h an idea l is called princip al . An ideal I ⊂ S ALGEBRA IN THE SUPEREXTENS IONS OF GR OUPS, I II 3 is called minimal if any ideal of S that lies in I coincides with I . By analogy we define minimal left and right ideals o f S . It is e asy to see that each minimal left (resp. right) ideal I is principal. Moreover, I = S x (resp. I = xS ) for e a c h x ∈ I . If S is a compact Hausdo rff right-topolog ical semigro up, then each minimal left ideal in S , b eing principal, is closed in S . By [HS, 2.6 ], each left ideal in S contains a minimal left ideal. The union of all minimal left ideals o f S coincides with the minimal ideal K ( S ) of S , [HS, 2.8]. By [HS, 2.11], all the minimal left ideals o f S are mutually homeomor phic. An element z o f a semigro up S is called a rig ht zer o in S if xz = z for all x ∈ S . It is clear that z ∈ S is a r igh t zero in S if and only if the singleton { z } is a (minimal) left ideal in S . In the sequel w e shall often use the following Lemma 1.1. L et X , Y b e c omp act righ t-top olo gic al semigr oups. If a semigr ou p homomorph ism h : X → Y is inje ct ive on some minimal left ide al of X , then h is inje ctive on e ach minimal left ide al of X . Pr o of. Assume that h is injective o n a minimal left idea l X a of X and take any other minimal left ideal X b of X . By [HS, 2.11 ], the r igh t shift r a : X → X , r a : x 7→ xa , is injectiv e on X b . Next, consider the right shift r h ( a ) : Y → Y , r h ( a ) : y 7→ y · h ( a ). It follows from the equality h ◦ r a = r h ( a ) ◦ h and the injectivit y of the maps r a | X b and h | X a that the map h | X b is injective.  2. Inclusio n hypersp aces and superextensions A family L o f subsets o f a set X is called a linke d system on X if A ∩ B 6 = ∅ for all A, B ∈ L . Such a linked system L is maximal linke d if L coincides with any linked sy stem L ′ on X that contains L . Each (ultra)filter on X is a (maximal) linked system. A linked sys tem L on X is maxima l linked if and only if for any partition X = A ∪ B e ither A o r B b elongs to L . By λ ( X ) w e denote the family of all maxima l linked systems on X . Since each ultrafilter on X is a maximal linked system, λ ( X ) contain the Stone- ˇ Cech extens io n β ( X ) of X . It is ea sy to see that each maxima l linked system on X is an inclusion hyperspace on X and hence λ ( X ) ⊂ G ( X ). Moreov er, it can be shown that λ ( X ) = {A ∈ G ( X ) : A = A ⊥ } . Let also N 2 ( X ) = {A ∈ G ( X ) : A ⊂ A ⊥ } denote the family o f all linked inclusion h yp erspaces on X . By [G 1 ] b oth the subspaces λ ( X ) and N 2 ( X ) are closed in the compact Ha usdorff space G ( X ). Each function f : X → Y b etw een sets X, Y induces a contin uo us map Gf : G ( X ) → G ( Y ) assigning to an inclusio n hyper space A ∈ G ( X ) the inclusion hy- per space Gf ( A ) = { B ⊂ Y : f − 1 ( B ) ∈ A} ∈ G ( Y ) . The function Gf maps λ ( X ) into λ ( Y ), so we can put λf = Gf | λ ( X ). Given any semigroup op eration ∗ : X × X → X on a set X we c an ex tend this op eration to G ( X ) letting U ∗ V = n [ x ∈ U x ∗ V x : U ∈ U , { V x } x ∈ U ⊂ V o for inclusion h yp e rspaces U , V ∈ G ( X ). Equiv alently , the pro duct U ∗ V can b e defined a s U ∗ V = { A ⊂ X : { x ∈ X : x − 1 A ∈ V } ∈ U } 4 T.BANAKH, V.GA VR YLKIV where x − 1 A = { z ∈ X : x ∗ z ∈ A } . By [G 2 ] the so-ex tended op eration tur ns G ( X ) int o a right-topolo gical semigr oup. The str ucture of this semigroup was studied in details in [G 2 ]. In pa rticular, it was shown that fo r each g roup X the minimal left ideals of G ( X ) are sing le tons containing invariant inclusion hype rspaces. W e call a n inclusio n h yp erspace A ∈ G ( X ) invariant if x A = A for all x ∈ X . More gener ally , given a subgro up H ⊂ X we define A to b e H - invari ant if x A = A for all x ∈ H . It follows from the definition of the topo logy on G ( X ) that the set ↔ G ( X ) of in- v ariant inclusin hypers paces is clo sed in G ( X ) and coincides with the minimal ideal K ( G ( X )) of the semigroup G ( X ). Consequently , K ( G ( X )) is a clo sed rectangular subsemigroup of G ( X ). The r e ctangularity of K ( G ( X )) means that A ◦ B = B for all A , B ∈ K ( G ( X )). 3. The minimal ideal of λ ( G ) for o dd groups In this section we characterize groups G whose super extension λ ( G ) has one- po in t minimal left ideals. F ollowing [BGN], we define a group G to b e o dd if the order of each element x of G is o dd. If G is a finite odd gro up, then the maximal linked system L = { A ⊂ G : | A | > | G | / 2 } is inv aria n t. In fact, a gro up G po ssesses a n in v ariant maximal linked system if and only if G is o dd, see Theo rem 3.2 of [BGN]. By Pro p osition 3.1 of [BGN], a maximal linked system Z ∈ λ ( G ) on a g roup G is in v ariant if and o nly if Z is a right zero of the semigr oup λ ( G ) if and only if the s ingleton {Z } is a minimal left ideal in λ ( G ). T aking into account that the inv ariant maximal linked systems form a closed rectangular subsemigr oup of λ ( G ), we obtain the main result o f this section. Theorem 3.1. A gr oup G is o dd if and only if al l t he minimal left ide als of λ ( G ) ar e singletons. In this c ase the minimal ide al K ( λ ( G )) of λ ( G ) is a close d r e ct angular semigr oup c onsisting of invariant max imal linke d syst ems. Given a subgro up H of a g roup G let G/H = { xH : x ∈ G } and π : G → G/H denote the q uotien t map. It induces a contin uo us map λπ : λ ( G ) → λ ( G/H ) betw een the corre s ponding sup erextensions. Lemma 3. 2. F or any H -invariant m aximal linke d system A ∈ λ ( H ) ⊂ λ ( G ) the r estriction of λπ : λ ( G ) → λ ( G/H ) t o t he princip al left ide al λ ( G ) ∗ A is inje ctive. Pr o of. Fix a section s : G/H → G o f π . F or every L ∈ λ ( G ) let e L = λπ ( L ) ∈ λ ( G/H ) be the pro jection of L onto G/H and M = λs ( e L ) ∈ λ ( G ) b e the lift of e L by the section s . W e claim that L ∗ A = M ∗ A . Since L ∗ A and M ∗ A are maximal link ed systems, it suffices to check that L ∗A ⊂ M ∗ A . T a k e an y set S x ∈ L x ∗ A x ∈ L ∗A where L ∈ L and { A x } x ∈ L ⊂ A . Consider the set M = s ◦ π ( L ) ∈ M . F or every p oin t y ∈ M find a p oin t x y ∈ L with y = sπ ( x y ) and o bserve that y H = π ( y ) = π ( x y ) = x y H , which implies y − 1 x y ∈ H and hence y − 1 x y A x y ∈ A by the H -inv ar ian tness of A . Since M ∗ A ∋ [ y ∈ M y ( y − 1 x y ∗ A x y ) = [ y ∈ M x y ∗ A x y ⊂ [ x ∈ L x ∗ A x ALGEBRA IN THE SUPEREXTENS IONS OF GR OUPS, I II 5 we conclude that S x ∈ L x ∗ A x ∈ M ∗ A . Now we a re able to prov e that λπ : λ ( G ) ∗ A → λ ( G/H ) is injective. T ake any t wo distinct elements L 1 ∗ A 6 = L 2 ∗ A of λ ( G ) ∗ A . F or every i ∈ { 1 , 2 } consider the maximal linked sy stems e L i = λπ ( L i ) = λπ ( L i ∗ A ) and M i = λs ( e L i ). It follows from M 1 ∗ A = L 1 ∗ A 6 = L 2 ∗ A = M 2 ∗ A that M 1 6 = M 2 and he nc e λπ ( L 1 ∗ A ) = e L 1 6 = e L 2 = λπ ( L 2 ∗ A ) .  Corollary 3.3. F or a normal o dd sub gr oup H of a gr oup G the map λπ : λ ( G ) → λ ( G/H ) is inje ctive on e ach minimal left ide al of λ ( G ) . Conse quent ly, every mini- mal left ide al of λ ( G ) is t op olo gic al ly isomorphic to a minimal left ide al of λ ( G/H ) . Pr o of. By Le mma 1.1, it suffices to sho w that λπ is injective on some minimal left ideal. The group H , b eing o dd, admits an H -inv ariant ma ximal linked system A ∈ λ ( H ) ⊂ λ ( G ). By Lemma 3.2 the homomorphism λπ is injective on the left ideal λ ( G ) ∗ A a nd hence is injectiv e o n any minimal left ideal contained in λ ( G ) ∗ A (it exists b e c ause λ ( G ) is a compact rig h t-top ologica l semigroup).  4. Maximal inv arian t linked systems on groups As we have seen in the pr eceding se c tio n, the pr operty o f a maxima l sy stem L ∈ λ ( G ) to be inv ar ian t is very stro ng and forces L to b e a right zero o f λ ( G ). Such maximal linked systems exis t only on o dd groups. On the o ther hand, maximal inv ar ian t linked s ystems exis t on each g roup. An inv aria n t linked inclusio n hypers pace L ∈ ↔ N 2 ( G ) is calle d a maximal inva riant linke d system if L = L ′ for a n y inv aria n t linked inclus io n hyperspa ce L ′ ∈ ↔ N 2 ( G ) enlarging L . By the Zor n Lemma, each in v ariant linked inclusion h yp erspace ca n be enlar ged to a ma ximal inv aria n t linked system. Prop osition 4. 1. F or any maximal invariant linke d system L 0 on a gr oup G the set ↑L 0 = { L ∈ λ ( G ) : L ⊃ L 0 } is a left ide al in λ ( G ) . Pr o of. Let A , B ∈ λ ( X ) b e max imal linked systems with L 0 ⊂ B . The n for every subset L ∈ L 0 we get L = [ x ∈ G x ( x − 1 L ) ∈ A ∗ L 0 ⊂ A ∗ B which means that L 0 ⊂ A ∗ B .  Observe that L 0 ⊂ L ⊂ L ⊥ 0 for every L ∈ ↑ L 0 . T he following theor em shows that the difference L ⊥ 0 \ L 0 (and consequently , L \ L 0 ) is relatively small (for the group G = Z it is co untable!). Theorem 4.2. If L 0 is a maximal invariant linke d system on an Ab elian gr oup G , then for any su bset A ∈ L ⊥ 0 \ L 0 ther e is a p oint x ∈ G su ch that xA = G \ A and c onse quent ly, A = x 2 A . 6 T.BANAKH, V.GA VR YLKIV Pr o of. Fix a subse t A ∈ L ⊥ 0 \ L 0 . W e claim tha t (2) aA ∩ A = ∅ for some a ∈ G . Assuming the converse, we would co nclude that the family { xA : x ∈ G } is linked a nd then the inv ariant linked sys tem L 0 ∪ { xA : x ∈ G } is strictly larger than L 0 , which imp ossible b ecause of the max imalit y of L 0 . Next, we find b ∈ G with (3) A ∪ bA = G. Assuming that no such a p oin t b exist, we conclude that for a n y x, y ∈ G the union xA ∪ yA 6 = G . Then ( G \ xA ) ∩ ( G \ y A ) = G \ ( xA ∪ y A ) 6 = ∅ , which means that the fa mily { G \ xA : x ∈ G } is linked and inv aria n t. W e claim that G \ A ∈ L ⊥ 0 . Assuming the conv erse, we would conclude tha t G \ A misses some set L ∈ L 0 . Then L ⊂ A and hence A ∈ L 0 which is not the case. Thus G \ A ∈ L ⊥ 0 and hence { G \ xA : x ∈ G } b ecause L ⊥ 0 is inv aria n t. Since L 0 ∪ { G \ xA : x ∈ G } is a n inv ar ian t linked system containing L 0 , the maximality of L 0 guarantees that G \ A ∈ L 0 which contradicts A ∈ L ⊥ 0 . Finally we show that G \ A = aA = bA . Obser ve that (2) a nd (3) imply that aA ⊂ bA and hence A ⊂ a − 1 bA . On the other hand, (2) and (3) are equiv alent to a − 1 A ∩ A = ∅ and b − 1 A ∪ A = G , which implies a − 1 A ⊂ b − 1 A a nd this y ie lds ba − 1 A ⊂ A . Unifying this inclusion with A ⊂ a − 1 bA = ba − 1 A , we conclude that b a − 1 A = A and hence bA = aA . Now lo oking at (2) and (3 ) we see tha t G \ A = aA = bA .  5. Minima l left ideals of λ ( Z ) In this section w e apply the results of the preceding sections to des c r ibe the structure of minimal left idea ls of the semigroup λ ( Z ). It turns out that they are isomor phic to minimal left ideals of the super extension λ ( Z 2 ) of the compact top ological g roup Z 2 of in teger 2-adic num b ers. W e recall that Z 2 = lim ← − C 2 k is a totally disco nnec ted co mpact metr izable Abelia n group, which is the limit o f the inv erse se quence · · · → C 2 n → · · · → C 8 → C 4 → C 2 of cyclic 2 -groups C 2 n . Let π : Z → Z 2 denote the canonic (injective) ho momor- phism of Z in to Z 2 (induced by the quotient maps π 2 k : Z → Z / 2 k Z = C 2 k , k ∈ N ). By the contin uity of the functor λ in the ca tegory of compact Hausdorff spaces (see [TZ, 2.3.2 ]), the sup erextension λ ( Z 2 ) can b e identified with the limit o f the inv erse se quence · · · → λ ( C 2 n ) → · · · → λ ( C 8 ) → λ ( C 4 ) → λ ( C 2 ) of finite semig roups λ ( C 2 k ). This implies that λ ( Z 2 ) is a metrizable zero-dimensio nal compact top ologica l semigr o up. Theorem 5 .1. The homomorphism λπ : λ ( Z ) → λ ( Z 2 ) is inje ctive on e ach m ini- mal left ide al of λ ( Z ) . Co nse quently, the minimal left ide als of the semigr oup λ ( Z ) ar e c omp act metrizable t op olo gic al semigr oups. Pr o of. By Lemma 1 .1, it suffices to chec k that the homomo rphism λπ is injectiv e on some minimal left ideal of λ ( Z ). Fix any maximal in v ariant linked system L 0 on Z (such a system exists by Zo rn Lemma). By Pro position 4.1 the set ↑L 0 = {L ∈ λ ( Z ) : L ⊃ L 0 } is a left ideal which necessarily contains a minimal left ideal ALGEBRA IN THE SUPEREXTENS IONS OF GR OUPS, I II 7 I of λ ( Z ). W e claim that the homomorphis m λπ : λ ( Z ) → λ ( Z 2 ) is injective on I . Given t wo different maximal linked system A , B ∈ I we need to c heck that λπ ( A ) 6 = λπ ( B ). Since the sup erextension λ ( Z 2 ) is the limit of the inv e rse sequence · · · → λ ( C 2 n ) → · · · → λ ( C 8 ) → λ ( C 4 ) → λ ( C 2 ) , the ine q ualit y λπ ( A ) 6 = λπ ( B ) will follow as so on as we find k ∈ N suc h tha t λπ 2 k ( A ) 6 = λπ 2 k ( B ) where λπ 2 k : λ ( Z ) → λ ( C 2 k ) is the homomorphism induced by the quo tien t ho momorphism π 2 k : Z → C 2 k . Pick any set A ∈ A \ B . Since A ∈ L ⊥ 0 \ L 0 , w e ca n apply Theor em 4.2 to conclude that A = 2 n + A for some p ositiv e num b er n ∈ Z . The later equality means that A = π − 1 2 n ( π 2 n ( A )) is the complete preimag e of the set π 2 n ( A ) under the quotient homomorphism π 2 n : Z → Z / 2 n Z = C 2 n . It follows that π 2 n ( A ) ∈ λπ 2 n ( A ) \ λπ 2 n ( B ) and hence λπ 2 n ( A ) 6 = λπ 2 n ( B ). W rite the num b er 2 n as the pro duct 2 n = 2 k · m for some o dd n umber m and find a (unique) subgr oup H ⊂ C 2 n of or der | H | = m . It follows that the quotient group C 2 n /H ca n be identified with the cy clic 2-gro up C 2 k so tha t q ◦ π 2 n = π 2 k where q : C 2 n → C 2 n /H = C 2 k is the quotient homomor phis m. Cor ollary 3.3 gua ran tees that the ho momorphism λq : λ ( C 2 n ) → λ ( C 2 k ) is injective on each minimal left ideal of λ ( C 2 n ). In particular , it is injective on the minimal left ideal λπ 2 n ( I ). Consequently , λπ 2 k ( A ) = λq ( ˜ A ) 6 = λq ( ˜ B ) = λπ 2 k ( B ). This completes the pro of o f the injectivity of λπ : λ ( Z ) → λ ( Z 2 ) on the left ideal I and co nsequen tly , on e ac h minimal left ideal J of λ ( Z ). Since minimal left ideals of λ ( Z ) a re compact, the restriction λπ | J is a topologica l isomorphism of J onto the minimal left idea l λπ ( J ) of λ ( Z 2 ). Since λ ( Z 2 ) is a metrizable top ologica l semigr oup, so are the semigr oups λπ ( J ) a nd J .  6. Some Open Problems W e saw in Theorem 3.1 tha t the minimal ideal K ( λ ( G )) of the sup erextension of an o dd group G is a compact to polog ic al semigr oup. Problem 6.1. Char acterize gr oups G s uch t ha t the minimal ide al K ( λ ( G )) is close d in λ ( G ) . Is the minimal ide al K ( λ ( Z )) close d in λ ( Z ) ? Is K ( λ ( Z )) a t op olo gic al semigr oup? Problem 6. 2. Char acterize gr oups G such that the minimal left ide als of λ ( G ) ar e (metrizable) top olo gic al semigr oups. 8 T.BANAKH, V.GA VR YLKIV References [BB] B. Balcar, A. 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Dep ar tmen t of Mat hema tics, Iv an Franko Natio nal University of L viv and Instytut Ma temat yki, Uniwersytet Huma nistyczno-Prz yrodniczy w Kielcach E-mail addr ess : tbanakh @yahoo.co m V asyl S tef anyk Precarp at hian Na tional University, I v ano-Frankivsk (Ukraine) E-mail addr ess : vgavryl kiv@yahoo .com