Algebra in superextensions of twinic groups

ALGEBRA IN SUPEREXTENSIONS OF TWINIC GROUPS T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Abstract. Giv en a group X we study the algebraic structure of the compact r i gh t-topological semigroup λ ( X ) consisting of maximal linked systems on X . Thi s semigroup con tains the semigroup β ( X ) of ultraﬁlters as a closed subsemigroup. W e construct a faithful represen tation of the semigroup λ ( X ) in the semigroup P ( X ) P ( X ) of all self-maps of the p ow er-set P ( X ) and show that the image of λ ( X ) i n P ( X ) P ( X ) coincides with the semigroup End λ ( P ( X )) of all functions f : P ( X ) → P ( X ) that are e quiv ar ian t, monotone and symmetric in t he sense th at f ( X \ A ) = X \ f ( A ) for all A ⊂ X . Using this represen tation we describe the mi nimal ideal K ( λ ( X )) and m i nimal left i deals of the sup erextension λ ( X ) of a t winic group X . A gr oup X is called twinic if it admits a left-inv ari an t ideal I ⊂ P ( X ) such that xA = I y A for any subset A ⊂ X and p oin ts x, y ∈ X with xA ⊂ I X \ A ⊂ I y A . The class of twinic groups includes all amenable groups and all groups with p erio dic comm utators but do es not include the fr ee group F 2 with tw o gene rators. W e prov e that f or an y twinic group X , the re i s a cardinal m suc h that all mi nimal left ideals of λ ( X ) are algebraically isomorphic to 2 m × Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,C 2 k ) 2 k for some cardinals q ( X, C 2 k ) and q ( X, Q 2 k ), k ∈ N ∪ {∞} . Here C 2 k is the cyclic group of order 2 k , C 2 ∞ is the quasicyclic 2-group and Q 2 k , k ∈ N ∪ {∞} , are the groups of generalized quat ernions. If the group X is abelian, then q ( X, Q 2 k ) = 0 for all k and q ( X , C 2 k ) is the num ber of subgroups H ⊂ X with quotien t X/H homeomorphic to C 2 k . If X is an Ab elian group (admitting no epimorphism onto C 2 ∞ ) then eac h minimal left i deal of the superextension λ ( X ) is algebraically (and topologically) is omorphic to the pro duct Q 1 ≤ k ≤∞ ( C 2 k × 2 2 k − 1 − k ) q ( X ,C 2 k ) where the cube 2 2 k − 1 − k (equal to 2 ω if k = ∞ ) is endo w ed with the left-zero m ultiplication. F or an ab elian group X all minimal left ideals of λ ( X ) are metrizable i f and only i f X has ﬁnite ranks r 0 ( X ) and r 2 ( X ) and admits no homomorphism on to the group C 2 ∞ ⊕ C 2 ∞ . Applying this result to the group Z of int egers, we prov e that each minimal left ideal of λ ( Z ) i s topologically isomorphic to 2 ω × Q ∞ k =1 C 2 k . Consequent ly , all subgroups i n the minim al ideal K ( λ ( Z )) of λ ( Z ) ar e proﬁnite abeli an groups. On the other hand, the sup er extension λ ( Z ) cont ains an isomorphic top ological copy of each second count able proﬁnite top ological semigroup. This results contrasts with the famous Zelen yuk’s Theorem s a ying tha t the semigroup β ( Z ) cont ains no ﬁnite subgroups. At the end of the paper w e describe the structure of mi nimal left ideals of ﬁnite groups X of order | X | ≤ 15. 1991 Mathematics Subje ct Classiﬁc ation. Pr imary 20M30; 20M12; 22A15; 22A25; 54D35. Key wor ds and phr ases. Compact right-topological semigr oup, sup erextension of a group, semigroup of maximal link ed systems, f aithful represen tation, minim al ideal, mini mal left ideal, minimal idemp oten t, wreath pro duct, twinic group, t win set. 1 2 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Contents 1. Int ro duction 3 2. Right-topolog ic al semigroups 4 3. Acts and their endomorphism monoids 5 4. The function r epresentation of the semigroup P 2 ( X ) 6 5. Twin and I -twin subsets of gro ups 10 6. Twinic gr oups 11 7. 2-Cogr oups 14 8. The characteristic group H ( K ) of a 2 - cogroup K 16 9. Twin-genera ted top ologie s on groups 17 10. The characteristic group H ( A ) o f a t win subset A 18 11. Charac ter izing functions that b elong to End I λ ( F ) 18 12. The H ( K )-act T K of a max imal 2-cogr oup K 19 13. I -inco mparable and I -indep endent families 21 14. The endomorphism mo noid End( T K ) of the H ( K )-act T K 22 15. The semigro up End λ ( T K ) 26 16. Constructing nice idemp otents in the semigroup End λ ( P ( X )) 30 17. The minimal idea l of the se mig roups λ ( X ) and End λ ( P ( X )) 32 18. Minimal left ideals of sup erextensions o f twinic g roups 32 19. The structure o f the sup erextensions o f ab elian groups 37 20. Compact reﬂex ions of groups 41 21. Some examples 43 21.1. The inﬁnite cy c lic gro up Z 43 21.2. The (quasi)cyc lic 2-groups C 2 n 43 21.3. The gr o ups of generaliz e d quaternions Q 2 n 44 21.4. The dihedral 2 -groups D 2 n 45 21.5. Super extensions of ﬁnite groups of o rder < 16 47 22. Some Op en P roblems 48 References 48 ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 3 1. Introduction After disc ov ering a topo logical pro of of Hindman’s theorem [10] (see [12, p.10 2], [11]), top olog ical metho ds b ecome a standard to ol in the moder n combinatorics of num bers, see [12], [18]. The crucia l p oint is tha t any semigroup op era tion deﬁned on a discrete spa ce X can b e extended to a right-top o logical semigr oup op eration o n β ( X ), the Stone- ˇ Cech compactiﬁcation of X . The extension of the oper ation from X to β ( X ) can b e deﬁned b y the simple formula: (1) A ◦ B =  A ⊂ X : { x ∈ X : x − 1 A ∈ B } ∈ A  , The Stone- ˇ Cech compa c tiﬁca tion β ( X ) of X is a subs pa ce of the double power-set P 2 ( X ) = P ( P ( X )), which can b e ident iﬁed with the Cantor discontin uum { 0 , 1 } P ( X ) and endow ed with the compact Haus dorﬀ topolo gy of the Tychonoﬀ pro duct. It turns out that the for m ula (1) applied to ar bitrary families A , B ∈ P 2 ( X ) of subse ts of a group X still deﬁnes a binary opera tion ◦ : P 2 ( X ) × P 2 ( X ) → P 2 ( X ) that turns the double p ow er-set P 2 ( X ) in to a co mpact Hausdorﬀ right-topolo gical semigroup that contains β ( X ) a s a closed subse mig roup. The semigroup β ( X ) lies in a bit larger subsemigroup λ ( X ) ⊂ P 2 ( X ) co nsisting of a ll ma x imal linked sys tems o n X . W e recall that a family L of subse ts of X is • linke d if any sets A, B ∈ L have non-empt y in tersection A ∩ B 6 = ∅ ; • maximal linke d if L coincides with each link ed system L ′ on X that contains L . The space λ ( X ) is w ell-known in General and Categ orial T opo logy as the su p er extension of X , see [15], [22]. The thoroug h study of a lgebraic pro per ties of the sup ere xtensions of g roups was started in [2 ] and contin ued in [3] and [4]. In particular, in [4] we pr ov ed tha t the minimal left ideals of the sup erextensio n λ ( Z ) are metrizable to p olo gical semigroups. In this paper we shall extend this result to the sup erextensio ns λ ( X ) of all ﬁnitely-generated ab elian groups X . The results obtained in this pap er completely r eveal the top ologic a l and algebraic str ucture of the minimal ideal and minimal left ideals of the sup erextension λ ( X ) o f a twinic gr oup X . A g roup X is deﬁned to b e t winic if it admits a left-in v ar iant ideal I of subsets of X such that for a n y subset A ⊂ X w ith xA ⊂ I X \ A ⊂ I y A for some x, y ∈ X w e hav e xA = I y A . Her e the sy m b o l A ⊂ I B mea ns that A \ B ∈ I and A = I B means that A ⊂ I B a nd B ⊂ I A . In Section 6 we shall prov e that the class o f twinic groups con tains a ll amenable gr oups and all groups with perio dic commutators (in particular, all torsion groups), but do es not contain the free group with tw o generato rs F 2 . W e need t o recall the notation for some s tandard 2-gro ups. By Q 8 we deno te the g roup of quater nions. It is a m ultiplicative subg roup { 1 , i, j, k , − 1 , − i, − j, − k } of the algebra of quaternions H (which contains the ﬁeld of complex nu mbers C as a subalgebra ). F or every k ∈ ω let C 2 k = { z ∈ C : z 2 k = 1 } be the cyclic g roup of or de r 2 k . The multiplicativ e subgr oup Q 2 k ⊂ H generated by the union C 2 k − 1 ∪ Q 8 is called the gr oup of gener alize d quaternions . T he unio n C 2 ∞ = S ∞ k =1 C 2 k is called the quasicyclic 2-gr oup a nd the union Q 2 ∞ = S ∞ k =3 Q 2 k is c alled t he inﬁn ite gr oup of gener alize d qu aternions . By Theor em 8 .1, a gro up G is iso mo rphic to C 2 n or Q 2 n for some n ∈ N ∪ { ∞} if and o nly if G is a 2 - group with a uniq ue 2- element subgroup. The following theorem desc r ibing the structur e of minimal left ideals of the sup erextesio ns of twinic groups can b e derived from Theorem 18.11 and P rop osition 19.1: Theorem 1. 1. F or e ach twinic gr oup X ther e ar e c ar dinals q ( X , C 2 k ) , q ( X , Q 2 k ) , k ∈ N ∪ {∞} , such that (1) e ach minimal left ide al of λ ( X ) is algebr aic al ly isomorphic to Z × Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k for some semigr oup Z of left zer os; (2) e ach maximal su b gr oup of t he minimal ide al of λ ( X ) is algebr aic al ly isomorphi c to Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k . (3) If q ( X , C 2 ∞ ) = q ( X , Q 2 ∞ ) = 0 , then e ach maximal sub gr oup of t he minimal ide al of λ ( X ) is top olo gic al ly isomor- phic to the c omp act t op olo gic al gr oup Y 1 ≤ k< ∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k< ∞ Q q ( X ,Q 2 k ) 2 k . If the gr oup X is ab elian, then (4) q ( X , Q 2 k ) = 0 for every k ∈ N ∪ { ∞} while q ( X , C 2 k ) is e qual to the nu mb er of su b gr oups H ⊂ X such that the quotient gr oup X/H is isomorphic to C 2 k ; 4 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV (5) for every k ∈ N q ( X , C 2 k ) = | hom( X, C 2 k ) | − | hom( X , C 2 k − 1 ) | 2 k − 1 , wher e hom( X , C 2 k ) is the gr oup of homomorphi sms fr om X into C 2 k . 2. Right-topological semigr oups In this section we recall so me informa tion from [12] r elated to right-top o logical semigroups . By deﬁnitio n, a right- top ological s e migroup is a top ologica l s pace S endow ed w ith a semigroup o per ation ∗ : S × S → S such that for every a ∈ S the right shift r a : S → S , r a : x 7→ x ∗ a , is contin uous. If the semigro up o pe r ation ∗ : S × S → S is (separately) contin uo us, then ( S, ∗ ) is a ( semi -) top olo gic al semigr oup . A t ypical example of a rig h t-top ologica l semigroup is the semigroup X X of a ll self-maps o f a top olog ical space X e ndowed with the Tychonoﬀ pro duct top olo gy and the binary op era tion of comp osition of functions. F rom no w on, S is a compa c t Hausdorﬀ right-top ological semigro up. W e shall rec all some known informatio n co ncerning ideals in S , s ee [12]. A non- e mpty subset I of S is called a left (resp. right ) ide al if S I ⊂ I (re s p. I S ⊂ I ). If I is both a left and r ight idea l in S , then I is called an ide al in S . Obs e rve that for ev ery x ∈ S the set S xS = { sxt : s, t ∈ S } (resp. S x = { sx : s ∈ S } , xS = { xs : s ∈ S } ) is an ideal (resp. left ideal, rig h t idea l) in S . Such an ideal is called princip al . An ideal I ⊂ S is called minimal if any ideal of S that lies in I co incides with I . By analo gy we deﬁne minimal left and right ideals of S . It is easy to see that ea ch minimal left (resp. right) ideal I is principal. Mor eov er, I = S x (resp. I = xS ) for each x ∈ I . This simple observ ation implies that each minimal left ideal in S , b eing principal, is clos ed in S . By [12, 2.6], ea ch left ideal in S c ontains a minimal left idea l. The union K ( S ) of all minimal left ideals of S coincides with the minimal ide a l of S , [12, 2.8]. All minimal left ideals of S are mutually homeomorphic and all maxima l groups of the minimal ideal K ( S ) are alg e - braically is o morphic. Mor eov er, if tw o maximal groups lie in the same minimal right ideal, then they ar e topo logically isomorphic. W e shall need the following known fact, see Theorem 2 .11(c) [12]. Prop ositio n 2.1. F or any two minimal left ide als A, B of a c omp act right-top olo gic al semigr oup S and any p oint b ∈ B the right shift r b : A → B , r b : x 7→ xb , is a home omorphism. This prop osition implies the following corollary , see [4, Lemma 1.1]. Corollary 2. 2. If a homomorphism h : S → S ′ b etwe en two c omp act right-top olo gic al semigr oups is inje ctive on s ome minimal left ide al of S , then h is inje ctive on e ach minimal left ide al of S . An elemen t z of a semigroup S is called a right zer o (resp. a left zer o ) in S if xz = z (res p. z x = z ) for all x ∈ S . It is clear that z ∈ S is a right (left) zero in S if and only if the sing leton { z } is a left (righ t) ideal in S . An element e ∈ S is called an idemp otent if ee = e . By Ellis’s Theorem [12, 2.5], the set E ( S ) of idemp otents of any compact right-topo logical semigroup is not empty . F or every idempo ten t e the set H e = { x ∈ S : ∃ x − 1 ∈ S ( xx − 1 x = x, x − 1 xx − 1 = x − 1 , xx − 1 = e = x − 1 x ) } is the lar g est subgroup of S co ntaining e . By [12, 1.4 8], for an idemp otent e ∈ E ( S ) the following conditions are equiv alen t: • e ∈ K ( S ); • K ( S ) = S e S ; • S e is a minimal left ideal in S ; • eS is a minimal right idea l in S ; • eS e is a s ubgroup of S . An idempotent e satisfying the ab ov e equiv alent conditions will be called a minimal idemp otent in S . By [12, 1 .64], for any minimal idempotent e ∈ S the set E ( S e ) = E ( S ) ∩ S e of idemp otents of the minimal left idea l S e is a semigroup of left zeros, which means that xy = x for all x, y ∈ E ( S e ). By the Rees-Suschk ewitsch Structure Theorem (see [12, 1.64]) the map ϕ : E ( S e ) × H e → S e, ϕ : ( x, y ) 7→ xy , is an alge br aic isomo rphism of the corre s po nding semig roups. If the minimal left ideal S e is a to po logical semigroup, then ϕ is a topo lo gical isomorphism. Now we see tha t all the information on the alg ebraic (a nd sometimes top ologica l) str ucture of the minimal left ideal S e is enco ded in the prop erties o f the left zero semigroup E ( S e ) and the maximal gro up H e . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 5 3. Acts and their endo morphism monoids In this sectio n we survey the information on acts tha t will be widely used in this pap er for describing the a lgebraic structure of minimal left ideals o f the super extensions of groups. F ollo wing the terminology of [13] b y an act we understand a se t X endow ed with a left a ction · : H × X → X o f a group H ca lled the structur e gr oup of the a c t. The action should satisfy tw o ax io ms: 1 x = x and g ( hx ) = ( g h ) x for all x ∈ X and g , h ∈ H . Acts with the structure gr oup H will be called H -acts or H - sp ac es . An a c t X is called fr e e if the stabilizer Fix( x ) = { h ∈ H : hx = x } of each p oint x ∈ X is trivial. F or a p oint x ∈ X by [ x ] = { hx : h ∈ H } w e denote its orbit a nd b y [ X ] = { [ x ] : x ∈ X } the orbit space of the act X . More gener ally , for each subset A ⊂ X we put [ A ] = { [ a ] : a ∈ A } . A function f : X → Y betw een tw o H - acts is ca lled e quivariant if f ( hx ) = hf ( x ) for all x ∈ X a nd h ∈ H . A function f : X → Y is ca lled an isomorphism of the H -acts X a nd Y if it is bijective and equiv ariant. An equiv a riant self-map f : X → X is called an endomorphism of the H -act X . If f is bijective, then f is a n automorphism o f X . The set End( X ) of endo morphisms of an H -a ct X , endowed with the oper ation of comp osition of functions, is a mo no id called the endomorphism monoid of X . Each free H -ac t X is is o morphic to the pro duct H × [ X ] endow ed with the action h · ( x, y ) = ( hx, y ). F or such an a c t the s e migroup End( X ) is is omorphic to the wreath pro duct H ≀ [ X ] [ X ] of the gr oup H a nd the semigr oup [ X ] [ X ] of all self-maps of the orbit space [ X ]. The wreath pro duct H ≀ A A of a gro up H and the s emigroup A A of self-maps of a set A is de ﬁned a s the semidirect pro duct H A ⋊ A A of the A -th p ow er of H with A A , endow ed with the semigr oup op era tion ( h, f ) ∗ ( h ′ , f ′ ) = ( h ′′ , f ′′ ) where f ′′ = f ◦ f ′ and h ′′ ( α ) = h ( f ′ ( α )) · h ′ ( α ) for α ∈ A . F or any subsemigr o up S ⊂ A A the subset H ≀ S = { ( h, f ) ∈ H A ⋊ A A : f ∈ S } is called the wr e ath pr o duct of H and S . If b oth H and S ar e g roups, then their wr eath pro duct H ≀ S is a g roup. Observe that the maximal subgroup of A A containing the identit y self-map of A coincides with the gro up S A of all bijectiv e functions f : A → A . Theorem 3. 1. L et H b e a gr oup and X b e a fr e e H - act. Then (1) the semigr oup End( X ) is isomorphic to the wr e ath pr o duct H ≀ [ X ] [ X ] ; (2) the minimal ide al K (End( X )) of End( X ) c oincides with the set { f ∈ End( X ) : ∀ x ∈ f ( X ) f ( X ) ⊂ [ x ] } ; (3) e ach minimal left ide al of E nd( X ) is isomorph ic to H × [ X ] wher e [ X ] is endow e d with the left zer o multiplic ation; (4) for e ach idemp otent f ∈ E nd( X ) the maximal sub gr oup H f ⊂ E nd( X ) is isomorphic to H ≀ S [ f ( X )] ; (5) for e ach minimal idemp otent f ∈ K (End( X )) the maximal gr oup H f = f · End( X ) · f is isomorphic to H . Pr o of. 1. Let π : X → [ X ], π : x 7→ [ x ], denote the orbit map and s : [ X ] → X b e a sectio n o f π , which means that π ◦ s ([ x ]) = [ x ] fo r all [ x ] ∈ [ X ]. Observe that ea ch equiv ariant map f : X → X induces a well-deﬁned map [ f ] : [ X ] → [ X ], [ f ] : [ x ] 7→ [ f ( x )], o f the orbit spaces. Since the action of H o n X is fre e , for every orbit [ x ] ∈ [ X ] we can ﬁnd a unique point f H ([ x ]) ∈ H such that f ◦ s ([ x ]) = ( f H ([ x ])) − 1 · s ([ f ( x )]). W e claim that the map Ψ : End( X ) → H ≀ [ X ] [ X ] , Ψ : f 7→ ( f H , [ f ]) , is a se mig roup isomorphism. First we check that the ma p Ψ is a homomorphism. Pick any t w o equiv ar ia nt functions f , g ∈ End( X ) a nd consider their images Ψ( f ) = ( f H , [ f ]) and Ψ ( g ) = ( g H , [ g ]) in H ≀ [ X ] [ X ] . Consider als o the c o mpo sition f ◦ g and its image Ψ( f ◦ g ) = (( f ◦ g ) H , [ f ◦ g ]). W e claim that (( f ◦ g ) H , [ f ◦ g ]) = ( f H , [ f ]) ∗ ( g H , [ g ]) = (( f H ◦ [ g ]) · g H , [ f ] ◦ [ g ]) . The equality [ f ◦ g ] = [ f ] ◦ [ g ] is clear . T o prov e that ( f ◦ g ) H = ( f H ◦ [ g ]) · g H , take any orbit [ x ] ∈ [ X ]. It follows from the deﬁnition o f ( f ◦ g ) H ([ x ]) that (( f ◦ g ) H ([ x ])) − 1 · s ([ f ◦ g ( x )]) = ( f ◦ g ) ◦ s ([ x ]) = f ( g ◦ s ([ x ])) = f  ( g H ([ x ])) − 1 · s ([ g ( x )])  = ( g H ([ x ])) − 1 · f ◦ s ([ g ( x )]) = ( g H ([ x ])) − 1 · ( f H ([ g ( x )])) − 1 · s ([ f ◦ g ( x )]) = ( f H ◦ [ g ]([ x ]) · g H ([ x ])) − 1 · s ([ f ◦ g ( x )]) which implies the desired equa lit y ( f ◦ g ) H = ( f H ◦ [ g ]) · g H . Next, we show that the homomo rphism Ψ is injective. Given t wo equiv aria nt functions f , g ∈ End( X ) with ( f H , [ f ]) = Ψ( f ) = Ψ ( g ) = ( g H , [ g ]), we need to show that f = g . Observe that for every orbit [ x ] ∈ [ X ] we get f ( s ([ x ])) = ( f H ([ x ])) − 1 · s ◦ [ f ]([ x ])) = ( g H ([ x ])) − 1 · s ◦ [ g ]([ x ]) = g ( s ([ x ])) . 6 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Now for each x ∈ X we can ﬁnd a unique h ∈ H with x = h · s ([ x ]) and apply the equiv ariantness of the functions f , g to conclude that f ( x ) = f ( h · s ([ x ])) = h · f ( s ([ x ])) = h · g ( s ([ x ])) = g ( h · s ([ x ])) = g ( x ) . Finally , we show that Ψ is surjective. Giv en any pair ( h, g ) ∈ H ≀ [ X ] [ X ] = H [ X ] × [ X ] [ X ] , we deﬁne an equiv ariant function f ∈ End( X ) with ( h, g ) = ( f H , [ f ]) as follows. Given any x ∈ X ﬁnd a unique y ∈ H with x = y · s ([ x ]) and let f ( x ) = y · h ([ x ]) − 1 · s ( g ([ x ])) . This formula determines a well-deﬁned equiv ariant function f : X → X with Ψ ( f ) = ( h, g ). Therefore, Ψ : End( X ) → H ≀ [ X ] [ X ] is a se mig roup isomorphism. 2. O bs erve that the set I = { f ∈ End( X ) : { [ f ( x )] : x ∈ X } is a single to n } is a (non-empty) ideal in End( X ). T o show that I is the minimal idea l of the semigroup End( X ), we need to chec k tha t I lies in any idea l J ⊂ End( X ). T ake any functions f ∈ I and g ∈ J . Find an or bit [ x ] ∈ [ X ] such that [ f ( z )] = [ x ] for all z ∈ X . Since the restrictio n g | [ x ] : [ x ] → [ g ( x )] is bijective and equiv ar iant, so is its in verse ( g | [ x ]) − 1 : [ g ( x )] → [ x ]. Extend this equiv a riant map to any equiv ariant ma p h : X → X . Then f = h ◦ g ◦ f ∈ E nd( X ) ◦ g ◦ End( X ) ⊂ J . 3. T ake any idemp otent f ∈ K (E nd( X )) and consider the minimal left ideal End( X ) · f . Fix any point z ∈ f ( X ) a nd observe that f ([ x ]) = [ z ] fo r all x ∈ X according to the preceding item. It follows that the set Z = f − 1 ( z ) meets each orbit [ x ], x ∈ X , at a sing le p oint. So, we can de ﬁne a unique section s : [ X ] → Z ⊂ X of the or bit ma p X → [ X ] s uch that f ◦ s ([ X ]) = { z } . T o ea ch e quiv ar ia nt map g ∈ End( X ) as sign a unique element g H ∈ H such that g ( x ) = g − 1 H · s ([ g ( x )]). It is ea sy to chec k that the map Φ : End( X ) · f → H × [ X ] , Φ : g 7→ ( g H , [ g ]([ x ])) , is a se mig roup homomorphism where the orbit space [ X ] is e ndow ed with the left zero m ultiplication. 4. T ake any idemp otent f ∈ End( X ) and consider the sur jectiv e semig roup homomorphism pr : End( X ) → [ X ] [ X ] , pr : g 7→ [ g ]. It follows that [ f ] is an idempo ten t of the semigr oup [ X ] [ X ] and the image pr( H f ) of the maximal gro up H f is a subg roup o f [ X ] [ X ] . It is e a sy to s e e that the maximal subgro up H [ f ] of the idemp otent [ f ] in [ X ] [ X ] coincides with S [ f ( X )] · [ f ]. The preimage pr − 1 ( H [ f ] ) of the maxima l subg roup H [ f ] = S [ f ( X )] · f is iso morphic to the wreath pro duct H ≀ H [ f ] and hence is a group. Now the maximality of H f guarantees that H f = pr − 1 ( H [ f ] ) and hence H f is isomorphic to H ≀ S [ f ( X )] . 5. If f ∈ K (End( X )) is a minimal idemp otent, then the set [ f ( X )] = { [ f ( x )] : x ∈ X } is a singleto n by the seco nd item. By the preceding item the maximal group H f is isomor phic to H ≀ S [ f ( X )] , which is isomorphic to the group H since [ f ( X )] is a sing leton.  F o r each g roup X the p ow er-set P ( X ) will b e co ns idered as an X - a ct endo wed with the left action · : X × P ( X ) → P ( X ) , · : ( x, A ) 7→ xA = { xa : a ∈ A } , of the gr oup X . This X -a c t P ( X ) and its endomorphism monoid End( P ( X )) will play a crucial ro le in our consider ations. 4. The function represent a tion of the semigroup P 2 ( X ) In this section g iven a gro up X we construct a top olog ical isomorphism Φ : P 2 ( X ) → E nd( P ( X )) called the function r epr esent ation of the s e migroup P 2 ( X ) in the e ndomorphism monoid of the X -act P ( X ). W e recall that the do uble power-set P 2 ( X ) = P ( P ( X )) o f the gr oup X is endo wed with the binary op eration A ◦ B =  A ⊂ X : { x ∈ X : x − 1 A ∈ B } ∈ A  . The isomorphism Φ a ssigns to each family A of s ubsets of X the function Φ A : P ( X ) → P ( X ) , Φ A : A 7→ { x ∈ X : x − 1 A ∈ A} , called the funct ion r epr esentation of A . In the following theorem by e we de no te the neutral elemen t of the g roup X . Theorem 4.1. F or any gr oup X t he map Φ : P 2 ( X ) → E nd( P ( X )) is a top olo gic al isomorphi sm with inverse Φ − 1 : ϕ 7→ { A ⊂ X : e ∈ ϕ ( A ) } . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 7 Pr o of. First o bserve that for any family A ∈ P 2 ( X ) the function Φ A is equiv a riant, because Φ A ( xA ) = { y ∈ X : y − 1 xA ∈ A} = { xz ∈ X : z − 1 A ∈ A} = x Φ A ( A ) for any x ∈ X and A ⊂ X . Thus the map Φ : P 2 ( X ) → E nd( P ( X )) is w ell-deﬁned. T o prove that Φ is a semigr oup homomorphism, take tw o families X , Y ∈ P 2 ( X ) and let Z = X ◦ Y . W e need to chec k that Φ Z ( A ) = Φ X ◦ Φ Y ( A ) for every A ⊂ X . Obs e r ve that Φ Z ( A ) = { z ∈ X : z − 1 A ∈ Z } = { z ∈ X : { x ∈ X : x − 1 z − 1 A ∈ Y } ∈ X } = = { z ∈ X : Φ Y ( z − 1 A ) ∈ X } = { z ∈ X : z − 1 Φ Y ( A ) ∈ X } = = Φ X (Φ Y ( A )) = Φ X ◦ Φ Y ( A ) . T o see that the map Φ is injective, take a ny tw o distinct families A , B ∈ P 2 ( X ). Without loss of gener ality , A \ B contains some set A ⊂ X . It follows that e ∈ Φ A ( A ) but e / ∈ Φ B ( A ) and hence Φ A 6 = Φ B . T o see that the map Φ is surjective, take any equiv ar iant function ϕ : P ( X ) → P ( X ) and consider the family A = { A ⊂ X : e ∈ ϕ ( A ) } . It follows that for every A ∈ P ( X ) Φ A ( A ) = { x ∈ X : x − 1 A ∈ A} = { x ∈ X : e ∈ ϕ ( x − 1 A ) } = = { x ∈ X : e ∈ x − 1 ϕ ( A ) } = { x ∈ X : x ∈ ϕ ( A ) } = ϕ ( A ) . T o pro ve that Φ : P 2 ( X ) → E nd( P ( X )) ⊂ P ( X ) P ( X ) is contin uous w e ﬁrst deﬁne a con venien t subba se of the topolo gy on the space s P ( X ) and P ( X ) P ( X ) . The pro duct top olo gy of P ( X ) is g enerated by the subbase consisting o f the sets x + = { A ⊂ X : x ∈ A } and x − = { A ⊂ X : x / ∈ A } where x ∈ X . O n the o ther hand, the pro duct topo logy on P ( X ) P ( X ) is gener ated b y the s ubba se consisting of the sets h x, A i + = { f ∈ P ( X ) P ( X ) : x ∈ f ( A ) } and h x, A i − = { f ∈ P ( X ) P ( X ) : x / ∈ f ( A ) } where A ∈ P ( X ) and x ∈ X . Now o bserve tha t the preimage Φ − 1 ( h x, A i + ) = {A ∈ P 2 ( X ) : x ∈ Φ A ( A ) } = {A ∈ P 2 ( X ) : x − 1 A ∈ A} is op en in P 2 ( X ). The same is true for the pr eimage Φ − 1 ( h x, A i − ) = {A ∈ P 2 ( X ) : x / ∈ Φ A ( A ) } = {A ∈ P 2 ( X ) : x − 1 A / ∈ A} which a lso is op en in P 2 ( X ). Since the spaces P 2 ( X ) ∼ = { 0 , 1 } P ( X ) and End( P ( X )) ⊂ P ( X ) P ( X ) are compact and Hausdor ﬀ, the contin uit y of the map Φ implies the contin uit y of its in verse Φ − 1 . Consequently , Φ : P 2 ( X ) → End( P ( X )) is a topo logical isomorphis m of compact right-topo logical semigroups.  Remark 4.2. The functions representations Φ A of some families A ⊂ P ( X ) have tr ansparent top ologic a l interpretations. F o r example, if A is the ﬁlter of neighborho o ds of the identit y element e of a left-top o logical gro up X and A ⊥ = { B ⊂ X : ∀ A ∈ A ( B ∩ A 6 = ∅ ) } , then for any subset B ⊂ X the set Φ A ( B ) coincides with the interior of the set B while Φ A ⊥ ( B ) with the closur e of B in X ! Theorem 4.1 has a strategical impo rtance b ecause it allows us to translate (usually diﬃcult) problems concerning the structure of the se mig roup P 2 ( X ) to (usua lly mor e tractable) problems abo ut the endo morphism monoid End( P ( X )). In particular, Theorem 4.1 implies “ for fre e ” that the binar y op eration on P 2 ( X ) is ass o ciative and r ight-topolog ic al and hence P 2 ( X ) indeed is a compact right-topolo gical semigroup. Now let us investigate the interpla y b et ween the prop erties o f a family A ∈ P 2 ( X ) and tho se of its function represen- tation Φ A . Let us deﬁne a family A ⊂ P ( X ) to b e • monotone if for any s ubs ets A ⊂ B ⊂ X the inclus ion A ∈ A implies B ∈ A ; • left-invariant if for any A ∈ A a nd x ∈ X we get xA ∈ A . Resp ectively , a function ϕ : P ( X ) → P ( X ) is called • monotone if ϕ ( A ) ⊂ ϕ ( B ) for a n y subsets A ⊂ B ⊂ X ; • symmetric if ϕ ( X \ A ) = X \ ϕ ( A ) for e very A ⊂ X . Prop ositio n 4.3. F or an e quivariant funct ion ϕ ∈ End( P ( X )) the family Φ − 1 ( ϕ ) = { A ⊂ X : e ∈ ϕ ( A ) } is (1) monotone if and only if ϕ is monotone; (2) left-invariant if and only if ϕ ( P ( X )) ⊂ { ∅ , X } ; 8 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV (3) maximal linke d if and only if ϕ is m onotone and symm et ric. Pr o of. Let A = Φ − 1 ( ϕ ). 1. If ϕ is monotone, then for any sets A ⊂ B with A ∈ A we get e ∈ ϕ ( A ) ⊂ ϕ ( B ) a nd hence B ∈ A , which means that the family A is monotone . Now assume conversely tha t the family A is monotone and ta ke an y sets A ⊂ B ⊂ X . Note that for any x ∈ X with xA ∈ A we g et xB ∈ A . Then ϕ ( A ) = { x ∈ X : x − 1 A ∈ A} ⊂ { x ∈ X : x − 1 B ∈ A} = ϕ ( B ) , witnessing that the function ϕ is monotone. 2. If the family A is left-inv aria nt, then for each A ∈ A w e get ϕ ( A ) = { x ∈ X : x − 1 A ∈ A} = X and for each A / ∈ A we g et ϕ ( A ) = { x ∈ X : x − 1 A ∈ A} = ∅ . Now a ssume conv ersely that ϕ ( P ( X )) ⊂ {∅ , X } . Then for e ach A ∈ A w e get e ∈ ϕ ( A ) = X and then for each x ∈ X , the eq uiv aria nce of ϕ guarantees that ϕ ( xA ) = xϕ ( A ) = xX = X ∋ e and th us xA ∈ A , witnessing that the family A is inv ariant. 3. Assume that the family A is maximal linked. By the maximality , A is monoto ne . Conseque ntly , its function representation ϕ is monoto ne . The maximal linked pro pe r t y o f A guara n tees that for any s ubset A ⊂ X we get ( A ∈ A ) ⇔ ( X \ A / ∈ A ). Then ϕ ( X \ A ) = { x ∈ X : x − 1 ( X \ A ) ∈ A} = { x ∈ X : X \ x − 1 A ∈ A} = = { x ∈ X : x − 1 A / ∈ A} = X \ { x ∈ X : x − 1 A ∈ A} = X \ ϕ ( A ) , which mea ns that the function ϕ is symmetr ic. Now assuming that the function ϕ is mo notone and sy mmetric, w e sha ll show that the family A = Φ − 1 ( ϕ ) is maximal linked. The statement (1) g uarantees that A is monoto ne. Assuming that A is not linked, w e could ﬁnd tw o dis joint sets A, B ∈ A . Since A is monotone, w e can assume tha t B = X \ A . Then e ∈ ϕ ( A ) ∩ ϕ ( X \ A ), which is imposs ible as ϕ ( X \ A ) = X \ ϕ ( A ). Thus A is link ed. T o show that A is maxima l linked, it s uﬃces to c heck that for each subset A ⊂ X either A or X \ A belo ng s to A . Since ϕ ( X \ A ) = X \ ϕ ( A ), either ϕ ( A ) or ϕ ( X \ A ) c ontains the neutral element e of the group X . In the ﬁrst case A ∈ A and in the se c ond case X \ A ∈ A .  Let us recall that the aim of this pape r is the description of the structure of minimal left ideals of the sup erextension λ ( X ) of a gr oup X . Instead of the semigroup λ ( X ) it will be more conv enient to consider its iso morphic copy End λ ( P ( X )) = Φ( λ ( X )) ⊂ End( P ( X )) called the funct ion r epr esentation of λ ( X ). Prop ositio n 4.3 implies Corollary 4.4. The function r epr esentation End λ ( P ( X )) of λ ( X ) c onsists of e quivaria nt m onotone symmetric functions ϕ : P ( X ) → P ( X ) . In o rder to describ e the structure of minimal left ideals of the semigr oup End λ ( P ( X )) we shall lo ok for a relatively small subfamily F ⊂ P ( X ) such that the restriction op erator R F : End λ ( P ( X )) → P ( X ) F , R F : ϕ 7→ ϕ | F , is injective on each minima l left ideal of the semigroup E nd λ ( P ( X )). Then the comp osition Φ F = R F ◦ Φ : λ ( X ) → P ( X ) F will b e injective on each minimal left idea l of the semigroup λ ( X ). By P r op osition 2.2, a homomorphis m b etw een semigroups is injective on each minimal left ideal if it is injective o n some minimal left ideal. Such a sp ecial minimal left ideal of the semigroup λ ( X ) will be fo und in the left ideal of the form λ I ( X ) for a suitable left-in v ar ia nt ideal I o f subsets of the group X . A family I of subsets o f X is called an ide al o n X if • X / ∈ I ; • A ∪ B ∈ I fo r any A, B ∈ I ; • for any A ∈ I and B ⊂ A w e get B ∈ I . Such an ideal I is called left-invariant (r eps. right-invariant ) if xA ∈ I (resp. Ax ∈ I ) for a ll A ∈ I and x ∈ X . An ideal I will be called invariant if it is bo th left-inv ariant and right-in v ariant. The smallest idea l on X is the trivia l ideal {∅} containing only the empty set. The smallest non-trivial le ft-inv aria nt ideal o n an inﬁnite group X is the ideal [ X ] <ω of ﬁnite s ubs ets of X . This idea l is inv ariant. F rom now o n we shall assume that I is a left-inv aria nt ideal o n a gr oup X . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 9 F o r subsets A, B ⊂ X we write • A ⊂ I B if A \ B ∈ I , a nd • A = I B if A ⊂ I B and B ⊂ I A . The deﬁnition of the ideal I implies that = I is an e q uiv alence relation on P ( X ). F or a subset A ⊂ X its equiv alence cla ss ¯ ¯ A I = { B ⊂ X : B = I A } is calle d the I -satu r ation of A . A family A of subsets of X is deﬁned to b e I -satura te d if ¯ ¯ A I ⊂ A for any A ∈ A . Let us o bserve that a monotone family A ⊂ P ( X ) is I -satura ted if and only if for any A ∈ A a nd B ∈ I we g et A \ B ∈ A . Resp ectively , a function ϕ : P ( X ) → P ( X ) is called I -satur ate d if ϕ ( A ) = ϕ ( B ) for any s ubsets A = I B o f X . Prop ositio n 4.5 . A family A ⊂ P ( X ) is I -s atu r ate d if and only if its function r epr esent ation Φ A : P ( X ) → P ( X ) is I -satura te d. Pr o of. Assume that A is I -satura ted and take t wo subsets A = I B of X . W e need to show that Φ A ( A ) = Φ A ( B ). The left-in v ar iance of the idea l I implies that for every x ∈ X we get xA = I xB a nd hence ( xA ∈ A ) ⇔ ( xB ∈ A ). Then Φ A ( A ) = { x ∈ X : x − 1 A ∈ A} = { x ∈ X : x − 1 B ∈ A} = Φ A ( B ) . Now assume conv ersely that the function representation Φ A is I -sa turated and tak e an y subsets A = I B with A ∈ A . Then e ∈ Φ A ( A ) = Φ A ( B ), which implies that B ∈ A .  F o r an left-in v ar iant ideal I on a g roup X let λ I ( X ) ⊂ λ ( X ) be the subspace of I -satura ted maximal linked systems on X a nd End I λ ( P ( X )) ⊂ End λ ( P ( X )) b e the subspa ce c o nsisting of I -s aturated monotone symmetr ic endomorphisms of the X -ac t P ( X ). It is clear that for any functions f , g : P ( X ) → P ( X ) the co mpo sition f ◦ g is I -satura ted provided so is the function g . This trivial r emark (and Lemma 4.7 below) imply: Prop ositio n 4.6. F or any ide al I t he fun ction r epr esentation Φ : λ I ( X ) → End I λ ( P ( X )) is a top olo gic al isomorphism b etwe en the close d left ide als λ I ( X ) and End I λ ( P ( X )) of the s emigr oups λ ( X ) and End λ ( P ( X )) , r esp e ctively. The following lemma (com bined with Z o rn’s Lemma) implies that the sets λ I ( X ) and End I λ ( P ( X )) are not empty . Lemma 4. 7. Each maximal I -satur ate d linke d syst em L on X is maximal linke d. Pr o of. W e need to show that each set A ⊂ X that meets a ll sets L ∈ L b elongs to L . W e claim that A / ∈ I . O therwise, taking any subset L ∈ L , w e get L \ A = I L and hence L \ A b elongs to L , which is not pos sible as L \ A misses the set A . Since A / ∈ I , the I -satura ted family ¯ ¯ A I is linked. W e claim that the I -satur ated family ¯ ¯ A I ∪ L is link ed. Assuming the c o nv erse, we would ﬁnd tw o disjoint sets A ′ ∈ ¯ ¯ A I and L ∈ L . Then L ∩ A = I L ∩ A ′ = ∅ and hence the set L \ A = I L belongs to L , whic h is not p oss ible a s this s et misses A . Now we see that the family ¯ ¯ A I ∪ L , b eing I -satura ted and linked, coinc ide s with the maximal I -satur ated linked system L . Then A ∈ ¯ ¯ A I ∪ L = L .  Given a subfamily F ⊂ P ( X ) consider the restriction op era to r R F : P ( X ) P ( X ) → P ( X ) F , R F : f 7→ f | F , and let End λ ( F ) = R F (End λ ( P ( X ))) and E nd I λ ( F ) = R F (End I λ ( P ( X )) for a left-inv ariant ideal I on X . The space E nd λ ( F ) is compact a nd Hausdorﬀ as a c ontin uo us image of a co mpa ct Hausdorﬀ space. A subfamily F ⊂ P ( X ) is called λ -invaria nt if Φ L ( F ) ⊂ F for each max imal link ed system L ∈ λ ( X ). By Corollar y 4.4, F is λ -inv ariant if and only if f ( F ) ⊂ F for each eq uiv aria n t monotone symmetric function f : P ( X ) → P ( X ). If a family F ⊂ P ( X ) is λ -inv ariant, then the space End λ ( F ) ⊂ F F is a co mpact r ig ht -top ologica l s e mig roup with resp ect to the op era tion o f comp ositio n of functions and the restrictio n op er a tor R F : End λ ( P ( X )) → End λ ( F ) is a surjective contin uous semigroup homomorphism. In this ca se the comp ositio n Φ F = R F ◦ Φ : λ ( X ) → End λ ( F ) also is a surjective contin uous s emigroup homomorphis m a nd End I λ ( F ) = Φ F ( λ I ( X )) is a left ideal in the semigroup End λ ( F ). In the following pro po sition w e characterize functions that b elong to the space End I λ ( F ) for an I -satura ted left-inv ariant symmetric subfamily F ⊂ P ( X ). A family F ⊂ P ( X ) is called symmetric if for each set A ∈ F the c o mplemen t X \ A ∈ F . Theorem 4 .8. F or a left-invariant ide al I on a gr oup X and a I -satur ate d symmetric left-invariant family F ⊂ P ( X ) , a function ϕ : F → P ( X ) b elongs to End I λ ( F ) if and only if ϕ is e quivariant, symmetric, monotone, and I -satur ate d. 10 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Pr o of. The “only if ” part follows immediately from Coro llary 4.4. T o prove the “if ” par t, ﬁx a ny eq uiv aria n t monoto ne symmetric I - s aturated function ϕ : F → P ( X ) a nd co nsider the the families L ϕ = { x − 1 A : A ∈ F , x ∈ ϕ ( A ) } and ¯ ¯ L I ϕ = [ A ∈L ¯ ¯ A I . W e claim that the family ¯ ¯ L I ϕ is linked. Assuming the co n verse, we could ﬁnd tw o sets A, B ∈ F and tw o p oints x ∈ ϕ ( A ) and y ∈ ϕ ( B ) suc h that x − 1 A ∩ y − 1 B ∈ I . Then y x − 1 A ⊂ I X \ B and hence y x − 1 A ⊂ ( X \ B ) ∪ C for some set C ∈ I . Since F is symmetric and I -sa turated, the set ( X \ B ) ∪ C = I X \ B b elongs to the fa mily F . Applying to the chain of the inclusions y x − 1 A ⊂ ( X \ B ) ∪ C = I X \ B the equiv ariant monotone s y mmetric I - saturated function ϕ , we get the c hain y x − 1 ϕ ( A ) ⊂ ϕ (( X \ B ) ∪ C ) = ϕ ( X \ B ) = X \ ϕ ( B ) . Then x − 1 ϕ ( A ) ⊂ X \ y − 1 ϕ ( B ), which is not p oss ible b ecause the neutra l element e of the group X b elongs to x − 1 ϕ ( A ) ∩ y − 1 ϕ ( B ). Enlarge the I -satura ted linked family ¯ ¯ L I ϕ to a max ima l I -sa turated linked family L , which is maximal link ed by Lemma 4.7 and thus L ∈ λ I ( X ). W e claim that Φ L | F = ϕ . Indeed, take an y set A ∈ F and observe tha t ϕ ( A ) ⊂ { x ∈ X : x − 1 A ∈ L ϕ } ⊂ { x ∈ X : x − 1 A ∈ L} = Φ L ( A ) . T o pro ve the reverse inclusion, observe that for any x ∈ X \ ϕ ( A ) = ϕ ( X \ A ) w e get x − 1 ( X \ A ) = X \ x − 1 A ∈ L ϕ ⊂ L . Since L is link ed, x − 1 A / ∈ L and hence x / ∈ Φ L ( A ).  Corollary 4 . 9. F or any symm et ric left-invariant family F ⊂ P ( X ) , a function ϕ : F → P ( X ) b elongs to End λ ( F ) if and only if ϕ is e quivariant, symmetric, and monotone. 5. Twin and I -twin subsets of groups In this section we star t studying very interesting ob jects called twin sets. F or a n ab elian (more generally , twinic) group X twin subsets of X form a subfamily T ⊂ P ( X ) for whic h the function repre s ent ation Φ T : λ ( X ) → End λ ( T ) is injectiv e on ea ch minimal left ideal of the s uper extension λ ( X ). The machinery related to twin sets will b e develop ed in Sections 5 – 1 5, after whic h we shall return back to studying minimal (left) ideals of the s emigroups λ ( X ) and E nd( X ). F o r a subset A o f a gr oup X consider the following three subsets of X : Fix( A ) = { x ∈ X : xA = A } , Fix − ( A ) = { x ∈ X : xA = X \ A } , Fix ± ( A ) = Fix( A ) ∪ Fix − ( A ) . Deﬁnition 5.1. A subset A ⊂ X is deﬁned to be • twin if xA = X \ A for some x ∈ X , • pr et win if xA ⊂ X \ A ⊂ y A for some p oints x, y ∈ X . The families of t win and pretwin subsets of X will b e denoted by T and pT , r e s pec tively . Observe that a set A ⊂ X is t win if a nd only if Fix − ( A ) is not empt y . The notion of a t win set has an obvious “ideal” version. F or a left-inv ariant ideal I of subsets of a group X , and a subset A ⊂ X co nsider the following subsets of X : I -Fix( A ) = { x ∈ X : xA = I A } , I -Fix − ( A ) = { x ∈ X : xA = I X \ A } , I -Fix ± ( A ) = I -Fix( A ) ∪ I -Fix − ( A ) . Deﬁnition 5.2. A subset A ⊂ X is deﬁned to be • I -t win if xA = I X \ A for some x ∈ X , • I -pr etwin if xA ⊂ I X \ A ⊂ I y A for some po in ts x, y ∈ X . The families of I -twin and I -pretwin subsets of X will be denoted by T I and pT I , resp ectively . It is clear that T {∅} = T and pT {∅} = pT . Prop ositio n 5.3. F or e ach su bset A ⊂ X t he set I - Fix ± ( A ) is a sub gr oup in X . The set A is I - twin if and only if I - Fix( A ) is a normal sub gr oup of index 2 in I - Fix ± ( A ) . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 11 Pr o of. If the s e t A is no t I -twin, then I -Fix − ( A ) = ∅ and then I -Fix ± ( A ) = I -Fix( A ) = { x ∈ X : xA = I A } is a subgroup of X by the transitivity and the left-inv ariance of the equiv a le nce relation = I . So, we assume that A is I -twin, which means that I -Fix − ( A ) 6 = ∅ . T o show that I - Fix ± ( A ) is a subgroup in X , take any t w o po in ts x, y ∈ I -Fix ± ( A ). W e claim that xy − 1 ∈ I -Fix ± ( A ). This is clear if x, y ∈ I -Fix( A ) ⊂ I -Fix ± ( A ). If x ∈ I -Fix( A ) and y ∈ I -Fix − ( A ), then xA = I A , y A = I X \ A and th us A = I X \ y − 1 A which implies y − 1 A = I X \ A . Then xy − 1 A = I x ( X \ A ) = X \ xA = I X \ A , which means that xy − 1 ∈ I -Fix − ( A ) ⊂ Fix ± ( A ). If x, y ∈ I -Fix − ( A ), then xA = I X \ A , y − 1 A = I X \ A . This implies that xy − 1 A = I x ( X \ A ) = I X \ xA = I X \ ( X \ A ) = I A and c o nsequently xy − 1 ∈ I -Fix( A ). T o show that I -Fix( A ) is a subgro up o f index 2 in Fix ± ( A ), ﬁx a ny element g ∈ I - Fix − ( A ). Then for every x ∈ I -Fix( A ) we get g xA = I g A = I X \ A and thus g x ∈ I -Fix − ( A ). This y ields I -Fix − ( A ) = g ( I -Fix( A )), which means that the subgroup I -Fix( A ) has index 2 in the gr oup I -Fix ± ( A ).  The following propo sition shows that the family T I of I -twin se ts of a gro up X is left-inv aria n t. Prop ositio n 5.4. F or any I -twin set A ⊂ X and any x ∈ X the set xA is I -twin and I - Fix − ( xA ) = x ( I - Fix − ( A )) x − 1 . Pr o of. T o see that xA is an I -twin s et, take an y z ∈ I -Fix − ( A ) and obs erve that X \ xA = x ( X \ A ) = I xz A = xz x − 1 xA, which mea ns that xz x − 1 ∈ I -Fix − ( xA ) for every z ∈ I -Fix − ( A ). Hence I -Fix − ( xA ) = x ( I -Fix − ( A )) x − 1 .  The pre c eding prop ositio n implies that the family T I of I -twin subsets of X can b e consider ed as an X -act with resp ect to the left action · : X × T I → T I , · : ( x, A ) 7→ xA of the group X . By [ A ] = { xA : x ∈ X } we denote the orbit of a I -twin set A ∈ T I and b y [ T I ] = { [ A ] : A ∈ T I } the orbit space. If I = {∅} is a trivial ideal, then we wr ite [ T ] instea d of [ T I ]. 6. Twinic groups A left-in v aria nt ideal I on a gr oup X is called twinic if for an y subset A ⊂ X a nd p oints x, y ∈ X with xA ⊂ I X \ A ⊂ I y A we get A = I B . In this case the families pT I and T I coincide. A group X is deﬁned to be t winic if it admits a t winic ideal I . It is clea r that in a twinic g roup X the intersection I I of all twinic idea ls is the smallest twinic ideal in X calle d the twinic ide al o f X . The structure of the twinic idea l I I can b e descr ib ed as follows. Let I I 0 = {∅} and for each n ∈ ω let I I n +1 be the idea l gene r ated by se ts of the form y A \ xA where xA ⊂ I I n X \ A ⊂ I I n y A for some A ⊂ X and x, y ∈ X . By induction it is easy to c heck that I I n ⊂ I I n +1 ⊂ I is an inv ar iant idea l a nd hence I I = S n ∈ ω I I n ⊂ I is a well-deﬁned (s ma llest) twinic ideal on X . This ideal I I is in v ar iant. In fact, the a bove constructiv e deﬁnition of the additive inv ariant family I I is v alid for each g roup X . How ev er, I I is an ideal if and only if the g roup X is twinic. W e shall sa y that a g roup X has trivia l twinic ide al if the trivia l idea l I = { ∅} is t winic. This happen if and only if for any subset A ⊂ X with xA ⊂ X \ A ⊂ y A we get xA = X \ A = y A . In this ca se the twinic ideal I I of X is tr ivial. The cla ss o f t winic gro ups is suﬃciently w ide . In particula r, it c ontains all amenable groups . Let us recall that a group X is called amenable if it admits a Bana ch measur e µ : P ( X ) → [0 , 1], which is a left-inv ariant proba bilit y measur e deﬁned on the family of all s ubsets of P ( X ). In this case the family N µ = { A ⊂ X : µ ( A ) = 0 } is an le ft-inv aria nt ideal in X . It is w ell-known that the class o f amenable groups co n tains all ab elia n groups and is closed with resp ect to man y op erations ov er groups, see [17 ]. A subset A of an amenable group X is called absolutely nul l if µ ( A ) = 0 for each Banach measure µ on X . The family N of all absolutely n ull subsets is an ideal on X . This ideal coincides with the intersection N = T µ N µ where µ runs ov er all Banach measur es of X . Theorem 6 .1. Each amenable gr oup X is twinic. The twinic ide al I I of X lies in the ide al N of absolute nul l subset s of X . Pr o of. It suﬃces to chec k that the ideal N is twinic. T ake any set A ⊂ X such that xA ⊂ N X \ A ⊂ N y A for some x, y ∈ X . W e need to show that µ ( y A \ xA ) = 0 for ea ch Banach measure µ on X . It follows from xA ⊂ N X \ A ⊂ N y A and the inv ariance o f the Banach measure µ that µ ( A ) = µ ( xA ) ≤ µ ( X \ A ) ≤ µ ( y A ) = µ ( A ) 12 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV and hence µ ( y A \ xA ) = µ ( A ) − µ ( A ) = 0.  Next, we show that the class of t winic groups contains also some non-amenable groups. The s implest example is the Burnside group B ( n, m ) for n ≥ 2 a nd o dd m ≥ 6 65. W e reca ll that the B urnside gro up B ( n, m ) is g e nerated by n elements a nd one rela tion x m = 1. Adia n [1] proved that fo r n ≥ 2 and any odd m ≥ 6 65 the Burnside g roup B ( n, m ) is not amenable, see also [16] for a strong e r version of this res ult. The following theo rem implies that e ach Burnside group, being a torsio n group, is twinic. Moreover, its t winic ideal I I is triv ial! Theorem 6.2. A gr oup X has trivial twinic ide al I I = {∅} if and only if t he pr o duct ab of any elements a, b ∈ X b elongs to the subsemigr oup of X gener ate d by the set b ± · a ± wher e a ± = { a, a − 1 } . Pr o of. T o prov e the “if ” pa rt, assume that I I 6 = {∅} . Then I I n +1 6 = {∅} = I I n for some n ∈ ω and we can ﬁnd a subset A ⊂ X a nd po in ts a, b ∈ X such that a − 1 A ⊂ X \ A ⊂ bA but a − 1 A 6 = bA . Consider the subsemig roup Fix ⊂ ( A ) = { x ∈ X : xA ⊂ A } ⊂ X and o bserve that b − 1 a − 1 ∈ Fix ⊂ ( A ). The inclusion a − 1 A ⊂ X \ A implies a − 1 A ∩ A = ∅ which is equiv a len t to A ∩ aA = ∅ and yie lds aA ⊂ X \ A ⊂ bA . Then b − 1 a ∈ Fix ⊂ ( A ). Now co nsider the chain of the equiv alences X \ A ⊂ b A ⇔ A ∪ bA = X ⇔ b − 1 A ∪ A = X ⇔ X \ A ⊂ b − 1 A and combine the la s t inclusion with aA ∪ a − 1 A ⊂ X \ A to o btain ba, ba − 1 ∈ Fix ⊂ ( A ). Now w e see that the subsemigr o up S of X ge nerated by the set { 1 , b a, ba − 1 , b − 1 a, b − 1 a − 1 } lies in Fix ⊂ ( A ). Obser ve that b − 1 a − 1 A $ A implies abA 6⊂ A , ab / ∈ Fix ⊂ ( A ) ⊃ S , and ﬁnally ab / ∈ S . This co mpletes the pro of of the “if ” part. T o prov e the “only if ” pa rt, assume that the gro up X contains elements a, b whos e pro duct ab do es not b elong to the subsemigro up generated by b ± a ± where a ± = { a, a − 1 } a nd b ± = { b, b − 1 } . Then ab do es not b elongs als o to the subsemigroup S gener ated b y { 1 } ∪ b ± a ± . Observe that a ± S = S − 1 a ± and b ± S − 1 = S b ± . W e claim that (2) S ∩ a ± S = ∅ a nd S ∩ S b ± = ∅ . Assuming that S ∩ a ± S 6 = ∅ we would ﬁnd a point s ∈ S such that as ∈ S o r a − 1 s ∈ S . If as ∈ S , then bs − 1 = b ( as ) − 1 a ∈ bS − 1 a ⊂ S b ± a ⊂ S and hence b = bs − 1 s ∈ S · S ⊂ S . Then a ± = b ( b − 1 a ± ) ⊂ S · S ⊂ S , b ± = ( b ± a ) a − 1 ∈ S · S ⊂ S and ﬁnally ab ∈ S · S ⊂ S , which contradicts ab / ∈ S . By analog y we can treat the c ase a − 1 s ∈ S and also prov e that S ∩ S b ± = ∅ . Consider the family P of all pairs ( A, B ) of disjoint subsets of X such that (a) a ± A ⊂ B and b ± B ⊂ A ; (b) S − 1 B ⊂ B ; (c) 1 ∈ A , ab ∈ B . The family P is partially order ed b y the relation ( A, B ) ≤ ( A ′ , B ′ ) deﬁned by A ⊂ A ′ and B ⊂ B ′ . W e claim that the pair ( A 0 , B 0 ) = ( S ∪ S b ± ab, S − 1 a ± ∪ S − 1 ab ) b elongs to P . I ndee d, a ± A 0 = a ± S ∪ a ± S b ± ab ⊂ S − 1 a ± ∪ S − 1 a ± b ± ab ⊂ S − 1 a ± ∪ S − 1 ab ⊂ B 0 . By analogy w e chec k that b ± B 0 ⊂ A 0 . The items (b), (c) triv ially follow from the deﬁnition of A 0 and B 0 . It rema ins to chec k that the sets A 0 and B 0 are disjoint. This will follow as so on as w e chec k that (d) S ∩ S − 1 a ± = ∅ , (e) S ∩ S − 1 ab = ∅ , (f ) S b ± ab ∩ S − 1 a ± = ∅ , (g) S b ± ab ∩ S − 1 ab = ∅ . The items (d) and (g) follow fro m (2 ). The item (e) follows fro m ab / ∈ S · S = S . By the sa me r eason, we ge t the item (f ) whic h is equiv alen t to ab / ∈ b ± S − 1 · S − 1 a ± = b ± S − 1 a ± = S b ± a ± ⊂ S . Thu s the par tially ordered set P is not empty and we can a pply Zorn’s Lemma to ﬁnd a maximal pair ( A, B ) ≥ ( A 0 , B 0 ) in P . W e claim that A ∪ B = X . Assuming the c onv erse, we co uld tak e an y p oint x ∈ X \ ( A ∪ B ) and put A ′ = A ∪ S x , B ′ = B ∪ a ± S x . It is clear that a ± A ′ ⊂ B ′ and b ± B ′ ⊂ A ′ , S − 1 B ′ = S − 1 B ∪ S − 1 a ± S x ⊂ B ∪ a ± S S x = B ′ , 1 ∈ A ⊂ A ′ and ab ∈ B ⊂ B ′ . Now w e see that the inclusion ( A ′ , B ′ ) ∈ P will follow as so o n as we chec k that A ′ ∩ B ′ = ∅ . The choice of x / ∈ B = S − 1 B guarantees that S x ∩ B = ∅ . Ass uming that a ± S x ∩ A 6 = ∅ , we w ould conclude that x ∈ S − 1 a ± A ⊂ S − 1 B ⊂ B , which contradicts the c hoice of x . Finally , the s e ts S x and a ± S x are dis jo in t bec ause of the prop erty (2) of S . Thus we obtain a cont radiction: ( A ′ , B ′ ) ∈ P is strictly gr eater than the maximal pair ( A, B ). This con tradiction shows that X = A ∪ B and cons equent ly , aA ⊂ X \ A = B ⊂ bA , which means that the s et A is pretwin and then bA \ aA ∈ I I 1 ⊂ I I . Since 1 ∈ A \ b − 1 a − 1 A , we conclude that bA \ a − 1 A ∋ b is not empty and thus I I 6 = {∅} .  ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 13 W e recall that a g roup X is p erio dic (o r else a t orsion gr oup ) if each e le men t x ∈ X has ﬁnite o r der (which means that x n = e for some n ∈ N ). W e shall say that a gr o up X has p erio dic c ommutators if for any x, y ∈ G the co mmutator [ x, y ] = xy x − 1 y − 1 has ﬁnite order in X . It is interesting to note tha t this condition is strictly weak er than the req uir ement for X to have perio dic commu tator subgroup X ′ (w e recall that the commutator subgr o up X ′ coincides with the set of ﬁnite pro ducts of comm utators), see [6]. Prop ositio n 6.3. Each gr oup X with p erio dic c ommutators has trivial twinic ide al I I = {∅} . Pr o of. Since X has p erio dic comm utators, for a n y points x, y ∈ X there is a num ber n ∈ N suc h that xy x − 1 y − 1 = ( y xy − 1 x − 1 ) − 1 = ( y xy − 1 x − 1 ) n and thus xy = ( y xy − 1 x − 1 ) n · y x b elongs to the semig roup generated by the set y ± · x ± . Applying Theor em 6.2, we conclude that the group X has trivia l t winic ideal I I = {∅} .  W e recall that a g roup G is called ab elian-by-ﬁnite (resp. ﬁnite-by-ab elian ) if G contains a normal Ab elian (resp. ﬁnite) subgr oup H ⊂ G with ﬁnite (resp. Ab elian) quotien t G/H . Observe that each ﬁnite-b y-ab elian g roup has p er io dic commutators and hence has trivial twinic ideal I I . In contrast, any a be lian-by-ﬁnite g roups, b eing amenable, is t winic but its twinic ideal I I need not b e trivial. The simplest counterexample is the isometry group Iso( Z ) of the group Z of integers endow ed with the Euclidean metr ic . Example 6.4. The abelia n-by-ﬁnite gr oup X = Iso( Z ) is twinic. Its twinic ideal I I coincides with the ideal [ X ] <ω of all ﬁnite subsets of X . Pr o of. Let a : x 7→ x + 1 b e the translation a nd b : x 7→ − x b e the inv ersion of the gr oup Z . It is easy to see that the element s a, b generate the isometry group X = Iso( Z ) a nd satisfy the relations b 2 = 1 and bab − 1 = a − 1 . Let Z = { a n : n ∈ Z } b e the cyclic subgroup of X generated b y the translation a . This subg roup Z has index 2 in X = Z ∪ Z b . First we s how that the ideal I = [ X ] <ω of ﬁnite subsets of X is twinic. Le t A ⊂ X b e a subset with xA ⊂ I X \ A ⊂ I y A for some x, y ∈ X . W e need to show that y A = I xA . W e consider three ca ses. 1) x, y ∈ Z . In this ca se the elements x, y commute. The I - inclusion xA ⊂ I y A implies y − 1 xA ⊂ I A . W e claim that y − 1 xA ⊃ I A . Obse rve that the I -inclus ion xA ⊂ I X \ A is equiv alen t to xA ∩ A ∈ I and to A ∩ x − 1 A ∈ I , which implies x − 1 A ⊂ I X \ A . By analogy , X \ A ⊂ I y A is equiv a lent to y A ∪ A = I X and to A ∪ y − 1 A = I X , which implies X \ A ⊂ I y − 1 A . Then x − 1 A ⊂ I X \ A ⊂ I y − 1 A implies y x − 1 A ⊂ I A and by the left-inv ariance of I , A ⊂ I xy − 1 A = y − 1 xA (we recall that the elemen ts x, y − 1 commute). Therefore, y − 1 xA = I A and henc e xA = I y A . 2) x ∈ Z and y ∈ X \ Z . Repea ting the a rgument fro m the pre ceding case, we can show tha t xA ⊂ I X \ A implies x − 1 A ⊂ I X \ A . Then we g et the chain of I -inclusio ns: xA ⊂ I X \ A ⊂ I y A ⊂ I y ( X \ xA ) = y x ( X \ A ) ⊂ I y xy A = I xA, where the las t I -equality follows from the case (1) since x, y xy ∈ Z . Now w e see that xA = I y A . 3) x / ∈ Z . Then xA ⊂ I X \ A ⊂ I y A implies x − 1 bA = bxb − 1 bA = bxA ⊂ I X \ bA ⊂ I by A = y − 1 bA. Since x − 1 b ∈ Z , the cases (1 ),(2 ) imply the I -equality x − 1 bA = I y − 1 bA . Shifting this equa lit y by b , we see tha t xA = b x − 1 bA = I by − 1 bA = y A . This completes the proo f of the twinic pro pe r ty o f the idea l I = [ X ] <ω . Then the twinic ideal I I ⊂ [ X ] <ω . Since [ X ] <ω is the smallest non-trivial left-inv ariant ideal on X , the e quality I I = [ X ] <ω will follow a s s o on as w e ﬁnd a non-empty set in the idea l I I . F o r this consider the s ubset A = { a n +1 , ba − n : n ≥ 0 } ⊂ X and obs e rve that X \ A = { a − n , ba n +1 : n ≥ 0 } = bA witnessing that A ∈ T . O bserve a ls o that aA = { a n +2 , aba − n : n ≥ 0 } = { a n +2 , ba − n − 1 : n ≥ 0 } $ A and thus baA ( X \ A = bA . Then ∅ 6 = baA \ bA ∈ I I 1 ⊂ I I witnesses that the t winic ideal I I is not trivial.  Next, we present (an exp ected) example of a g roup, which is not twinic. Example 6 . 5. The free gro up F 2 with tw o generator s is not twinic. Pr o of. Assume that the gr o up X = F 2 is t winic and let I I be the twinic ideal of F 2 . Let a, b be the generator s of the free group F 2 . Ea ch element w ∈ F 2 can b e represented b y a word in the alphab e t { a, a − 1 , b, b − 1 } . The word of the s ma llest length representing w is called the irr e ducible r epr esentation of w . The irreducible w ord represent ing the neutral element of F 2 is the empty word. Let A (resp. B ) b e the set o f words whose irreducible r epresentation start with letter a o r a − 1 (resp. b or b − 1 ). Consider the subset C =  a 2 n w : w ∈ B ∪ { e } , n ∈ Z  ⊂ F 2 14 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV and observe that abaC ⊂ X \ C = aC . The n a C \ a baC ∈ I I 1 by the deﬁnition of the subideal I I 1 ⊂ I I . Obser ve that a 3 baC ⊂ aC \ abaC and thus a 3 baC ∈ I I 1 . Then als o C ∈ I I 1 and X \ C = aC ∈ I I 1 by the left-inv ariance of I I 1 . By the additivity of I I 1 , we ﬁnally get X = C ∪ ( X \ C ) ∈ I I 1 ⊂ I I , whic h is the desired contradiction.  Next, we prov e so me per manence prop erties o f the class of twinic groups . Prop ositio n 6.6. L et f : X → Y b e a surje ctive gr oup homomorphism. If t he gr oup X is twinic, then so is the gr oup Y . Pr o of. Let I I b e the t winic idea l of X . It is easy to see that I = { B ⊂ Y : f − 1 ( B ) ∈ I I } is a left-inv aria nt ideal on the group Y . W e claim that it is twinic. Given any subset A ⊂ Y with xA ⊂ I Y \ A ⊂ I y A for some x, y ∈ Y , le t B = f − 1 ( A ) and observe that x ′ B ⊂ I I X \ B ⊂ I I y ′ B for some points x ′ ∈ f − 1 ( x ) and y ′ ∈ f − 1 ( y ). The t winic prop erty of the t winic ideal I I guar antees that f − 1 ( y A \ xA ) = y ′ B \ x ′ B ∈ I I , whic h implies y A \ xA ∈ I and hence xA = I Y \ A = I y A .  Problem 6 .7. Is a subg roup of a twinic group twinic? Is the pro duct of tw o t winic groups twinic? F o r gro ups with tr ivial twinic ideal the ﬁr st part o f this problem has an aﬃrma tive solution, whic h follows from the characterization Theorem 6.2. Prop ositio n 6.8. (1) The class of gr oups with trivial t winic ide al is close d with r esp e ct to taking sub gr oups and quotient gr oups. (2) A gr ou p X has trivial twinic ide al if and only if any 2-gener ate d su b gr oup of X has t rivial twinic ide al. 7. 2-Cogroups It follows from P rop osition 5.3 that for a t win subset A of a group X the stabilizer Fix( A ) of A is completely determined by the subset Fix − ( A ) becaus e Fix( A ) = x · Fix − ( A ) for eac h x ∈ Fix − ( A ). Ther efore, the subset Fix − ( A ) carries a ll the information ab out the pair (Fix ± ( A ) , Fix ( A )). The s ets Fix − ( A ) are par ticular cases of so-called 2-co groups deﬁned as follows. Deﬁnition 7 . 1. A subset K of a group X is called a 2-c o gr oup if for every x ∈ K the shift xK = K x is a subgroup of X , disjoint with K . By the index of a 2-co group K in X we understand the cardinality | X/K | of the set X/K = { K x : x ∈ X } . 2-Cogr oups can be characterized a s follows. Prop ositio n 7.2. A subset K of a gr oup X is a 2-c o gr oup in X if and only if ther e is a (unique) sub gr oup H ± of X and a sub gr oup H ⊂ H ± of index 2 su ch that K = H ± \ H and H = K · K . Pr o of. If K is a 2-cog roup, then for every x ∈ K the shift H = xK = K x is a s ubgroup of X disjoint with K . It follows that K = x − 1 H = H x − 1 . Since x − 1 ∈ x − 1 H = K , the shift x − 1 K = K x − 1 is a subgr oup of X a ccording to the deﬁnitio n of a 2- cogro up. Consequently , x − 1 K x − 1 K = x − 1 K , whic h implies K x − 1 K = K and H x − 1 x − 1 H x − 1 = K x − 1 K = K = H x − 1 . This implies x − 2 ∈ H a nd x 2 ∈ H . Co nsequently , xH = x − 1 x 2 H = x − 1 H = K = H x − 1 = H x 2 x − 1 = H x . Now we are able to s how tha t H ± = H ∪ K is a group. Indeed, ( H ∪ K ) · ( H ∪ K ) − 1 ⊂ H H − 1 ∪ H K − 1 ∪ K H − 1 ∪ K K − 1 ⊂ ⊂ H ∪ H H x ∪ xH H ∪ H x − 1 xH = H ∪ K ∪ K ∪ H = H ± . Since K = H x = xH , the subgro up H = K · K has index 2 in H ± . The uniqueness of the pair ( H ± , H ) follows from the fact that H = K · K and H ± = K K ∪ K . This completes the pro of of the “o nly if ” part. T o prove the “if ” part, assume that H ± is a s ubgroup o f X and H ⊂ H ± is a s ubgroup o f index 2 such that K = H ± \ H . Then for every x ∈ K the shift xK = K x = H is a subgro up of X disjoint with K . This means that K is a 2-c o group.  Prop ositio n 7.2 implies that for each 2-co group K ⊂ X the set K ± = K ∪ K K is a subgroup of X a nd K K is a subgroup of index 2 in K ± . By K we shall deno te the family of all 2- cogroups in X . It is partially ordered by the inclusion rela tion ⊂ and is considered as an X - act endo wed w ith the conjugating a ction · : X × K → K , · : ( x, K ) 7→ xK x − 1 , of the g roup X . F or each 2 - cogroup K ∈ K let Stab( K ) = { x ∈ X : xK x − 1 = K } b e the stabilizer of K and [ K ] = { xK x − 1 : x ∈ X } b e the orbit o f K . By [ K ] = { [ K ] : K ∈ K} be denote the orbit space of K b y the a ction of the group X . A cogr oup K ∈ K is called normal if xK x − 1 = K for a ll x ∈ X . This is equiv alen t to saying tha t Stab( K ) = X . Since for each t win subset A ⊂ X the s et Fix − ( A ) is a 2 -cogro up, the function Fix − : T → K , Fix − : A 7→ Fix − ( A ) , ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 15 is well-deﬁned and equiv ar iant a ccording to Pro po sition 5.4. A similar equiv ariant function I -Fix − : T I → K , I - Fix − : A 7→ I -Fix − ( A ) , can b e deﬁned fo r any left-in v ariant ideal I on a group X . Let b K denote the s et of ma ximal elements o f the partially ordered set ( K , ⊂ ). The following prop ositio n implies that the set b K lies in the image Fix − ( T ) and is coﬁnal in K . Prop ositio n 7.3. (1) F or any line arly or der e d family C ⊂ K of 2-c o gr oups in X the un ion ∪C is a 2-c o gr oup in X . (2) Each 2-c o gr oup K ∈ K lies in a maximal 2-c o gr ou p b K ∈ b K . (3) F or e ach maximal 2-c o gr oup K ∈ b K ther e is a twin su bset A ∈ T with K = Fix − ( A ) . Pr o of. 1. L e t C ⊂ K be a linearly ordere d family o f 2 - cogro ups of X . Since each 2-cogro up C ∈ C is disjoint with the group C · C a nd C = C · C · C , we get that the union K = ∪C is disjoint with the union S C ∈ C C · C = K · K and K = S C ∈ C C = S C ∈ C C · C · C = K · K · K witnessing that K is a 2-cogro up. 2. Since each chain in K is upp er b ounded, Zorn’s Lemma gua rantees that each 2-cogr oup o f X lies in a max imal 2-cogr oup. 3. Given a max imal 2-cog r oup K ∈ b K , co nsider the subgroups K · K and K ± = K ∪ K K o f X and choose a subset S ⊂ G meeting ea ch coset K ± x , x ∈ X , at a single p oint. Consider the set A = K K · S and note tha t X \ A = K S = xA for each x ∈ K , which means that K ⊂ Fix − ( A ). The maximality of K g uarantees that K = Fix − ( A ).  It should b e mentioned that in g eneral, Fix − ( T ) 6 = K . Example 7.4. F or any twin subset A in the 4 -element gro up X = C 2 ⊕ C 2 the group Fix( A ) is not trivial. Consequently , each singleton { a } ⊂ X \ { e } is a 2 -cogro up that does not b elong to the image Fix − ( T ). A left-inv ariant subfamily F ⊂ T is calle d • b K -c overing if b K ⊂ Fix − ( F ) (this means that for e a ch maximal 2-co group K ∈ b K ther e is a twin set A ∈ F with Fix − ( A ) = K ); • minimal b K - c overing if F coincides with each left-inv ariant b K - cov ering s ubfa mily o f F . Prop ositio n 7.3(3) implies that the family b T = { A ∈ T : Fix − ( A ) ∈ b K} is b K - cov ering. Prop ositio n 7.5. F or any function f ∈ End λ ( P ( X )) the family f ( b T ) is b K - c overing. Pr o of. The equiv aria nce of the function f a nd the left-inv aria nc e of the family b T imply the left-inv ariance of the family f ( b T ). T o see that f ( b T ) is b K -covering, ﬁx any maximal 2- c o group K ∈ b K and using Prop ositio n 7 .3, ﬁnd a t win set A ⊂ X with Fix − ( A ) = K . W e c la im tha t Fix − ( f ( A )) = Fix − ( A ) = K . By Cor ollary 4 .4, the function f is equiv ariant and symmetric. Then for every x ∈ Fix − ( A ), applying f to the equality xA = X \ A , w e obtain x f ( A ) = f ( xA ) = f ( X \ A ) = X \ f ( A ) , which means that x ∈ Fix − ( f ( A )) and th us Fix − ( A ) ⊂ Fix − ( f ( A )). Now the maximality o f the 2-cog roup Fix − ( A ) guarantees that Fix − ( f ( A )) = Fix − ( A ).  Remark 7. 6. In Theorem 17 .1 we shall show that for a t winic gr oup X and a function f ∈ K  End λ ( P ( X ))  from the minimal ideal of End λ ( P ( X )) the family f ( b T ) is minimal b K - cov ering. F o r each 2 -cogro up K ⊂ X consider the families T K = { A ∈ T : Fix − ( A ) = K } and T [ K ] = { A ∈ T : ∃ x ∈ X with Fix − ( xA ) = K } . The following prop osition des cribing the s tructure o f minimal b K - cov ering families c a n b e ea sily der ived from the deﬁnitions. Prop ositio n 7.7. A left-invariant subfamily F ⊂ b T is minimal b K -c overing if and only if for e ach K ∈ b K ther e is a set A ∈ F such that F ∩ T [ K ] = [ A ] . 16 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV 8. The cha racteristic group H ( K ) o f a 2-cogr oup K In this sec tio n w e in troduce an impo rtant notion of the c haracteristic group H ( K ) of a 2-cogroup K in a group X a nd reveal the algebraic structure of characteris tic gro ups of maximal 2-c o groups. Observe that for each 2-cogro up K ⊂ X its sta bilizer Stab( K ) = { x ∈ X : xK x − 1 = K } contains K K as a nor mal subgroup. So , w e can consider the q uotient group H ( K ) = Stab( K ) /K K ca lled the char acteristic gr oup of the 2-cogr o up K . Chara cteristic gro ups will play a crucial role for description of the structure of ma ximal subgroups o f the minimal ideal of the semigroup λ ( X ). Observe that for a norma l 2-co group K ∈ K the characteristic group H ( K ) is equal to the quotient group X/K K . The characteristic gro up H ( K ) of each maximal 2- cogro up K ⊂ X has a remark able algebr aic prop erty: it is a 2 -group with a unique 2-element subgro up. Let us recall that a gr oup G is called a 2-gr oup if the order of each elemen t of G is a power o f 2. Let us recall some standard ex amples of 2-gr oups. By Q 8 = { 1 , i , j, k , − 1 , − i, − j, − k } w e denote the group o f qua ter nions. It is a multiplicativ e subgroup o f the algebra of quater nions H . The algebra H contains the ﬁeld o f complex num b ers C as a suba lg ebra. F or each n ∈ ω let C 2 n = { z ∈ C : z 2 n = 1 } be the cyclic group of order 2 n . The m ultiplicative subg roup Q 2 n ⊂ H generated by the set C 2 n − 1 ∪ Q 8 is c a lled the gr oup of gener alize d quaternions , see [19, § 5.3]. The subg roup C 2 n − 1 has index 2 in Q 2 n and all element of Q 2 n \ C 2 n − 1 hav e order 4. Accor ding to our deﬁnition, Q 2 n = Q 8 for n ≤ 3. F or n ≥ 3 the gr oup Q 2 n has the pre s en tation h x, y | x 2 = y 2 n − 2 , x 4 = 1 , xy x − 1 = y − 1 i . The unions C 2 ∞ = [ n ∈ ω C 2 n and Q 2 ∞ = [ n ∈ ω Q 2 n are called the quasicyclic 2-gr oup and the inﬁnite gr oup of gener ali ze d quaternions , resp ectively . Theorem 8.1. A gr oup G is isomorphic to C 2 n or Q 2 n for some 1 ≤ n ≤ ∞ if and only if G is a 2-gr oup with a unique element of or der 2. Pr o of. The “only if ” pa rt is trivial. T o prov e the “if ” part, assume that G is a 2-g roup with a unique e lemen t of order 2. Denote this element by − 1 and let 1 b e the neutra l elemen t of G . If the group G is ﬁnite, then by Theo rem 5.3.6 o f [19], G is is omorphic to C 2 n or Q 2 n for some n ∈ N . So, w e assume that G is inﬁnite. Since − 1 is a unique element of order 2 , the cyclic subgroup {− 1 , 1 } is the ma ximal 2- element ary subg roup o f G (w e recall that a gro up is 2-elementary if it can be written as the direct sum of 2-element cyclic gro ups). Now Theorem 2 of [21] implies that the group G is ˇ Cerniko v and hence contains a nor mal ab elian subgroup H of ﬁnite index. Since G is inﬁnite, so is the subgro up H . Let ˜ H b e a maximal subgro up o f G that contains H . W e c la im that H = ˜ H and H is isomorphic to the quasicyclic 2 -group C 2 ∞ . Since H is a 2-gr oup, the unique element − 1 of the g roup G b elongs to H . Let f : {− 1 , 1 } → C 2 be the unique isomo r phism. Since the group C 2 ∞ is injective, b y Baer’s Theo rem [19, 4.1 .2], the homomo r phism f : {− 1 , 1 } → C 2 ⊂ C 2 ∞ extends to a homomorphism ¯ f : ˜ H → C 2 ∞ . W e claim that ¯ f is an isomor phis m. Indeed, the kernel ¯ f − 1 (1) of ¯ f is tr ivial since it is a 2-gr oup and contains no elemen t of order 2. So, ¯ f is injective and then ¯ f ( ˜ H ) coincides with C 2 ∞ , b eing a n inﬁnite subg roup of C 2 ∞ . By the same rea son, ¯ f ( H ) = C 2 ∞ . Consequently , H = ˜ H is isomor phic to C 2 ∞ . If G = H , then G is isomorphic to C 2 ∞ . So, it re ma ins to consider the case of non-ab elian g roup G 6 = H . Claim 8.2. F or every a ∈ H and b ∈ G \ H we get b 2 = − 1 and bab − 1 = a − 1 . Pr o of. The maximality of the ab elian subgroup H implies tha t bx 6 = xb for so me element of H . Since H is quas ic yclic, we ca n assume that the elemen t x ha s order ≥ 8 and a b elong s to the cyclic subgroup genera ted h x i . Using the fact that the maximal ab elian subgroup H has ﬁnite index in G , one can s how that the group G is lo cally ﬁnite. Consequently , the subg roup F = h b, x i generated by the set { b, x } is ﬁnite. By Theorem 5 .3.6 [19], this subg roup is isomorphic to Q 2 n for some n ≥ 4 . Analy z ing the pro per ties of the group Q 2 n we see that b 2 = − 1 and by b − 1 = y − 1 for all y ∈ h x i . In par ticula r, bab − 1 = a − 1 .  Next, we show that the subg r oup H has index 2 in G . This will follow as so on a s we show tha t for ea ch x, y ∈ G \ H we get xy ∈ H . Obser ve that for every a ∈ H we get xy ay − 1 x − 1 = xa − 1 x − 1 = a , which means that xy commut es with each element of H and hence xy ∈ H by the maximality o f H . Now take any element s b ∈ G \ H a nd q ∈ Q 2 ∞ \ C 2 ∞ . Extend the iso mo rphism ¯ f : H → C 2 ∞ to a map ˜ f : G → Q 2 ∞ letting ˜ f ( bh ) = q · ¯ f ( h ) for h ∈ H . Claim 8.2 implies that ˜ f is a well-deﬁned isomorphism b etw e en G = H ∪ bH and Q 2 ∞ .  Theorem 8.3. F or e ach max imal 2-c o gr oup K ∈ b K in a gr oup X the char acteristic gr oup H ( K ) = Stab( K ) / K K is isomorphi c either to C 2 n or to Q 2 n for some 1 ≤ n ≤ ∞ . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 17 Pr o of. This theor em will fo llow from Theorem 8 .1 as soo n as we ch eck that H ( K ) is a 2-g roup with a unique element of order 2. Let q : Stab( K ) → H ( K ) b e the quotient homomorphism. T ake any element x ∈ K and cons ider its image d = q ( x ). Since K = xK K , the image q ( K ) = { d } is a singleto n. T aking into acco un t that x / ∈ K K and x 2 ∈ K K , we see that the element d has or der 2 in H ( K ). W e claim that any other element a of order 2 in H ( K ) is equal to d . Ass ume conv ersely that some element a 6 = d of H ( K ) has or der 2. Let C ± be the subgroup of H ( K ) g enerated by the elements a, d and C b e the cyclic s ubgroup generated b y the pro duct ad . W e cla im that d / ∈ C . Assuming conv ersely that d ∈ C , we conclude that d = ( ad ) n for some n ∈ Z . Then a = add = ad ( ad ) n = ( ad ) n +1 ∈ C a nd co nsequently a = d (b ecause cyclic groups con tain at mo st o ne ele ment o f or der 2). Therefore d / ∈ C . It is cle a r that C ± = C ∪ dC , which means that the subgr oup C has index 2 in C ± . Consider the subgr oups H ± = q − 1 ( C ± ), H = q − 1 ( C ) and o bserve that the 2 -cogro up H ± \ H is strictly lar g er than K , which contradicts K ∈ b K . Since d is a unique ele men t o f order 2 in H ( K ), the cyclic subg roup D = { d, d 2 } ge ne r ated by d is normal in H ( K ). Consequently , for e a ch non-tr ivial subgroup G ⊂ H ( K ) the pro duct D · G = G · D is a subgroup in H ( K ). Now we see that G m ust co n tain d . O therwise, dG would be a 2-cogr oup in H ( K ) and its preimag e q − 1 ( dG ) w ould be a 2-cogro up in X that contains the 2-cogro up K as a prop er subset, which is impo ssible a s K is a maximal 2 -cogro up in X . Therefore each no n-trivial subgro up of H ( K ) contains d . This implies that ea ch element x ∈ H ( K ) has ﬁnite or der which is a p ow er of 2, witnessing that H ( K ) is a 2-gr o up with a single element of order 2.  9. Twin-genera ted topologies o n groups In this s ection we study so-ca lle d twin-generated topo lo gies on groups. The information obta ine d in this section will b e used in Section 20 for studying the top olo gical structure of maximal s ubgroups of the minimal ideal of the supere x tension λ ( X ). Given a t win subset A o f a group X consider the top ology τ A on X generated b y the s ubbase consisting o f the right shifts Ax , x ∈ X . In the fo llowing pro p os ition b y the weight of a top olog ical space we unders tand the smalles t c a rdinality of a subba s e of its top olog y . Prop ositio n 9.1. (1) The top olo gy τ A turns X into a right-top olo gic al gr oup. (2) If Ax = xA for al l x ∈ Fix − ( A ) , then the top olo gy τ A is zer o-dimensional. (3) The top olo gy τ A is T 1 if and only if the interse ction T x ∈ A Ax − 1 is a singleton. (4) The weight of the sp ac e ( X, τ A ) do es n ot exc e e d the index of the sub gr oup Fix( A − 1 ) in X . Pr o of. 1. It is clear that the top olog y τ A is right-in v aria nt . 2. If Ax = xA for all x ∈ Fix − ( A ), then the set X \ A is op en in the top olog y τ A bec ause X \ A = xA = Ax for any x ∈ Fix − ( A ). Consequently , A is an op en-and-clo s ed subbasic set. Now w e see that the space ( X , τ A ) has a base consisting of op en-a nd-closed subsets, which means that it is zero-dimensio nal. 3. If the top olog y τ A is T 1 , then the intersection T a ∈ A Aa − 1 of all ope n neighborho o ds of the neutral element e of X consists of a single point e . Assuming c o nv ersely that T a ∈ A Aa − 1 is a singleton { e } , for any t wo distinct points x, y ∈ X we can ﬁnd a shift Aa − 1 , a ∈ A , that c o nt ains the neutral element e but not y x − 1 . Then the shift Aa − 1 x is an op en subset of ( X , τ ) tha t contains x but not y , witness ing that the space ( X, τ A ) is T 1 . 4. T o estimate the weight o f the space ( X , τ A ), choose a subs et S ⊂ X meeting each coset x Fix( A − 1 ), x ∈ X , at a single p oint (here Fix( A − 1 ) = { x ∈ X : xA − 1 = A − 1 } ). Then the set S − 1 meets each coset Fix( A ) x , x ∈ X , at a sing le po in t. It is ea sy to see that the family { Ax : x ∈ S − 1 } forms a subbase of the top ology of τ a nd henc e the weigh t of ( X, τ ) do e s not exceed | X/ Fix( A − 1 ) | .  Deﬁnition 9.2. A top olo gy τ on a group X will be calle d twin-gener ate d if τ is equal to the top olog y τ A generated by some twin subset A ⊂ X , i.e., τ is generated by the subbase { Ax : x ∈ X } . Because of Theorem 8.3, we sha ll b e espec ially in terested in twin-generated top olog ies on the qua si-cyclic gr oup C 2 ∞ and the inﬁnite quaternion group Q 2 ∞ . First we consider some e x amples. Example 9 . 3. In the circle T = { z ∈ C : | z | = 1 } co nsider the twin subset C x = { e iϕ : 0 ≤ ϕ < π } . (1) F or each z ∈ T \ C 2 ∞ the twin set C 2 ∞ ∩ z C x generates the E uclidean top ology on C 2 ∞ . (2) F or ea ch z ∈ C 2 ∞ the twin set C 2 ∞ ∩ z C x generates the Sorg enfrey top ology o n C 2 ∞ . This topolo gy turns C 2 ∞ int o a par a top ological group with discontin uous inv ersion. A simila r s ituation ho lds for the group Q 2 ∞ . Its c lo sure in the alg ebra o f quater nions H c oincides with the multiplicativ e subgroup T ∪ T j o f H , where j ∈ Q 8 \ C is one o f non-complex quaternion units. Example 9 . 4. In the gro up T ∪ T j ⊂ H consider the twin subset Q x = C x ∪ C x j . 18 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV (1) F or each z ∈ T \ C 2 ∞ the twin set Q 2 ∞ ∩ z Q x generates the E uclidean top ology on Q 2 ∞ . (2) F or each z ∈ C 2 ∞ the t win s e t Q 2 ∞ ∩ z Q x generates the Sorgenfre y top ology on C 2 ∞ . This top olo g y turns Q 2 ∞ int o a rig h t-top ologica l group with discontin uous in verse and discontin uous left shifts l x : Q 2 ∞ → Q 2 ∞ for x ∈ Q 2 ∞ \ C 2 ∞ . In the following prop osition by τ E we deno te the E uc lide a n top o logy on C 2 ∞ . Theorem 9. 5. Each metrizable right-invariant t op olo gy τ ⊃ τ E on the gr oup C 2 ∞ (or Q 2 ∞ ) is twin-gener ate d. Pr o of. First we c onsider the ca se of the group C 2 ∞ . Let E 0 = C 2 ∞ ∩ { e iϕ : − π / 3 < ϕ < 2 π / 3 } b e the t win subset generating the E uclidean top o logy τ E on C 2 ∞ and E n = C 2 ∞ ∩ { e iϕ : | ϕ | < 3 − n − 1 π } for n ≥ 1. F or every n ∈ N le t ϕ n = P n k =1 π / 4 n and observe that ϕ ∞ = P ∞ k =1 π / 4 n = π / 3. Let τ ⊃ τ E be a ny metrizable r ight-in v ariant top olog y on C 2 ∞ . The metriza ble space ( C 2 ∞ , τ ) is count able and hence zero-dimensio nal. Since τ ⊃ τ E , there exists a neighborho o d base { U n } ∞ n =1 ⊂ τ at the unit 1 s uch that each set U n is closed and op en in τ and U n ⊂ E n for all n ∈ N . The int erested rea der can chec k that the twin subset A = ( E 0 \ ∞ [ n =1 e iϕ n E n ) ∪ ∞ [ n =1 e iϕ n U n ∪ ∞ [ n =1 e i ( π + ϕ n ) E n \ U n generates the top olo gy τ . Next, ass ume that τ ⊃ τ E is a right-in v ar iant top olog y on the gr oup Q 2 ∞ . This group c a n b e written as Q 2 ∞ = C 2 ∞ ∪ C 2 ∞ j , where j ∈ Q 8 \ C is a non-complex quater nio n unit. Since C 2 ∞ ∈ τ E ⊂ τ , the subgr oup C 2 ∞ is o pen in Q 2 ∞ . By the preceding item, the top o logy τ ∩ P ( C 2 ∞ ) on the gro up C 2 ∞ is g enerated b y a twin set A ⊂ C 2 ∞ \ { e iϕ : ϕ ∈ ( 2 π 3 , π ) ∪ ( 4 π 3 , 5 π 3 ) } . A simple g e ometric arg umen t s hows that the top olo gy τ is generated by the twin subset A ∪ A j of Q 2 ∞ .  Problem 9 .6. Are all metr iz able right-in v aria nt topo lo gy on C 2 ∞ and Q 2 ∞ t win-genera ted? 10. The characteristic gr oup H ( A ) of a twin subset A In this section, g iven a t win subset A ∈ T of a gro up X we in tro duce a t win-genera ted to p olo gy on the characteristic group H ( K ) of the 2-co group K = Fix − ( A ). Consider the intersection B = A ∩ Stab( K ) = B · K K and the ima ge A ′ = q A ( B ) of the set B under the quotient homomorphism q A : Stab( K ) → H ( K ) = Stab( K ) /K K . W e claim that A ′ is a twin subset of H ( K ). Indeed, for every x ∈ Fix − ( A ) = K ⊂ Stab( K ) we get X \ A = xA and c onsequently , Stab( K ) \ B = xB and H ( A ) \ A ′ = z A ′ where z ∈ q A ( x ). Now it is lega l to endow the gro up H ( K ) with the to po logy τ A ′ generated by the twin s ubs et A ′ . This top olog y is generated by the subbase { A ′ x : x ∈ H ( K ) } . B y Pr o po sition 9.1 the top olog y τ A ′ turns the characteristic group H ( K ) int o a r ig ht -top ologica l gr oup, which will b e called the char acteristic gr oup o f A and will b e denoted by H ( A ). B y Prop ositio n 9.1, the c haracter istic g roup H ( A ) is a T 1 -space and its w eight does not exceed the car dinality o f H ( A ). The reader should b e c o nscious of the fact that fo r t wo t win subsets A, B ∈ T with Fix − ( A ) = Fix − ( B ) the characteristic group H ( A ) a nd H ( B ) a re algebr aically isomorphic but top olo gically they can b e distinct, see Examples 9.3 and 9.4. 11. Characterizing functions tha t belong to End I λ ( F ) In this section for a twinic ideal I on a group X and a left-inv ariant subfamily F ⊂ b T we characterize functions f : F → P ( X ) that belong to the space End I λ ( F ). W e reca ll that End I λ ( F ) is the pro jection of End I λ ( P ( X )) onto the fa ce P ( X ) F . Theorem 11.1. F or a left-invariant twinic ide al I on a gr oup X and a left-invariant subfamily F ⊂ T a function ϕ : F → P ( X ) b elongs t o the sp ac e E nd I λ ( F ) if and only if ϕ is e quivariant, I -satur ate d, and Fix − ( A ) ⊂ Fix − ( ϕ ( A )) for al l A ∈ F . Pr o of. T o prov e the “o nly if ” part, take any function ϕ ∈ End I λ ( F ) a nd ﬁnd a function ψ ∈ End I λ ( P ( X )) such that ϕ = ψ | F . By Theorem 4 .8, the function ψ is equiv a riant, monotone, symmetric , and I -satura ted. Consequently , its restriction ϕ = ψ | F is equiv ariant and I -satur a ted. Now ﬁx any subset A ∈ F and take any p oint x ∈ Fix − ( A ). The left-in v ar iance of F g uarantees that X \ A = xA ∈ F , which means that the family F is symmetric. Applying the equiv ariant s ymmetric function ψ to the equality xA = X \ A , w e get x ϕ ( A ) = x ψ ( A ) = ψ ( xA ) = ψ ( X \ A ) = X \ ψ ( A ) = X \ ϕ ( A ) and thus x ∈ Fix − ( ϕ ( A )) and Fix − ( A ) ⊂ Fix − ( ϕ ( A )). ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 19 T o prov e the “if ” part, ﬁx any eq uiv aria n t I -saturated function ϕ : F → P ( X ) s uc h that Fix − ( A ) ⊂ Fix − ( ϕ ( A )). In order to apply Theorem 4.8, we need to e x tend the function ϕ to some s ymmetric I -sa turated family . This can b e done as follows. Consider the I -sa turization ¯ ¯ F I = S A ∈ F ¯ ¯ A I of F . Next, extend the function ϕ to the function ¯ ϕ : ¯ ¯ F → F assigning to each set B ∈ ¯ ¯ F I the s e t ϕ ( A ) where A ∈ F ∩ ¯ ¯ B I . Since ϕ is I -sa turated, so deﬁned extensio n ¯ ϕ of ϕ is well-deﬁned and I -sa tur ated. The equiv ar iance of ϕ implies the equiv aria nce of its extensio n ¯ ϕ . Let us chec k that the function ¯ ϕ : ¯ ¯ F I → F is symmetric and monotone. T o see that ¯ ϕ is symmetric, ta ke any set B ∈ ¯ ¯ F I and ﬁnd a set A ∈ F ∩ ¯ ¯ B I . Fix a n y p oint x ∈ Fix − ( A ). By our hypothesis x ∈ Fix − ( A ) ⊂ Fix − ( ϕ ( A )). It follows from A = I B that X \ A = I X \ B and hence ¯ ϕ ( X \ B ) = ϕ ( X \ A ) = ϕ ( xA ) = xϕ ( A ) = X \ ϕ ( A ) = X \ ¯ ϕ ( B ) , which mea ns that the function ¯ ϕ is symmetric. The monoto nic ity of ¯ ϕ will fo llow as so on as we ch eck that ϕ ( A ) = ϕ ( B ) for any se ts A, B ∈ F with A ⊂ I B . P ick po in ts a ∈ Fix − ( A ), b ∈ Fix − ( B ). Since the idea l I is twinic, the c hain of I - inclusions bB = X \ B ⊂ I X \ A = aA ⊂ I aB implies the chain of I -e q ualities bB = I X \ B = I X \ A = aA = I aB , which yields A = I B and ϕ ( A ) = ϕ ( B ) a s ϕ is I -sa tur ated. Therefore ¯ ϕ : ¯ ¯ F I → F is a left-inv ariant sy mmetric monotone I -satura ted function deﬁned on a I -satura ted left- inv ariant symmetric family ¯ ¯ F I . By Theo rem 4 .8, ¯ ϕ b elongs to E nd I λ ( ¯ ¯ F I ) and then its r e striction ϕ = ¯ ϕ | F b elong s to End I λ ( F ).  Let us reca ll that b T = { A ⊂ X : Fix − ( A ) ∈ b K} . Corollary 1 1.2. F or a left-invariant twinic ide al I on a gr oup X the sp ac e End I λ ( b T ) c onsists of al l e quivariant I -satu r ate d functions ϕ : b T → b T such that Fix − ( ϕ ( A )) = Fix − ( A ) for al l A ∈ b T . A similar characteriza tion holds for functions that b e long to the spa ce End I λ ( T K ) for K ∈ b K (let us observe that Theorem 4.8 is not applicable to the family T K bec ause it is not left-inv ariant). A function ϕ : T K → T K is Stab( K ) - e quivariant if ϕ ( xA ) = xϕ ( A ) for all A ∈ T K and x ∈ Stab( K ). Prop ositio n 1 1.3. F or any maximal 2-c o gr oup K ⊂ X and a left-invariant twinic ide al I on a gr oup X a fun ction ϕ : T K → T K b elongs to t he s p ac e End I λ ( T K ) if and only if ϕ is Sta b( K ) -e quivariant and I -satur ate d. Pr o of. The “only if ” par t follows fro m Theorem 4.8. T o prove the “if part” , assume that a function ϕ : T K → T K is Stab( K )-inv aria n t and I -satura ted. F or any A ∈ T K and x ∈ K = Fix − ( A ) = Fix − ( ϕ ( A )) w e get ϕ ( X \ A ) = ϕ ( xA ) = xϕ ( A ) = X \ ϕ ( A ), whic h means that the function ϕ is symmetric. Now co nsider the families L ϕ = { x − 1 A : A ∈ F , x ∈ ϕ ( A ) } a nd ¯ ¯ L I ϕ = [ A ∈L ϕ ¯ ¯ A I . W e claim that the family ¯ ¯ L I ϕ is linked. Assuming the co n verse, we could ﬁnd tw o sets A, B ∈ F and tw o p oints x ∈ ϕ ( A ) and y ∈ ϕ ( B ) such that x − 1 A ∩ y − 1 B ∈ I . Then y x − 1 A ⊂ I X \ B . Let us show that the p oint c = y x − 1 belo ngs to the subgroup Sta b( K ) of X . Given any p oint z ∈ K , we need to prov e that c − 1 z c ∈ K . T aking in to account that z ∈ K = Fix − ( B ) = Fix − ( A ), we see that cA ⊂ I X \ B implies that cA ⊂ I X \ B = z B ⊂ I z c ( X \ A ) = z cz A. Since the idea l I is twinic, we get c A = I X \ B = I z c z A , which implies c − 1 z c z ∈ I -Fix( A ). The ma x imality of the 2-cogr oup K = Fix − ( A ) ⊂ I -Fix − ( A ) guara n tees that I -Fix − ( A ) = K and I -Fix ( A ) = I -Fix − ( A ) · I -Fix − ( A ) = K K . Therefore c − 1 z c z ∈ K K and c − 1 z c ∈ K K z − 1 = K . Now we see that y x − 1 = c ∈ Stab( K ). So it is legal to a pply the Stab( K )-inv ariant I -satura ted function ϕ to the I -equality y x − 1 A = I X \ B and o btain y x − 1 ϕ ( A ) = ϕ ( X \ B ) = X \ ϕ ( B ). Then x − 1 ϕ ( A ) ⊂ X \ y − 1 ϕ ( B ), which is no t possible b ecause the neutral elemen t e of the group X b elong s to x − 1 ϕ ( A ) ∩ y − 1 ϕ ( B ). F ur ther we con tin ue as in the pro of of Theor em 4.8.  12. The H ( K ) -act T K of a maximal 2-cogr oup K In this section, given a max ima l 2-cogr oup K in a g r oup X we study the structur e of the s ubspace T K = { A ∈ P ( X ) : Fix − ( A ) = K } ⊂ P ( X ) of the compact Hausdorﬀ spa ce P ( X ). The latter space is naturally homeomor phic to the C a nt or discontin uum 2 X where the ordina l 2 = { 0 , 1 } is endow ed with the discrete top olo gy . 20 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Prop ositio n 12 .1. F or any 2-c o gr oup K ⊂ X the subsp ac e T K of P ( X ) is home omorph ic to the Cantor disc ontinu um 2 X/K ± wher e X/K ± = { K ± x : x ∈ X } . Pr o of. Cho ose any subset S ⊂ X that meets each coset K ± x , x ∈ X , a t a single p oint, and consider the bijective function Ψ : P ( S ) → T K assigning to each subset A ⊂ S the t win set T A = K K A ∪ K ( S \ A ). Let us show that the function Ψ is contin uous. The s ubbase of the topolog y of T K consists of the sets h x i + = { B ∈ T K : x ∈ B } and h x i − = { B ∈ T K : x / ∈ B } where x ∈ X . Obs erve that for every z ∈ K we get h x i − = { B ∈ T K : x ∈ X \ B = z B } = h z − 1 x i + , which means that the se ts h x i + , x ∈ X , form a subba se of the topolo gy of T K . Now the co n tinu ity of the map Ψ w ill follow as so on as we check that for every x ∈ X the set Ψ − 1 ( h x i + ) = { A ∈ P ( S ) : x ∈ T A } is open in P ( S ). Fix any subset A ∈ Ψ − 1 ( h x i + ) and let s b e the unique p oint of the int ersection S ∩ K ± x . Consider the o pe n neighborho o d O ( A ) = { A ′ ∈ P ( S ) : A ′ ∩ { s } = A ∩ { s }} of A in the spac e P ( S ). W e claim that O ( A ) ⊂ Ψ − 1 ( h x i + ). Fix any A ′ ∈ O ( A ) and consider tw o cases: (i) If s ∈ A , then s ∈ A ′ and x ∈ T A ∩ K ± s = K K s ⊂ T A ′ . (ii) If s ∈ S \ A , then s ∈ S \ A ′ and x ∈ T A ∩ K ± s = K s ⊂ K ( S \ A ′ ) ⊂ T A ′ . In b oth ca ses Ψ( A ′ ) = T A ′ ∈ h x i + . Now we see that Ψ : P ( S ) → T K , b eing a contin uous bijectiv e map deﬁned on the compact Hausdor ﬀ spa ce P ( S ), is a homeomorphism. It remains to obser ve that P ( S ) is ho meo morphic to 2 X/K ± .  Let us obser ve that in general the subfamily T K ⊂ P ( X ) is not left-in v aria nt . Indeed, for any A ∈ T K and x ∈ X the shift xA b elongs to T K if a nd only if K = Fix − ( xA ) = x Fix − ( A ) x − 1 = xK x − 1 if a nd only if x ∈ Stab( K ). Thu s the family T K can b e consider ed as an act endowed with the left action o f the gr oup Stab( K ). F o r any t win set A ∈ T K its stabilizer Fix( A ) = { x ∈ X : xA = A } is equal to Fix − ( A ) · Fix − ( A ) = K K and hence is a normal subgroup of Stab( K ). This implies that the c haracteristic group H ( K ) = Stab( K ) /K K a cts fre e ly on the space T K . Therefore, we can (and will) consider the spa ce T K as a free H ( K )-act. F or each set A ∈ T K by ⌊ A ⌋ = [ A ] ∩ T K = { xA : x ∈ Sta b( K ) } = { hA : h ∈ H ( K ) } we denote the orbit of A in T K and by [ T K ] = {⌊ A ⌋ : A ∈ T K } the orbit s pace of the H ( K )-act T K , endow ed with the quotient top olog y . By Theor em 3.1, the H ( K )-act T K is iso morphic to [ T K ] × H ( K ). In some ca ses the iso morphism betw een the H ( K )-acts T K and [ T K ] × H ( K ) is top olo gical. Prop ositio n 12 .2. The orbit sp ac e [ T K ] is a T 1 -sp ac e if and only if t he char acteristic gr oup H ( K ) is ﬁnite. In this c ase [ T K ] is a c omp act Hausdorﬀ sp ac e and the orbit map q : T K → [ T K ] has a c ontinuous se ction s : [ T K ] → T K , which implies t hat T K is home omorph ic to the pr o duct [ T K ] × H ( K ) wher e the (ﬁnite) gr oup H ( K ) is endowe d with the discr ete top olo gy. Pr o of. By Theorem 8.3, the characteristic gr oup H ( K ) is at most countable. Since T K is a free H ( K )-act, eac h orbit ⌊ A ⌋ , A ∈ T K , has cardina lity |⌊ A ⌋| = | H ( K ) | and hence is a t most countable. Note that the o rbit ⌊ A ⌋ admits a transitive action of the group H ( K ) and hence is top olog ically homogeneous . If [ T K ] is a T 1 -space, then each orbit ⌊ A ⌋ , A ∈ T K , is closed in the co mpact Hausdorﬀ space T K . Now Bair e theorem implies that ⌊ A ⌋ has an isola ted p oint and is discrete (b eing top olog ic ally homogeneo us). T aking into acc o unt that ⌊ A ⌋ is compact a nd discrete, we conclude that it is ﬁnite. Consequently | H ( K ) | = |⌊ A ⌋| < ℵ 0 . Now assume that the characteristic gr oup H ( K ) is ﬁnite. Let q : T K → [ T K ] denote the o r bit map. T o show that the or bit space [ T K ] is Hausdor ﬀ, pick tw o distinct orbits ⌊ A ⌋ a nd ⌊ B ⌋ . Since H ( K ) is ﬁnite and xA 6 = y B for any x, y ∈ H ( K ), we can ﬁnd t wo neighborho o ds O ( A ) and O ( B ) of A, B in T K such that xO ( A ) ∩ y O ( B ) = ∅ . Then O ( ⌊ A ⌋ ) = S x ∈ H ( K ) xO ( A ) a nd O ( ⌊ B ⌋ ) = S y ∈ H ( K ) y O ( B ) are tw o dis jo in t op en H ( K )-inv ariant subsets in T K . Their images q ( O ( ⌊ A ⌋ )) a nd q ( O ( ⌊ B ⌋ )) a r e dis join t open neighborho o ds of ⌊ A ⌋ , ⌊ B ⌋ in [ T K ], whic h means that the orbit s pace [ T K ] is Hausdo r ﬀ. This space is compact and zero-dimensiona l as the imag e of the compact z e ro-dimensiona l s pace T K under the op en con tinuous map q : T K → [ T K ]. Using the zero- dimensionality of [ T K ] and the ﬁniteness of H ( K ) it is easy to construct a contin uo us section s : [ T K ] → T K of the ma p q and prove that T K is homeo mo rphic to H ( K ) × [ T K ].  Let us recall that a subfamily F ⊂ P ( X ) is λ -inv ariant if f ( F ) ⊂ F for an y equiv a riant symmetric monotone function f : P ( X ) → P ( X ). F or a λ -inv ariant subfamily F ⊂ P ( X ) the pro jection End λ ( F ) = { f | F : f ∈ End λ ( P ( X )) } is a subs e migroup of the s e migroup F F of all self-ma ppings of F . Prop ositio n 12 .3. F or any maximal 2-c o gr oup K ⊂ X the family T K is λ -invariant and henc e End λ ( T K ) is a c omp act right-top olo gic al semigr oup. ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 21 Pr o of. Given any function f ∈ E nd λ ( P ( X )) and a set A ∈ T K we need to show that f ( A ) ∈ T K . By Corolla ry 4 .4, the function f is eq uiv aria nt and symmetric. Then for a n y x ∈ K = Fix − ( A ) we get xA = X \ A and hence xϕ ( A ) = ϕ ( xA ) = ϕ ( X \ A ) = X \ ϕ ( A ), which means that x ∈ Fix − ( ϕ ( A )) and K ⊂ Fix − ( ϕ ( A )). The max imalit y of the 2-c o group K guarantees that K = Fix − ( ϕ ( A )) and thus ϕ ( A ) ∈ T K . So, the family T K is λ -inv ariant.  13. I -incomp arable and I -independent f am il ies Let I b e a left-inv ariant ideal on a gro up X . A fa mily F ⊂ P ( X ) is called • I -inc omp ar able if ∀ A, B ∈ F ( A ⊂ I B ⇒ A = I B ); • I -indep endent if ∀ A, B ∈ F ( A = I B ⇒ A = B ). Prop ositio n 13.1. A left-invariant ide al I on a gr oup X is twinic if and only if the family pT I of I -pr etwin sets is I -inc omp ar able. Pr o of. First assume tha t the family pT I is I -incompar able. T o show that the ideal I is twinic, take any subset A ⊂ X with xA ⊂ I X \ A ⊂ I y A for some x, y ∈ X . Then A ∈ pT I and also xA, y A ∈ pT I . Since xA ⊂ I y A , the I -inco mparability of the family pT I implies that xA = I y A and then xA = I X \ A = I y A , which means that the idea l I is t winic. Now assume conv ersely that I is twinic and take tw o I -pretwin sets A ⊂ I B . Since the sets A, B are I -pretwin, there are elements x, y ∈ X such that xB ⊂ I X \ B a nd X \ A ⊂ I y A . T aking in to acco unt that xB ⊂ I X \ B ⊂ I X \ A ⊂ I y A ⊂ I y B , and I is t winic, we conclude that X \ B = I X \ A and hence A = I B .  Corollary 13.2 . F or e ach twinic left-invariant ide al I on a gr oup X the family T of twin set s is I -inc omp ar able. Prop ositio n 13.3. F or a left-invariant ide al I on a gr oup X the family b T is I -indep endent if and only if I ∩ b K = ∅ . Pr o of. T o prov e the “only if ” par t, assume that the idea l I contains some maximal 2- cogroup K ∈ b K . Since I is left-in v ar iant, fo r each x ∈ K , K K = xK ∈ I a nd hence K ± = K ∪ K K ∈ I . Cho ose a subset S ⊂ X that contains the neutral element e of the gr oup X a nd meets ea ch cose t K ± x , x ∈ X , at a single po in t. Then A = K K S and B = K K ( S \ { e } ) ∪ K a re t wo distinct twin sets with K ⊂ Fix − ( A ) ∩ Fix − ( B ). By the maximality of K , K = Fix − ( A ) = Fix − ( B ) and hence A, B ∈ b T . Since the symmetric diﬀerenc e A △ B = K K ∪ K = K ± ∈ I , we get A = I B , whic h means that the family b T fails to be I -indep endent . T o prov e the “if ” part, a s sume that the family b T is not I - independent and ﬁnd t w o subsets A, B ∈ b T such that A 6 = B but A = I B . The 2-cogroup Fix − ( A ) o f A is maximal and hence coincides with the 2-co group I -Fix − ( A ) ⊃ Fix − ( A ). By the s a me rea son, Fix − ( B ) = I -Fix − ( B ). The I -equa lit y A = I B implies I -Fix − ( A ) = I -Fix − ( B ). Denote the ma ximal 2-cogr oup Fix − ( A ) = I -Fix − ( A ) = I -Fix − ( B ) = Fix − ( B ) by K . Then Fix( A ) = Fix − ( A ) · Fix − ( A ) = K K = Fix( B ) a nd hence A = K K A and B = K K B . Now we see that the symmetric diﬀerence A △ B = K K A △ K K B contains a subset K K x for some x ∈ X . Then for any y ∈ K , w e get K y x = K K x ⊂ A △ B ∈ I and hence K y x ∈ I . Finally o bserve tha t the set K ′ = x − 1 y − 1 K y x is a maxima l 2 -cogro up and by the left inv ariance of the ideal I , K ′ = x − 1 y − 1 K y x ∈ I . So, b K ∩ I 6 = ∅ .  Prop ositio n 13.4. A su bfamily F ⊂ P ( X ) is I -indep endent for any left-invariant ide al I on X if for e ach set A ∈ F the sub gr oup Fix( A ) has ﬁnite index in X . Pr o of. Assume that A, B ∈ F b e tw o subsets with A = I B for some left-inv ariant idea l I . Since the subgroups Fix( A ) and Fix( B ) hav e ﬁnite indices in X , their intersection Fix( A ) ∩ Fix( B ) also has ﬁnite index in X and cont ains a nor mal subgroup H ⊂ X o f ﬁnite index in X , see [14, I.Ex.9(a)]. Then X = F H for some ﬁnite s ubset F ⊂ X . Assuming that A 6 = B , we can ﬁnd a po int x ∈ A △ B a nd conclude that xH = H x ⊂ H A △ H B = A △ B ∈ I and X = F H ∈ I by the left-in v ar iance of the idea l I . This contradiction completes the pro o f.  Prop ositio n 13.5. Each minimal b K -c overing subfamily e T ⊂ b T is I -indep endent. Pr o of. Fix any tw o s ets A, B ∈ e T with A = I B . Repeating the argument from the pro of of Pro po sition 13 .3, we can prov e that I -Fix − ( A ) = Fix − ( A ) = I -Fix − ( B ) = Fix − ( B ) = K for some maximal 2- cogroup K ∈ b K . Since the fa mily e T ∋ A, B is minimal b K -covering, the s ets A, B lie in the sa me orbit and hence A = xB for some x ∈ X . It follows from B = I A = xB that x ∈ I -Fix( B ) = Fix( B ) and thus A = xB = B .  22 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV 14. The endomorphism monoid End( T K ) of the H ( K ) -act T K F o r any maximal 2 -cogro up K in a group X the compact rig ht -top ologica l semigro up End λ ( T K ) is a subsemigro up of the endomor phism mono id End( T K ) of the free H ( K )-act T K . The endomorphism monoid E nd( T K ) is the s pace of all (not neces sarily contin uous) functions f : T K → T K that a r e equiv ariant in the sense that f ( xA ) = xf ( A ) fo r all A ∈ T K and x ∈ Sta b( K ). It is easy to c heck that End( T K ) is a closed subsemig roup of the co mpact Hausdor ﬀ right-topolog ical semigroup T T K K of a ll self-maps of the compact Hausdor ﬀ space T K . So, End( T K ) is a compact Hausdor ﬀ rig h t-top ologica l semigroup that co n tains End λ ( T K ) as a c lo sed subsemigro up. If I is a left-inv ariant idea l on the group X , then the left ideal E nd I λ ( T K ) of End λ ( T K ) lies in the left ideal End I ( T K ) ⊂ End( T K ) consisting of all equiv ar iant functions f : T K → T K , which a re I -satura ted in the sens e that f ( A ) = f ( B ) for all A, B ∈ T K with A = I B . In the following theor em we describ e some algebra ic a nd topo logical pr op erties o f the endomorphis m monoid End( T K ). Theorem 14 . 1. L et K b e a maximal 2-c o gr oup in a gr oup X . Then: (1) End I ( T K ) = End I λ ( T K ) ⊂ End λ ( T K ) ⊂ End( T K ) for any twinic ide al I on X ; (2) End I ( T K ) = End( T K ) for any left-invariant ide al I on X s uch that I ∩ b K = ∅ ; (3) the semigr oup End( T K ) is algebr aic al ly isomorphi c t o the wr e ath pr o duct H ( K ) ≀ [ T K ] [ T K ] ; (4) for e ach idemp otent f ∈ E nd( T K ) the maximal s ub gr oup H f ⊂ E nd( T K ) c ont aining f is isomorphic to H ( K ) ≀ S [ f ( T K )] ; (5) the minimal ide al K (End( T K )) = { f ∈ E nd( T K ) : ∀ A ∈ f ( T K ) , f ( T K ) ⊂ ⌊ A ⌋} ; (6) e ach minimal left ide al of the semigr oup End( T K ) is algebr aic al ly isomorphic t o H ( K ) × [ T K ] wher e the orbit sp ac e [ T K ] is endowe d with the left zer o mu ltiplic ation; (7) e ach maximal su b gr oup of t he minimal ide al K (End( T K )) is algebr aic al ly isomorphic to H ( K ) ; (8) e ach minimal left ide al of the semigr oup End( T K ) is home omorphic to T K ; (9) for e ach minimal idemp otent f ∈ K (End( T K )) the maximal su b gr oup H f = f ◦ End( T K ) ◦ f is top olo gic al ly isomorphic to the twin-gener ate d gr oup H ( A ) wher e A ∈ f ( T K ) ; Pr o of. 1,2. The ﬁr st statement follo ws from Pro po sition 11.3 and the seco nd o ne from Pr op osition 13.3. 3–7. Since T K is a free H ( K )-act, the (algebraic) statements (3)–(7) follow from Theorem 3.1. 8. Given a minimal idemp otent f ∈ E nd( T K ), we need to prove that the minimal left ideal L f = End( T K ) ◦ f is homeomorphic to T K ⊂ P ( X ). F or this ﬁx any set B ∈ f ( T K ) and observe that f ( T K ) ⊂ ⌊ B ⌋ according to the statemen t (5). W e claim that the map Ψ : L f → T K , Ψ : g 7→ g ( B ) , is a homeomorphism. The deﬁnition of the top ology (of p oint wise convergence) on End( T K ) implies that the map Ψ is contin uous. Next, we show that the map Ψ is bijective. T o show that Ψ is injective, ﬁx any tw o distinct functions g , h ∈ L f and ﬁnd a set A ∈ T K such tha t g ( A ) 6 = h ( A ). Since f ( T K ) ⊂ ⌊ B ⌋ , there is x ∈ X such that f ( A ) = xB . Then xg ( B ) = g ( xB ) = g f ( A ) = g ( A ) 6 = h ( A ) = hf ( A ) = h ( xB ) = xh ( B ) and hence Ψ( g ) = g ( A ) 6 = h ( A ) = Ψ( h ). T o show that Ψ is surjective, take a ny subset C ∈ T K and choose any equiv ar iant map ϕ : [ B ] → [ C ] such that ϕ ( B ) = C . Then the function g = ϕ ◦ f b elongs to L f and has image Ψ( g ) = g ( B ) = C witnessing that the map Ψ is surjective. Since L f is compact, the bijective contin uo us ma p Ψ : L f → T K is a ho meo morphism. By Prop osition 12.1, the s pace T K is homeomorphic to the cube 2 X/K ± . 9. Given a minimal idemp otent f ∈ End( T K ) we shall show that the max imal subg roup H f = f ◦ E nd( T K ) ◦ f is top ologically isomo rphic to the characteristic group H ( A ) of any twin set A ∈ f ( T K ). W e rec a ll that H ( A ) is the characteristic gr oup H ( K ) of the 2-co group K = Fix − ( A ), endowed with the to po logy generated by the twin s et q ( A ∩ Sta b( K )) wher e q : Stab( K ) → H ( K ) = Stab( K ) /K K is the quotien t homomorphism. W e deﬁne a top olo g ical is omorphism Θ A : H f → H ( A ) in the following wa y . Since f is a minima l idemp otent, g ( A ) = f g f ( A ) ∈ f ( T K ) ⊂ ⌊ A ⌋ . So we can ﬁnd x ∈ Stab( K ) with f g f ( A ) = x − 1 A . Now deﬁne Θ A ( g ) as the ima g e q ( x ) = xK K = K K x of x under the quotient homomorphism q : Stab( K ) → H ( K ) = H ( A ). It remains to prov e that Θ A : H f → H ( A ) is a well-deﬁned top olog ical isomorphism of the right-topologic al g roups. First w e c heck that Θ A is well-deﬁned, that is Θ A ( g ) = q ( x ) do es not depend on the c hoice of the p oint x . Indeed, for any other p oint y ∈ X with g ( A ) = y − 1 A we get x − 1 A = y − 1 A and thus y x − 1 ∈ Fix( A ) = K · K where K = Fix − ( A ). Consequently , q ( x ) = K K x = K K y = q ( y ). Next, w e pro ve that Θ A is a gr oup ho momorphism. Given t w o functions g , h ∈ H f , ﬁnd elements x g , x h ∈ Fix( A ) suc h that h ( A ) = x − 1 h A and g ( A ) = x − 1 g A . It fo llows that g ◦ h ( A ) = g ( x − 1 h A ) = x − 1 h g ( A ) = x − 1 h x − 1 g A = ( x g x h ) − 1 A , which implies that Θ A ( g ◦ h ) = x g x h K K = Θ A ( g ) · Θ A ( h ). ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 23 Now, we ca lculate the k ernel of the homomorphism Θ A . T ake a ny function g ∈ H f with Θ A ( g ) = e , whic h means that g ( A ) = f g f ( A ) = A . Then for every A ′ ∈ T K we ca n ﬁnd x ∈ X with f ( A ′ ) = xA and conclude that g ( A ′ ) = f g f ( A ′ ) = f g ( xA ) = xf g ( A ) = xf g f ( A ) = xA = f ( A ′ ) witnessing that g = f g f = f . This mea ns that the homomo rphism Θ A is one-to-one . T o s ee that Θ A is onto, ﬁrst obser ve tha t each ele men t of the characteristic group H ( A ) can b e written as [ y ] = y K K = K K y ∈ H ( K ) for some y ∈ Stab( K ). Given s uch an element [ y ] ∈ H ( A ), co ns ider the equiv ariant function s [ y ] : ⌊ A ⌋ → ⌊ A ⌋ , s [ y ] : z A 7→ z y − 1 A = z y − 1 K K A . Let us sho w that this function is well deﬁned. Indeed, for each p oint u ∈ X with z A = u A , we get u − 1 z ∈ Fix( A ) and hence, y u − 1 z y − 1 ∈ y Fix( A ) y − 1 = y K K y − 1 = K K = Fix( A ). Then y u − 1 z y − 1 A = A and hence z y − 1 A = uy − 1 A . It follows fro m s [ y ] ◦ f = f ◦ s [ y ] ◦ f that the function s [ y ] ◦ f b elongs to the maximal gro up H f . Since s [ y ] ◦ f ( A ) = s [ y ] ( A ) = y − 1 A , the imag e Θ A ( s [ y ] ◦ f ) = [ y ]. So, Θ A ( H f ) = H ( A ) and Θ A : H f → H ( A ) is an a lgebraic isomor phis m. It rema ins to prove that this iso morphism is top olo gical. O bserve that for every [ y ] ∈ H ( A ) we g et s [ y ] ◦ f ( A ) = s [ y ] ( A ) = y − 1 K K A = y − 1 A . Conseq ue ntly , x ∈ s [ y ] ◦ f ( A ) iﬀ x ∈ y − 1 A iﬀ y ∈ Ax − 1 . T o see that the ma p Θ A : H f → H ( A ) is contin uous, take an y sub-basic op en s e t U x = { [ y ] ∈ H ( A ) : y ∈ Ax − 1 } , x ∈ Stab( K ) , in H ( A ) and obser ve that Θ − 1 A ( U x ) = { s [ y ] ◦ f : [ y ] ∈ U x } = { s [ y ] ◦ f : y ∈ Ax − 1 } = { s [ y ] ◦ f : x ∈ s [ y ] ◦ f ( A ) } is a sub-basic op en set in H ( f ). T o see that the inv erse map Θ − 1 A : H ( A ) → H f is contin uous, take a n y sub-ba sic op en set V x,T = { g ∈ H ( f ) : x ∈ g ( T ) } wher e x ∈ X and T ∈ T K . It follows that f ( T ) = x T A for so me x T ∈ X . The n Θ A ( V x,T ) = { [ y ] ∈ H ( A ) : x ∈ s [ y ] ◦ f ( T ) } = { [ y ] ∈ H ( A ) : x ∈ s [ y ] ( x T A ) } = = { [ y ] ∈ H ( A ) : x − 1 T x ∈ s [ y ] ( A ) } = { [ y ] ∈ H ( A ) : y ∈ Ax − 1 x T } is a sub- basic ope n set in H ( A ).  In the following pro po s ition we calculate the cardinalities of the ob jects app earing in Theore m 14.1. W e shall say that a cardinal n ≥ 1 divides a cardinal m ≥ 1 if there is a car dinal k s uc h that m = k × n . The smalle st cardinal k w ith this prop erty is denoted by m n . Prop ositio n 14.2. If K ∈ b K is a maximal 2-c o gr oup in a gr oup X , then (1) | T K | = 2 | X/K ± | ; (2) | H ( K ) | ∈ { 2 k : k ∈ N } ∪ {ℵ 0 } and | H ( K ) | divides the index | X/K | of K in X ; (3) | [ T K ] | = | T K | | H ( K ) | = 2 | X/K ± | | H ( K ) | ; (4) | H ( K ) | = | X/ K | if the 2-c o gr oup K is normal in X . Pr o of. Cho ose an y subset S ⊂ X that meets each coset K ± x , x ∈ X , o f the gr oup K ± = K ∪ K K at a single p oint. It is clear that | S | = | X/K ± | . 1. The equality | T K | = 2 | X/K ± | follows from Prop osition 12.1. 2. By Theor em 8.3, | H ( K ) | ∈ { 2 n : n ∈ N } ∪ {ℵ 0 } . Since Stab( K ) is a subg roup of X , | H ( K ) | = | Stab( K ) /K K | divides | X/K K | = | X/K | . 3. Since T K is a free H ( K )- a ct, | [ T K ] | = | T K | | H ( K ) | . This equality is clear if H ( K ) is ﬁnite. If H ( K ) is inﬁnite, then | H ( K ) | = ℵ 0 and the index | X/K ± | of the g roup K ± in X is inﬁnite. In this case | T K | = 2 | X/K ± | > ℵ 0 and thus | [ T K ] | = 2 | X/K ± | ℵ 0 = 2 | X/K ± | . 4. If the 2-cog roup K is normal in X , then Stab( K ) = X and H ( K ) = X/K K . In this cas e | H ( K ) | = | X/K K | = | X/K | .  By Theor em 14 .1(6), for any maximal 2-cog roup K ⊂ X each minimal left ideal of the semigro up End( T K ) is alge- braically iso mo rphic to H ( K ) × [ T K ]. It turns out that in some cas es this isomo rphism is top olo gical. W e recall that the o rbit space [ T K ] = T K / H ( K ) is endow ed with the quotient top ology . By Prop ositio n 12.2, the orbit space [ T K ] is compact and Hausdor ﬀ if and only if the characteristic group H ( K ) is ﬁnite. Since T K is a compa ct Hausdo rﬀ space, the T yc honoﬀ p ower T T K K is a compact Hausdorﬀ r ight top olo gical semig roup (endow ed with the op eration o f comp osition of functions). This semig roup contains the subse migroup C ( T K , T K ) co n- sisting of all contin uous maps f : T K → T K . It is e a sy to chec k that the semigroup C ( T K , T K ) is se mito po lo gical (which means that the semigroup op eration is sepa rately contin uous). W e reca ll that a right-topolo gical semig roup S is called semitop olo gic al if the semigroup op era tion S × S → S is separately contin uous. If the semigro up op eratio n is contin uous, then S is called a top olo gic al semigr oup . 24 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Theorem 14. 3 . L et K b e a maximal 2-c o gr oup in a gr oup X . F or a minimal idemp otent f in the semigr oup End( T K ) and its minimal left ide al L f = End( T K ) ◦ f the fol lowing c onditions ar e e quivalent: (1) L f is a top olo gic al semigr oup; (2) L f is top olo gic al ly isomorphi c to t he t op olo gic al semigr oup [ T K ] × H ( K ) wher e the orbit sp ac e [ T K ] is endowe d with the left zero multiplic ation; (3) L f is a semitop olo gic al semigr oup; (4) the left shift l f : L f → L f , l f : g 7→ f ◦ g , is c ont inu ous; (5) f is c ont inuous; (6) L f ⊂ C ( T K , T K ) ; (7) H ( K ) is ﬁ nite and the idemp otent b and E ( L f ) of L f is c omp act; Pr o of. The implicatio ns (2) ⇒ (1 ) ⇒ (3) ⇒ (4) ar e trivial. (4) ⇒ (5) Assume that the le ft shift l f : L f → L f is contin uous. W e ne e d to chec k that f is contin uous. First we show that for any set B ∈ f ( T K ) the preimage Z = f − 1 ( B ) is closed in T K . Assume conv ersely that f − 1 ( B ) is not close d and ﬁnd a p oint A 0 ∈ Z \ Z . It follows that the set B 0 = f ( A 0 ) is not equal to B . Let ϕ : ⌊ B ⌋ → [ A 0 ] b e a unique equiv ar iant function such that ϕ ( B ) = A 0 . Then the function g 0 = ϕ ◦ f b elongs to the minimal left ideal L f . Observe tha t f ◦ g 0 ( B ) = f ( A 0 ) = B 0 6 = B . Since the left shift l f is con tinuous, for the neighbo rho o d O ( f ◦ g 0 ) = { h ∈ L f : h ( B ) 6 = B } of f ◦ g 0 = l f ( g 0 ) there is a neighborho o d O ( g 0 ) ⊂ L f such that f ◦ g ⊂ O ( f ◦ g 0 ) for ev ery g ∈ O ( g 0 ). It follows fro m the equiv ar iantness of g 0 = g 0 ◦ f and the deﬁnition of the top olog y (of p oint wise conv ergence) on L f ⊂ T T K K that the po in t g 0 ( B ) = A 0 of T K has a neighborho o d O ( A 0 ) ⊂ T K such that each function g ∈ L f with g ( B ) ∈ O ( A 0 ) b elongs to the neighborho o d O ( g 0 ). Since A 0 is a limit po in t of the set Z , there is a set A ∈ O ( A 0 ) ∩ Z . F or this set ﬁnd an equiv ariant function g = g ◦ f such that g ( B ) = A . Then g ∈ O ( g 0 ) and hence f ◦ g ( B ) 6 = B , which con tradicts g ( B ) = A ∈ f − 1 ( B ). This contradiction prov es that all preimag es f − 1 ( B ), B ∈ f ( T K ), are clo sed in T K . Next, we s how that each or bit ⌊ A ⌋ , A ∈ T K , is discrete. Assume conversely that so me orbit ⌊ A ⌋ is not discrete and consider its closure ⌊ A ⌋ in the compact Hausdorﬀ space T K . The orbit ⌊ A ⌋ has no isola ted p oints, be ing no n-discrete and to p olo gically ho mogeneous. Fix any B ∈ f ( T K ). By Theo rems 3.1(2) and 8.3, the ima g e f ( T K ) has cardinality | f ( T K ) | = |⌊ B ⌋| = | H ( K ) | ≤ ℵ 0 . Then we can write the compa c t space ⌊ A ⌋ as a coun table union ⌊ A ⌋ = [ B ∈ f ( T K ) f − 1 ( B ) ∩ ⌊ A ⌋ of closed subsets. By Bair e’s Theorem, for so me B ∈ f ( T K ) the set ⌊ A ⌋ ∩ f − 1 ( B ) has no n-empt y interior in ⌊ A ⌋ . Since the orbit ⌊ A ⌋ has no isolated p oints, the intersection ⌊ A ⌋ ∩ f − 1 ( B ) is inﬁnite, whic h is not p ossible as f is equiv a riant. Finally , w e show that for every B ∈ f ( T K ) the pre image f − 1 ( B ) is op en in L f . Assuming the opp os ite, we can ﬁnd a po in t A 0 ∈ f − 1 ( B ) that lies in the closure o f the set T K \ f − 1 ( B ). Cho ose any equiv ar iant function g 0 ∈ L f such that g 0 ( B ) = A 0 and obse r ve that f ◦ g 0 ( B ) = B . Since the orbit ⌊ B ⌋ of B is discrete, we can ﬁnd an op en neigh bo rho o d O ( B ) ⊂ T K of B such that O ( B ) ∩ ⌊ B ⌋ = { B } . This neighbo r ho o d determines a neighbo r ho o d O ( f ◦ g 0 ) = { g ∈ L f : g ( B ) ∈ O ( B ) } of the function f ◦ g 0 in L f ⊂ T T K K . Since the left shift l f : L f → L f is contin uous, the function g 0 has a neighbo r ho o d O ( g 0 ) ⊂ L f such that l f ( O ( g 0 )) ⊂ O ( f ◦ g 0 ). By the deﬁnition of the top ology (of p oint wise conv ergence) on L f , there is a neighborho o d O ( g 0 ( B )) ⊂ T K such that each function g ∈ L f with g ( B ) ∈ O ( g 0 ( B )) b elongs to O ( g 0 ). By the choice of the po in t A 0 = g 0 ( B ), there is a set A ∈ O ( g 0 ( B )) \ f − 1 ( B ). F or this set choo se an equiv ariant function g ∈ L f such that g ( B ) = A . This function g belo ngs to O ( g 0 ) and thus f ◦ g ∈ O ( f ◦ g 0 ), which means that f ◦ g ( B ) = B . But this co nt radicts g ( B ) = A / ∈ f − 1 ( B ). Thu s for each B ∈ f ( T K ) the preima ge f − 1 ( B ) is op en in T K , which implies that the function f : T K → T K is contin uous. (5) ⇒ (6) Assume that f is contin uous. Then fo r any B ∈ T K the o rbit ⌊ B ⌋ = f ( T K ) is co mpact (as a contin uous image of the co mpact space T K ). Being a compact top olog ically homogeneous space of cardina lit y |⌊ B ⌋| ≤ | H ( K ) | ≤ ℵ 0 , the orbit ⌊ B ⌋ = f ( T K ) is ﬁnite. Then for each g ∈ L f the restrictio n g |⌊ B ⌋ is contin uous a nd hence g = g ◦ f is contin uous as the co mpos ition of tw o cont inuous maps f and g |⌊ B ⌋ . (6) ⇒ (7) Assume that L f ⊂ C ( T K , T K ). Then f is contin uous. Rep ea ting the argument from the preceding item, w e can show that the characteristic group H ( K ) is ﬁnite. By the contin uit y of f , for every B ∈ T K the pr eimage f − 1 ( B ) is closed in T K . In the fo llowing claim E ( L f ) stands for the idempo ten t band of the semigroup L f . Claim 14.4. E ( L f ) = { g ∈ L f : f ◦ g ( B ) = B } . Pr o of. If g ∈ L f is a n idemp otent, then for the unique p oint C ∈ g ( T K ) ∩ f − 1 ( B ) we g et C = g ( C ) and then B = f ( C ) = f g ( C ) = f g f ( C ) = f g ( B ). ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 25 Now ass ume conv ersely that g ∈ L f is a function with f g ( B ) = B . Let C = g ( B ) ∈ g ( T K ). Then g ( C ) = g f ( C ) = g f g ( B ) = g ( B ) = C . F or every A ∈ T K we can ﬁnd x ∈ X such that g ( A ) = xC and then g g ( A ) = g ( xC ) = xg ( C ) = xC = g ( A ), which means that g is an idempo ten t.  Since the set f − 1 ( B ) ⊂ T K is clos ed and the calculatio n map c B : L f → T K , c B : g 7→ g ( B ) , is con tinuous, the preimage c − 1 B ( f − 1 ( B )) is closed in L f . By Claim 14.4, this preima g e is equal to the idemp otent ba nd E ( L f ) of the se mig roup L f . (7) ⇒ (2) Assume that the gro up H ( K ) is ﬁnite and the idemp otent band E ( L f ) is compact. By Pro po sition 12.2, the orbit space [ T K ] is compact, Hausdorﬀ and zero - dimensional, and the quo tien t map q : T K → [ T K ] is contin uous a nd op en. W e claim that for every B ∈ f ( T K ) the preimage f − 1 ( B ) ⊂ T K is co mpact. Since the idemp otent ba nd E ( L f ) is compact and the ca lc ulation map c B : L f → T K , c B : g 7→ g ( B ), is contin uous, the imag e c B ( E ( L f )) is compact. By Claim 14.4, c B ( E ( L f )) ⊂ f − 1 ( B ). T o show the reverse inclusion, ﬁx any subset A ∈ f − 1 ( B ) and c ho ose a ny equiv aria n t map ϕ : ⌊ B ⌋ → ⌊ A ⌋ such that ϕ ( B ) = A . Then the map g = ϕ ◦ f b elongs to L f and is an idemp otent by Claim 14.4. Since A = g ( B ), we see that f − 1 ( B ) ⊂ c B ( E ( L f )) and hence f − 1 ( B ) = c B ( E ( L f )) is co mpact. Fix any set B ∈ f ( T K ). Since | f ( T K ) | = |⌊ B ⌋| = | H ( K ) | < ℵ 0 , the preimag e Z = f − 1 ( B ) = T K \ [ B 6 = A ∈⌊ B ⌋ f − 1 ( A ) is op en-a nd- closed in T K . Since the compact space Z mee ts ea ch orbit ⌊ A ⌋ , A ∈ T K , a t a single p oint, the restric tio n q |Z : Z → [ T K ], b eing contin uous and bijectiv e, is a homeomor phism. So, it suﬃces to prov e that L f is top ologic a lly isomorphic to Z × H ( K ) where the space Z is endow ed with the le ft z e r o multiplication. Deﬁne an isomorphis m Φ : Z × H ( K ) → L f assigning to each pair ( Z , x ) ∈ Z × H ( K ) the function g Z,x ◦ f where g Z,x : ⌊ B ⌋ → ⌊ Z ⌋ is the unique equiv ar iant function such that g Z,x ( B ) = x − 1 Z . It is eas y to chec k that Φ is a top olog ical isomor phism b etw een Z × H ( K ) and L f .  In the following prop osition we prov e the existence of co nt inuous o r discontin uous minimal idemp otents in the semigr oup End( T K ). Let us recall that for a left-in v ar iant idea l I on a g roup X by End I ( T K ) we denote the left idea l in End( T K ) consisting of all equiv ar iant I - saturated functions. Prop ositio n 14.5. L et I b e a left- invariant ide al on a gr oup X and assum e that a maximal 2-c o gr oup K ⊂ X has ﬁnite char acteristic gr oup H ( K ) . Then the semigr oup End I ( T K ) c ontains: (1) a c ontinuous minimal idemp otent if K x / ∈ I for al l x ∈ X ; (2) no c ontinuous function if K x ∈ I for al l x ∈ X ; (3) no disc ont inuous function (which is a minimal idemp otent) if (and only if ) for e ach A ∈ T K the set ¯ ¯ A I ∩ T K is op en in T K . Pr o of. By Prop ositio n 12.2, the orbit space [ T K ] is c o mpact, Hausdo rﬀ, and zero -dimensional and the orbit map q : T K → [ T K ] has a con tinu ous section s : [ T K ] → T K . Then Z = s ([ T K ]) is a clo s ed s ubset of T K that meets each orbit ⌊ A ⌋ , A ∈ T K , at a single p oint. Pick any B ∈ Z a nd deﬁne a co n tinuous minimal idempo ten t f : T K → T K letting f ( xZ ) = xB for each x ∈ H ( K ) and Z ∈ Z . 1. Assuming that K x / ∈ I for all x ∈ X , we shall show that the function f is I -saturated and hence b e longs to End I ( T K ). Given any sets A, B ∈ T K with A = I B , we need to show that f ( A ) = f ( B ). W e shall prove mo re: A = B . Assume co n versely that A 6 = B and ﬁnd a p oint x ∈ A △ B in the s y mmetric diﬀerence A △ B = ( A \ B ) ∪ ( B \ A ). Since K K A = Fix( A ) A = A a nd K K B = Fix( B ) B = B , we get K K x ∈ A △ B ∈ I and then for e very y ∈ K , we g et K y x = K K x ∈ I , which contradicts our assumption. 2. Now assume that K x ∈ I for all x ∈ X . W e shall pr ov e tha t no function g ∈ End I ( T K ) is c o nt inuous. F or this we show that for each A ∈ T K the set ¯ ¯ A I ∩ T K is dense in T K . Given a n y set C ∈ T K and a neighborho o d O ( C ) of C in T K , w e need to ﬁnd a set B ∈ O ( C ) such that B = I A . By the deﬁnition of the to p olo gy on T K ⊂ P ( X ), there is a ﬁnite subset F ⊂ X such that O ( C ) ⊃ { B ∈ T K : B ∩ F = C ∩ F } . Now we see that the set B = ( A \ K ± F ) ∪ ( K ± F ∩ C ) ∈ T K belo ngs to the neig hborho o d O ( C ) and B = I A b ecause A △ B ⊂ K ± F ∈ I . Assuming that some I -satura ted equiv ar ia nt function g : T K → T K , is con tinuous, w e conclude tha t the preima ge g − 1 ( f ( A )) ⊃ ¯ ¯ A I ∩ T K coincides with T K , b eing a closed dense subset o f T K . So, g is constant. Since the actio n of the (non-triv ial) gro up H ( K ) on T K is fr ee, the constant map g ca nnot be equiv ar iant. 3. If for ev ery A ∈ T K the set ¯ ¯ A I ∩ T K is op en in T K , then each I -sa turated function is lo cally constant and hence contin uous. So, End I ( T K ) contains no discontin uous function. 26 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Now ass uming that for some A ∈ T K the set A = ¯ ¯ A I ∩ T K is not op en in T K , we sha ll construct a discontin uous minimal idemp otent f ∈ End I ( T K ). T ake any minimal idemp o tent f ∈ End I ( T K ). If f is disco nt inuous, we are done. So assume that f is contin uous a nd ﬁx a n y s et B ∈ f ( T K ). By Theorem 1 4.1(5), the imag e f ( T K ) = ⌊ B ⌋ is ﬁnite. So, the preimage Z = f − 1 ( f ( B )) is op en-and-clo s ed in T K . T ak e any x ∈ Sta b( K ) \ K K and consider the subse t Z ′ = ( Z \ A ) ∪ x A which is not compact as A is not open in Z . Then the I -satura ted minimal idemp otent g : T K → T K deﬁned by g ( xZ ) = xB for x ∈ H ( K ) and Z ∈ Z ′ is disco ntin uous (beca use g − 1 ( B ) = Z ′ is not closed in T K ).  Corollary 14.6. F or a maximal 2-c o gr oup K ⊂ X and a left-invariant ide al I on X the fol lowing c onditions ar e e quivalent: (1) e ach minimal left ide al of End I ( T K ) is a t op olo gic al semigr oup; (2) e ach minimal left ide al of End I ( T K ) is a s emitop olo gic al semigr oup; (3) for e ach A ∈ T K the set ¯ ¯ A I ∩ T K is op en in T K . If the ide al I is right-invariant, then the c onditions (1)–(3) ar e e quivalent to (4) K has ﬁnite index in X ; (5) the semigr oup End( T K ) is ﬁnite. Pr o of. The equiv alence (1) ⇔ (2) ⇔ (3 ) follows from Theorem 14.3 and Pr op osition 14.5. Now a ssume that the left-inv ariant ideal I is right-in v ar iant. (3) ⇒ (4) Assume tha t for each A ∈ T K the set ¯ ¯ A I ∩ T K is op en in T K . Then it is also clo sed in T K being the complements of the union of o p en subse ts ¯ ¯ B I ∩ T K for B 6 = I A . By P rop osition 1 4.5, K x / ∈ I for s ome x ∈ X . Since the ideal I is right-in v ar iant, K x / ∈ I for all x ∈ X . W e claim that the set ¯ ¯ A I ∩ T K = { A } . Assuming that ¯ ¯ A I ∩ T K contains a set B distinct from A , we can ﬁnd a p oint x ∈ A △ B . Since Fix( A ) = K K = Fix( B ), we get K K x ⊂ K K A △ K K B = A △ B ∈ I and thus for any po in t z ∈ K , we arrive to the abs ur d conclusion K z x = K K x ∈ I . Since the s ingleton ¯ ¯ A I ∩ T K = { A } in op en in the s pa ce T K , which is homeomorphic to 2 X/K ± , the index of the group K ± in X is ﬁnite and so is the index of K in X . The implicatio ns (4) ⇒ (5) ⇒ (1) a re trivial.  15. The semigroup End λ ( T K ) In the preceding s ection we s tudied the contin uit y of the semigro up opera tion on minimal left ideals o f the semigro up End( T K ). In this sectio n we shall b e interested in the c o nt inuit y of the se mig roup op eration on the se mig roup End λ ( T K ) ⊂ End( T K ). This will b e done in a more gener al c o nt ext of upper subfamilies F ⊂ T . W e deﬁne a family F ⊂ T to b e upp er if for any t win set A ∈ F a nd a twin subset B ⊂ X with Fix − ( A ) ⊂ Fix − ( B ), w e get B ∈ F . Let us rema rk that b T is a n upper subfamily of T while T K is a minimal upper subfamily of T for every K ∈ b K . Prop ositio n 15.1 . Each u pp er subfamily F ⊂ T is symmetric and λ -invariant. Conse quently, End λ ( F ) is a c omp act right-top olo gic al semigr oup. Pr o of. T o prov e that F ⊂ T is symmetr ic , given any set A ∈ F c hoose a p oint x ∈ Fix − ( A ). By Prop os ition 5.4, Fix − ( xA ) = x Fix − ( A ) x − 1 = Fix − ( A ) and hence X \ A = xA ∈ F . T o see tha t F is λ -in v ar iant, we need to sho w that ϕ ( F ) ⊂ F for any function ϕ ∈ End λ ( P ( X )). By Co rollary 4.4, the function f is sy mmetr ic and left-in v aria n t. Then for each A ∈ F and x ∈ Fix − ( A ) we get xϕ ( A ) = ϕ ( xA ) = ϕ ( X \ A ) = X \ ϕ ( A ) a nd hence x ∈ Fix − ( ϕ ( A )). Since Fix − ( A ) ⊂ Fix − ( ϕ ( A )), the set ϕ ( A ) belo ngs to F by the deﬁnition of a n upper family .  Theorem 15 . 2. F or an upp er subfamily F ⊂ T the fol lowing c onditions ar e e quivalent: (1) End λ ( F ) is a top olo gic al semigr oup; (2) End λ ( F ) is a semitop olo gic al semigr oup; (3) for e ach t win set A ∈ F the sub gr oup Fix( A ) has ﬁnite index in X . Pr o of. (3) ⇒ (1) Ass ume that for ea ch twin set A ∈ F the s tabilizer Fix( A ) has ﬁnite index in X . T o show that the semigroup o per ation ◦ : End λ ( F ) × End λ ( F ) → End λ ( F ) is c o nt inuous, ﬁx any tw o functions f , g ∈ E nd λ ( F ) a nd a neighborho o d O ( f ◦ g ) of their comp osition. W e should show tha t the functions f , g have neighbo r ho o ds O ( f ) , O ( g ) ⊂ End λ ( F ) suc h that O ( f ) ◦ O ( g ) ⊂ O ( f ◦ g ). W e lose no generality assuming that the neigh bor ho o d O ( f , g ) is of sub-basic form: O ( f ◦ g ) = { h ∈ End λ ( F ) : x ∈ h ( A ) } for some x ∈ X and some t win set A ∈ F . Let B = g ( A ). It fo llows from f ◦ g ∈ O ( f ◦ g ) that x ∈ f ◦ g ( A ) = f ( B ). Let O ( f ) = { h ∈ End λ ( F ) : x ∈ h ( B ) } . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 27 The deﬁnition of a neighborho o d O ( g ) is a bit more complicated. By our hypothesis, the stabilize r Fix ( A ) has ﬁnite index in X . Let S ⊂ X b e a (ﬁnite) subset meeting each coset Fix( A ) z , z ∈ X , at a single point. Co nsider the following op en neighbor ho o d of g in End λ ( F ): O ( g ) = { g ′ ∈ End λ ( F ) : ∀ s ∈ S ( s ∈ B ⇔ s ∈ g ′ ( A )) } . W e claim that O ( f ) ◦ O ( g ) ⊂ O ( f ◦ g ). Indeed, take any functions f ′ ∈ O ( f ) and g ′ ∈ O ( g ). By Theorem 11.1, Fix − ( A ) ⊂ Fix − ( g ′ ( A )) a nd hence Fix( A ) ⊂ Fix( g ′ ( A )). Then g ′ ( A ) = Fix( A ) · ( S ∩ g ′ ( A )) = Fix( A ) · ( S ∩ B ) = B and th us x ∈ f ′ ( B ) = f ′ ◦ g ′ ( A ) witnessing that f ′ ◦ g ′ ∈ O ( f ◦ g ). The implication (1) ⇒ (2) is trivial. (2) ⇒ (3) As s ume that X contains a twin subset T 0 ∈ F whose s ta bilizer Fix( T 0 ) has inﬁnite index in X . Then the subgroup H = Fix ± ( T 0 ) also has inﬁnite index in X . B y Theorem 15.5 of [1 8], X 6 = F H F for any ﬁnite subset F ⊂ X . Lemma 15. 3. Ther e ar e c ountable sets A, B ⊂ X such that (1) xB ∩ y B = ∅ for any distinct x, y ∈ A ; (2) | AB ∩ H z | ≤ 1 for al l z ∈ X ; (3) e ∈ A , AB ∩ H = ∅ . Pr o of. Let a 0 = e and B < 0 = { e } . Inductively we shall co nstruct sequences A = { a n : n ∈ ω } and B = { b n : n ∈ ω } suc h that • b n / ∈ A − 1 ≤ n H A ≤ n B i . Then a n b i = a m b j implies that b j = a − 1 m a n b i ∈ A − 1 ≤ j A ≤ j B m . W e claim that b k / ∈ a − 1 n U ∪ a − 1 m U . Otherwise, b k ∈ a − 1 n a i ( V i ∩ B ≥ i ) ∪ a − 1 m a i ( V i ∩ B ≥ i ) for some i ∈ ω and hence b k = a − 1 n a i b j or b k = a − 1 m a i b j for some even j ≥ i . If k > j , then b oth the equalities ar e forbidden by the choice of b k / ∈ A − 1 ≤ k A ≤ k B ℵ 0 . So, Fix( T ) has ﬁnite index in X and the implication (3) ⇒ (1) of Theorem 15.2 gua rantees that End λ ( F ) is a top ological semigr oup. Now we show that this semigr o up is metr izable. First observe that for every T ∈ F the s et End λ ( { T } ) = { ϕ |{ T } : ϕ ∈ End λ ( P ( X )) } ha s ﬁnite c a rdinality | End λ ( { T } ) | = |{ ϕ ( T ) : ϕ ∈ End λ ( P ( X )) }| ≤ |{ A ∈ T : Fix( A ) ⊃ Fix( T ) }| . Since the fa mily F is coun table, the spac e E nd λ ( F ) ⊂ Q T ∈ F End λ ( { T } ) is metriza ble, being a subspace of the countable pro duct of ﬁnite discrete spaces. The implication (2) ⇒ (1) is trivial. (1) ⇒ (3) Ass uming that the family F is not countable, w e s hall show that the space End λ ( F ) is not metrizable. W e consider tw o cases. ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 29 (a) F or some t win set T ∈ F the stabilizer Fix( T ) has inﬁnite index. Then we can ﬁnd an inﬁnite set S ⊂ X tha t int ersects ea ch coset Fix ± ( T ) x , x ∈ X , a t a single p oint. As we already know, for each subse t E ⊂ S the s e t T E = Fix ( T ) · E ∪ Fix − ( T ) · ( S \ E ) belo ngs to the family F . No w take any tw o distinct ultra ﬁlters U , V ∈ β ( S ) ⊂ β ( X ) and consider their function rep- resentations f U = Φ F ( U ) and f V = Φ F ( V ). Since U 6 = V , there is a subs e t E ⊂ S such that E ∈ U \ V . It follows that T E ∈ U and T S \ E ∈ V , which implies T E / ∈ V and hence e ∈ f U ( T E ) \ f V ( T E ). This means that f U 6 = f V and consequently , | End λ ( F ) | ≥ | β ( S ) | ≥ 2 c , which implies that the co mpact space End λ ( F ) is not metrizable (be cause each metrizable compact spa ce has cardina lit y ≤ c ). (b) F or each T ∈ F the subgroup Fix( T ) ha s ﬁnite index in X . Then each set T ∈ F has ﬁnite orbit [ T ] = { xT : x ∈ X } . Consider the smallest left-inv ariant family ¯ F = S T ∈ F [ T ] that contains F . By P rop osition 1 3.4, the fa mily ¯ F is {∅} - independent. Since e a ch orbit [ T ], T ∈ ¯ F , is ﬁnite a nd F is uncountable, the or bit space [ ¯ F ] = { [ T ] : T ∈ F } also is uncountable. It follows fr om Theorem 11 .1 that the space End λ ( ¯ F ) is ho meomorphic to the pro duct Q [ T ] ∈ [ ¯ F ] End λ ([ T ]) where ea ch space E nd λ ([ T ]) cont ains a t lea st tw o eq uiv aria n t functions: ident ity i : [ T ] → [ T ] , i : A 7→ A and ant ip o dal α : [ T ] → [ T ], α : A 7→ X \ A . Since the orbit space [ ¯ F ] is uncountable, the pro duct Q [ T ] ∈ [ ¯ F ] End λ ([ T ]) is non-metrizable and so is its top ologica l cop y End λ ( ¯ F ). It rema ins to observe that the restriction map R : End λ ( ¯ F ) → End λ ( F ) is injective and th us a homeomo rphism. Indeed, given tw o distinct equiv ariant functions f , g ∈ End λ ( F ), w e can ﬁnd a set A ∈ ¯ F with f ( A ) 6 = g ( A ). Since [ A ] ∩ F 6 = ∅ , there is x ∈ X such that xA ∈ F . Then f ( xA ) = xf ( A ) 6 = xg ( A ) = g ( xA ) and thus f | F 6 = g | F .  The following prop osition characterizes groups co nt aining only co un tably many twin subsets. F ollowing [2 ], we deﬁne a group X to b e o dd if each e le men t x ∈ X has o dd order. Prop ositio n 15. 6. The family T of twin subsets of a gr oup X is at most c ountable if and only if e ach sub gr oup of inﬁnite index in X is o dd. Pr o of. Assume that each subg roup of inﬁnite index in X is o dd. W e claim that for e very A ∈ T the subgroup Fix( A ) has ﬁnite index in X . T ake any p oint c ∈ Fix − ( A ) and co nsider the cyclic subgr oup c Z = { c n : n ∈ Z } g e nerated by c . The subgroup c Z has ﬁnite index in X , being non-o dd. Since c 2 Z = { c 2 n : n ∈ Z } ⊂ Fix( A ), we conclude that Fix( A ) also has ﬁnite index in X . Next, we show that the family { Fix( A ) : A ∈ T } is at most countable. This is trivially true if T = ∅ . If T 6 = ∅ , then we can tak e any A ∈ T and c hoos e a po int c ∈ Fix − ( A ). The cyclic subgro up c Z generated b y c is not o dd and hence has ﬁnite index in X . Consequently , the group X is at most co untable. Now it remains to c hec k tha t for every x ∈ X the set T x = { A ∈ T : x ∈ Fix − ( A ) } is ﬁnite. If the set T x is not empt y , then the cyclic subgroup x Z generated by x is not o dd and hence has ﬁnite index in X . Cons ider the subgroup x 2 Z of index 2 in x Z . It is clear that x 2 Z ⊂ Fix( A ) . Let S ⊂ X be a ﬁnite set containing the neutral element of X and meeting each coset x 2 Z z , z ∈ X at a s ingle point. It follows fro m x 2 Z ⊂ Fix( A ) that A = x 2 Z · ( S ∩ A ) and consequently | T x | ≤ 2 | S | < ∞ . Now assume that some s ubgroup H of inﬁnite index in X is no t odd. Then H contains an element c ∈ H such that the sets c 2 Z = { c 2 n : n ∈ Z } and c 2 Z +1 = { c 2 n +1 : n ∈ Z } are disjoint. The union c 2 Z ∪ c 2 Z +1 coincides with the cyclic subgroup c Z of H ge nerated b y c . Find a set S ⊂ X that in tersects each coset c Z x , x ∈ X , at a single p oint. Since c 2 Z has inﬁnite index in X , the set S is inﬁnite. Now observe that for every E ⊂ S the union T E = c 2 Z · E ∪ c 2 Z +1 · ( S \ E ) is a twin set with c ∈ Fix − ( T E ). Consequently , T ⊃ { T E : E ⊂ S } has ca rdinality | T | ≥ |{ T E : E ⊂ S }| ≥ | 2 S | ≥ c > ℵ 0 .  Now we shall apply the ab ov e results to the minimal upp er s ubfamilies T K with K ∈ b K . By Theorem 14 .1(1), fo r a maximal 2- cogroup K in a group X minimal left ideals of End( T K ) are metrizable if and only if | X/ K | ≤ ℵ 0 . The metrizability of the whole semig r oup End( T K ) is equiv alen t to | X/K | < ℵ 0 . Theorem 15 . 7. F or a maximal 2-c o gr oup K of a gr oup X the fol lowing c onditions ar e e quivalent: (1) End λ ( T K ) is metrizable; (2) End λ ( T K ) is a s emitop olo gic al semigr oup; (3) End λ ( T K ) is a ﬁ nite semigr oup; (4) End λ ( T K ) is isomorph ic t o C 2 k ≀ m m or Q 2 k ≀ m m for some 1 ≤ k ≤ m < ∞ ; (5) K has ﬁnite index in X . 30 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Pr o of. The implicatio ns (1) ⇒ (2 ) ⇒ (5) follo w from Theorems 15.5 and 15.2. (5) ⇒ (4) Assume that K has ﬁnite index in X . Then the characteristic group H ( K ) of K is ﬁnite and hence is isomorphic to C 2 k or Q 2 k fo r some k ∈ N , see Theor em 8.3. Also the set T K is ﬁnite and so is the orbit space [ T K ]. By Theorem 14.1(3), the semig roup End λ ( T K ) is isomor phic to H ( K ) ≀ [ T K ] [ T K ] and the la tter semigr oup is isomorphic to C 2 k ≀ m m or Q 2 k ≀ m m for m = | [ T K ] | . The implications (4 ) ⇒ (3) ⇒ (1) a re trivial.  16. Co n s tructing nice idempotents in the semigroup End λ ( P ( X )) In this section we prov e the existence so me specia l ide mp otents in the semigro up End λ ( P ( X )). These ide mp otents will help us to describ e the str ucture of the minimal ideal of the s emigroup End λ ( P ( X )) and λ ( X ) in Theorems 17.1 and Corollar y 17.2. In this section w e assume that I is a left-inv ariant ideal in a g roup X . W e recall that pT I and T I denote the families of I -pretwin and I -twin subsets o f X , resp ectively . A function f : F → P ( X ) deﬁned on a subfamily F ⊂ P ( X ) is called I -satura te d if f ( A ) = f ( B ) for an y sets A = I B in F . Prop ositio n 16.1. Ther e is an idemp otent e I ∈ End λ ( P ( X )) su ch that • e I ( P ( X ) \ pT I ) ⊂ {∅ , X } ; • e I | pT I = id | pT I ; • the function e I r est ricte d to P ( X ) \ pT I is I -satur ate d. Pr o of. Consider the family ↔ N I 2 ( X ) ⊂ P 2 ( X ) of left in v ar iant I -satur ated link ed s ystems on X , partially order e d b y the inclusion relation. This set is not empt y because it co nt ains the inv aria nt I -sa turated link ed system { X \ A : A ∈ I } . By Zorn’s Lemma, the pa rtially o rdered set ↔ N I 2 ( X ) contains a maximal element L , whic h is a maximal inv ariant I -satur a ted linked s ystem on the group X . By the maxima lit y , the system L is monotone. Now c o nsider the family L ⊥ = { A ⊂ X : ∀ L ∈ L ( A ∩ L 6 = ∅ ) } . Claim 16.2. L ⊥ \ L ⊂ pT I . Pr o of. Fix a ny set A ∈ L ⊥ \ L . First we chec k that xA ∩ A ∈ I for some x ∈ X . A ssuming the conv erse, we would conclude that the family A = { A ′ ⊂ X : ∃ x ∈ X ( A ′ = I xA ) } is inv aria n t, I -saturated and linked, and so is the union A ∪ L , whic h is not p oss ible by the maximality o f L . So, there is x ∈ X with xA ∩ A ∈ I , which is equiv alent to xA ⊂ I X \ A . Next, we ﬁnd y ∈ X such that A ∪ y A = I X , whic h is equiv alen t to X \ A ⊂ I y A . Assuming that no suc h a p oint y exists, we conclude that for a n y x, y ∈ X the union xA ∪ y A 6 = I X . Then ( X \ xA ) ∩ ( X \ y A ) = X \ ( xA ∪ y A ) / ∈ I , which mea ns that the family B = { B ⊂ X : ∃ x ∈ X ( B = I X \ xA ) } is inv ariant I - saturated and linked. W e claim that X \ A ∈ L ⊥ . Assuming the conv erse, we would conclude that X \ A misses some set L ∈ L . Then L ⊂ A and hence A ∈ L whic h is not the case . Thus X \ A ∈ L ⊥ . Since L is inv ariant and I -saturated, B ⊂ L ⊥ and consequently , the union B ∪ L , b eing an inv aria n t I -satur ated linked system, coincides with L . Then X \ A ∈ L , whic h contradicts A ∈ L ⊥ . This contradiction shows that X \ A ⊂ I y A for some y ∈ X . Since xA ⊂ I X \ A ⊂ I y A , the set A is I -pretwin.  Consider the function repres e ntation Φ L : P ( X ) → P ( X ) of L . By Pr o po sitions 4.3 and 4 .5, the function Φ L is equiv ar iant, mo notone, I -satura ted, and Φ L ( P ( X )) ⊂ {∅ , X } . It is clear that the function e I : P ( X ) → P ( X ) deﬁned b y e I ( A ) = ( A if A ∈ pT I , Φ L ( A ) otherwise has prop erties (1 )–(3) of Pr op osition 16.1. It is also clear that e I = e I ◦ e I is an idemp otent. W e claim that e I ∈ End λ ( P ( X )). By Corollary 4 .4, we need to check that e I is e q uiv aria nt , mo notone and symmetric. The equiv ariance of e I follows from the equiv a riance of the ma ps Φ L and id. T o show that e I is monoto ne, take any t wo subsets A ⊂ B of X and cons ide r four cases. 1) If A, B / ∈ pT I , then e I ( A ) = Φ L ( A ) ⊂ Φ L ( B ) = e I ( B ) by the monotonicity of the function representation Φ L of the monotone family L . 2) If A, B ∈ pT I , then e I ( A ) = A ⊂ B = e I ( B ). 3) A ∈ pT I and B / ∈ pT I . W e cla im that B ∈ L . Assuming that B / ∈ L and applying Cla im 16.2, we get B / ∈ L ⊥ . Then B do e s not intersect some set L ∈ L and then A ∩ L = ∅ . It follows that the set X \ A ⊃ L belo ngs to the maxima l ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 31 inv ariant I - saturated linked system and so do es the set y A ⊃ I X \ A for some y ∈ X (which exists as A ∈ pT I ). By the left-in v ar iance o f L , we ge t A ∈ L which contradicts X \ A ∈ L and the link edness o f L . This cont radiction prov es that B ∈ L . In this cas e e I ( A ) = A ⊂ X = Φ L ( B ) = e I ( B ). 4) A / ∈ pT I and B ∈ pT I . In this ca se w e prov e that A / ∈ L . Assuming conv ersely that A ∈ L , we g e t B ∈ L . Since B ∈ pT I , there is a po in t x ∈ X with xB ⊂ I X \ B . Since L is left-inv ariant, monotone and I - saturated, we conc lude that X \ B ∈ L whic h contradicts B ∈ L . Thus A / ∈ L and e I ( A ) = Φ L ( A ) = ∅ ⊂ e I ( B ). Finally , we show that the function e I is symmetric . If A ∈ pT I , then X \ A ∈ pT I and then e I ( X \ A ) = X \ A = X \ e I ( A ). Next, assume that A / ∈ pT I . If A ∈ L , then X \ A / ∈ L b y the linkedness of L . In this case e I ( X \ A ) = ∅ = X \ X = X \ e I ( A ). If A / ∈ L , then by Claim 1 6.2, A / ∈ L ⊥ and thus A is disjoint with so me set L ∈ L , whic h implies that X \ A ∈ L . Then e I ( X \ A ) = Φ L ( X \ A ) = X = X \ ∅ = X \ Φ L ( A ) = X \ e I ( A ).  Our second sp ecial idempotent dep ends on a s ubfa mily e T of the family b T = { A ∈ T : Fix − ( A ) ∈ b K} of twin sets with maximal 2 -cogro up. Theorem 16 . 3. If the ide al I is twinic, then for any I - indep endent b K - c overing subfamily e T ⊂ b T ther e is an idemp otent e e T ∈ End I λ ( P ( X )) su ch that (1) e e T ( P ( X ) \ T I ) ⊂ {∅ , X } ; (2) e e T ( T I ) = e T ; (3) e e T |{∅ , X } ∪ e T = id . Pr o of. Let e I : P ( X ) → {∅ , X } ∪ pT I be the idemp otent from Pro po sition 16 .1. Since the ideal I is twinic, pT I = T I . The idemp otent e e T will b e deﬁned as the comp ositio n e e T = ϕ ◦ e I where ϕ : {∅ , X } ∪ T I → {∅ , X } ∪ e T is a n equiv ar iant I -sa tur ated function such that (1) ϕ ◦ ϕ = ϕ ; (2) ϕ |{∅ , X } ∪ e T = id; (3) ϕ ( T I ) ⊂ e T ; (4) I - Fix − ( A ) ⊂ Fix − ( ϕ ( A )) for all A ∈ T I . T o construct such a function ϕ , co nsider the family F of all p os s ible functions ϕ : D ϕ → {∅ , X } ∪ e T such that (a) {∅ , X } ∪ e T ⊂ D ϕ ⊂ {∅ , X } ∪ T I ; (b) the set D ϕ is left-inv ariant; (c) ϕ is equiv ariant and I -saturated; (d) ϕ |{∅ , X } ∪ e T = id; (e) I -Fix − ( A ) ⊂ Fix − ( ϕ ( A )) for all A ∈ D ϕ . The family F is par tia lly ordered by the relatio n ϕ ≤ ψ deﬁned b y ψ | D ϕ = ϕ . The set F is not empty b eca use it contains the iden tit y function id of {∅ , X } ∪ e T , whic h is I -sa turated beca use of the I -indep endence o f the family e T . B y Zor n’s Lemma, the family F contains a maximal element ϕ : D ϕ → {∅ , X } ∪ e T . W e claim that D ϕ = { ∅ , X } ∪ T I . Assuming the conv erse, ﬁx a set A ∈ T I \ D ϕ and deﬁne a family D ψ = D ϕ ∪ { xA : x ∈ X } . Next, we shall extend the function ϕ to a function ψ : D ψ → {∅ , X } ∪ e T . W e consider tw o cases. 1) Assume that A = I B for some B ∈ D ϕ . Then also xA = I xB for a ll x ∈ X . In this case we deﬁne the function ψ : D ψ → {∅ , X } ∪ e T assigning to each set C ∈ D ψ the set ϕ ( D ) where D ∈ D ϕ is an y set with D = I C . It can be shown that the function ψ : D ψ → {∅ , X } ∪ e T b elongs to the family F , which contradicts the maximality of ϕ . 2) Assume that A 6 = I B for all B ∈ D ϕ . By Pr op o sition 7.3, the 2-cog roup I -Fix − ( A ) lies in a maximal 2-cogr oup K ∈ b K . Since the fa mily e T is b K -covering, there is a twin set B ∈ e T such that Fix − ( B ) = K . In this case deﬁne the function ψ : D ψ → b T by the formula ψ ( C ) = ( ϕ ( C ) if C ∈ D ϕ ; xB if C = I xA for some x ∈ X . If xA = I y A for some x, y ∈ X , then y − 1 x ∈ I -Fix − ( A ) ⊂ K = Fix − ( B ) and thus xB = y B , which means that the function ψ is w ell-deﬁned and I -saturated. Also it is clear that ψ is equiv ariant and hence belongs to the family F , which is forbidden by the maximality of ϕ . 32 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Thu s the maxima l function ϕ is deﬁned on D ϕ = {∅ , X } ∪ T I and we ca n put e e T = ϕ ◦ e I where e I : P ( X ) → {∅ , X } ∪ pT I = {∅ , X } ∪ T I is the idempo ten t constructed in Pro po sition 16.1. It fo llows from the prop erties o f the functions ϕ and e I that the function e e T is equiv ar iant a nd I -satura ted. Since the ideal I is twinic, the family T I = pT I is I -incompar able (b y Pr op osition 13.1) and hence the monotonicity of the function ϕ follo ws automatica lly from its I -satur ated prop erty . Then e e T is monoto ne as the comp os itio n of t wo monotone functions. By Coro llary 4.4, e e T ∈ End λ ( P ( X )).  Theorem 16.3 and P rop osition 13.5 imply: Corollary 16. 4. If the ide al I is twinic, then for e ach minimal b K -c overing family e T ⊂ b T ther e is an idemp otent e e T ∈ End I λ ( P ( X )) su ch that (1) e e T ( P ( X ) \ T I ) ⊂ {∅ , X } ; (2) e e T ( T I ) = e T ; (3) e e T |{∅ , X } ∪ e T = id . 17. The minimal ideal of the semigroups λ ( X ) and E nd λ ( P ( X )) In this sectio n we apply Cor ollary 16.4 to describ e the structure o f the minimal ideals the semigr o ups λ ( X ) and End λ ( P ( X )). Theorem 1 7.1. F or a twinic gr oup X a fu n ction f ∈ End λ ( P ( X )) b elongs to the minimal ide al K (End λ ( P ( X )) of t he semigr oup End λ ( P ( X )) if and only if the fol lowing two c onditions hold: (1) the family f ( b T ) is minimal b K - c overing; (2) f ( P ( X )) ⊂ { ∅ , X } ∪ f ( b T ) . Pr o of. Let e T ⊂ b T b e a minimal b K -covering left-inv ariant family and e e T ∈ E nd λ ( P ( X )) be an ide mp otent satisfying the conditions (1)–(3) o f in Coro llary 16.4. By P rop ositions 13.1 a nd 13.5, the family e T is I - incomparable and I -indep endent for any t winic ideal I on X . T o prov e the “if ” part of the theorem, assume that f satisﬁes the conditions (1), (2). T o show that f b elongs to the minimal ideal K (E nd λ ( P ( X ))), it suﬃces for each g ∈ End λ ( P ( X )) to ﬁnd h ∈ End λ ( P ( X )) suc h that h ◦ g ◦ f = f . The minimality a nd the left-inv aria nce of the b K -covering subfamily f ( b T ) imply tha t the equiv ar iant function ψ = e e T ◦ g |{∅ , X } ∪ f ( b T ) : {∅ , X } ∪ f ( b T ) → {∅ , X } ∪ e T is bijectiv e. So, w e can consider the inv erse function ψ − 1 : {∅ , X } ∪ e T → {∅ , X } ∪ f ( b T ) suc h that ψ − 1 ◦ ψ = id |{∅ , X } ∪ f ( b T ). This function is eq uiv aria n t, symmetric, and monotone b ecause so is ψ and the family e T is I -incompar able and I -indep endent. Then the function ϕ = ψ − 1 ◦ e e T : P ( X ) → {∅ , X } ∪ f ( b T ) is well-deﬁned and b elongs to End I λ ( P ( X )) b y Cor ollary 4.4. Since ( ϕ ◦ e e T ) ◦ g ◦ f = ψ − 1 ◦ e e T ◦ e e T ◦ g ◦ f = ψ − 1 ◦ e e T ◦ g ◦ f = ψ − 1 ◦ ψ ◦ f = f , the function f b elongs to the minimal ideal of the se migroup End λ ( P ( X )). T o prov e the “only if ” pa rt, take any function f ∈ K (End λ ( P ( X ))) a nd for the idemp otent e e T ∈ End λ ( P ( X )) ﬁnd a function g ∈ End λ ( P ( X )) such that f = g ◦ e e T ◦ f . Now the prop erties (1), (2) of the function f follow from the corres p onding pro per ties of the idemp otent e e T .  Since the sup erextensio n λ ( X ) of a g roup X is top olo gically isomorphic to the s emigroup End λ ( P ( X )), Theorem 17.1 implies the fo llowing description of the minimal ideal K ( λ ( X )) o f λ ( X ). Corollary 1 7 .2. F or a twinic gr oup X a maximal linke d system L ∈ λ ( X ) b elongs to the minimal ide al K ( λ ( X )) of the sup er extension λ ( X ) if and only if its function r epr esentation Φ L satisﬁes two c onditions: (1) the family Φ L ( b T ) is minimal b K -c overing; (2) Φ L ( P ( X )) ⊂ {∅ , X } ∪ Φ L ( b T ) . 18. Minimal left ideals of superextensions of twinic groups After elab ora ting the necessa ry too ls in Section 5-17, we now return to descr ibing the structure of minimal left ideals of the sup erextensio n λ ( X ) of a twinic group X . In this sectio n w e as sume that X is a group. Our ﬁrst aim is to show that if X is twinic, then the restr iction op erator R b T : End λ ( P ( X )) → End λ ( b T ) is injectiv e on all minimal left ideals o f the semigroup End λ ( P ( X )). Since b T = S K ∈ b K T K , Pro po sition 12.3 implies that the family b T is λ -in v ar iant and hence End λ ( b T ) is a c o mpact right-topolo gical semigroup. F or each left-in v ar iant idea l I on the group X the semigr o up End λ ( b T ) contains a left ideal E nd I λ ( b T ) cons is ting of all left-inv aria n t monotone I - saturated functions, see ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 33 Theorem 4.8. If I is a twinic ideal with I ∩ b K = ∅ , then the family b T is I -indep endent (see Pr op osition 13.3) and hence End I λ ( b T ) = End λ ( b T ). Prop ositio n 18.1. If the gr oup X is twinic, then the r est riction op er ator R b T : End λ ( P ( X )) → End λ ( b T ) , R b T : f 7→ f | b T , is inje ctive on e ach minimal left ide al of the semigr oup End λ ( P ( X )) . If I I ∩ b K = ∅ , then for some idemp otent e b T ∈ End I I λ ( P ( X )) the r estriction R b T | End λ ( P ( X )) ◦ e b T is a top olo gic al isomorphism b etwe en t he princip al left ide al End λ ( P ( X )) ◦ e b T and End λ ( b T ) . Pr o of. Let e T ⊂ b T b e any minimal K -covering left-in v ariant subfamily . By P rop ositions 1 3 .5, the family e T is I -indep endent. By Theorem 16.3, there is an idemp otent e e T ∈ End I λ ( P ( X )) suc h tha t e e T ( P ( X ) \ T I ) ⊂ {∅ , X } and e e T ( T I ) = e T ⊂ b T . The latter prop erty of e e T implies that the restriction op erator R b T is injective o n the principal left ideal End λ ( P ( X )) ◦ e e T and consequently , is injective on each minimal left ideal of the semigr o up End λ ( P ( X )) according to Pro po sition 2.2. If I I ∩ b K = ∅ , then by Prop osition 13.3 the family b T is I I -indep endent and we can rep eat the above argument for the idempo ten t e b T .  Now let us loo k at the structure of the semigr oup End λ ( b T ). Obs e rve that b T = S [ K ] ∈ [ b K ] T [ K ] , whe r e T [ K ] = { A ⊂ X : Fix − ( A ) ∈ [ K ] } , [ b K ] = { [ K ] : K ∈ b K} and [ K ] = { xK x − 1 : x ∈ X } for K ∈ b K . It follows that the restr iction op erators R T [ K ] : E nd λ ( b T ) → End λ ( T [ K ] ), [ K ] ∈ [ b K ], comp ose a n injective semig r oup homomorphism R T [ b K ] : End λ ( b T ) → Y [ K ] ∈ [ b K ] End λ ( T [ K ] ) , R T [ b K ] : ϕ 7→ ( ϕ | T [ K ] ) [ K ] ∈ [ b K ] . Theorem 11.1 implies Lemma 18. 2. F or any twinic ide al I on the gr oup X we get R [ b K ] (End I λ ( b T )) = Q [ K ] ∈ [ e K ] End I λ ( T [ K ] ) . Next, we study the structure o f the semigro ups End I λ ( T [ K ] ) for [ K ] ∈ [ b K ]. Lemma 18. 3. F or any maximal 2-c o gr oup K ∈ b K the r estriction m ap R T K : End λ ( T [ K ] ) → End λ ( T K ) , R T K : ϕ 7→ ϕ | T K , is a top olo gic al isomorphism. Pr o of. Because of the compactness of the semigr oup End λ ( T [ K ] ) it suﬃces to chec k that the res triction ope r ator R T K : End I λ ( T [ K ] ) → E nd I λ ( T K ) is one-to -one. Given tw o distinct functions f , g ∈ End I λ ( T [ K ] ) ﬁnd a t win set A ∈ T [ K ] such tha t f ( A ) 6 = g ( A ). Since Fix − ( A ) ∈ [ K ], there is a po in t x ∈ X such that Fix − ( xA ) = x Fix − ( A ) x − 1 = K . By P r op osition 5.4, xA ∈ T K and f ( xA ) = xf ( A ) 6 = xg ( A ) = g ( xA ) witness ing that f | T K 6 = g | T K .  A subfamily e K ⊂ b K is ca lled a [ b K ] -sele ctor if e K has one-p oint intersection with ea c h or bit [ K ] = { xK x − 1 : x ∈ X } , K ∈ b K . In the following theorems w e assume that e K ⊂ b K is a [ b K ]- selector. All preceding discussion culminates in the following theor em, which can b e considered as the main result o f this pap er. Theorem 18 . 4. Given a [ b K ] -sele ctor e K ⊂ b K , c onsider the op er ator R e K : End λ ( P ( X )) → Y K ∈ e K End( T K ) , R e K : f 7→ ( f | T K ) K ∈ e K . If I is a left-invariant twinic ide al on X , then (1) R e K  End I λ ( P ( X ))  = Q K ∈ e K End I ( T K ) ; (2) the op er ator R e K maps isomorphi c al ly e ach minimal left ide al of the semigr oup End λ ( P ( X )) ont o some minimal left ide al of the semigr oup Q K ∈ e K End( T K ) . (3) If I ∩ b K = ∅ , t hen for some idemp otent ˆ e ∈ End I λ ( P ( X )) and t he princip al left ide al L ˆ e = End λ ( P ( X )) ◦ ˆ e t he r est riction R e K | L ˆ e : L ˆ e → Y K ∈ e K End( T K ) is a top olo gic al isomorphism. 34 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Pr o of. W rite the op era tor R e K as the co mpos ition R e K = R b T e K ◦ R b T of tw o op erators : R b T : End λ ( P ( X )) → End λ ( b T ) , R b T : f 7→ f | b T , and R b T e K : End λ ( b T ) → Y K ∈ e K End λ ( T K ) , R b T e K : f 7→ ( f | T K ) K ∈ e K . By Lemma 1 8.3, the op erator R b T e K is injective. 1. It follows from Lemmas 18.2 and 1 8 .3 that R e K (End I λ ( P ( X )) = R b T e K (End I λ ( b T )) = Y K ∈ e K End I ( T K ) . 2. T o prov e the second item, ﬁx any function f ∈ K (End λ ( P ( X ))) and consider the minimal left ideal L f = E nd λ ( P ( X )) ◦ f . W e need to show that R e K ( L f ) is a minimal left ideal in Q K ∈ e K End( T K ). F o r this pick any function g ∈ K (End I λ ( P ( X ))) and consider the minimal left ideal L g = End λ ( P ( X )) ◦ g = End I λ ( P ( X )) ◦ g . By Prop os ition 18.1, the op erator R b T : End λ ( P ( X )) → End λ ( b T ) is injectiv e on ea ch minimal left idea l. Consequently , the op erator R e K = R b T e K ◦ R b T also is injective on each minimal left ideal of the semigroup End λ ( P ( X )). In particular , R e K is injective on the minimal left ideals L f and L g . Since L g is a minimal left ideal o f the semig roup End I λ ( P ( X )) its image R b T ( L g ) is a minimal left ideal o f the semig roup End I λ ( b T ). By Lemmas 1 8.2 and 18.3, R b T e K maps isomorphically the semigroup End I λ ( b T ) on to Q K ∈ e K End I ( T K ), the image R e K ( L g ) is a minimal left ideal of the semigr oup Q K ∈ e K End I ( T K ). Since the latter semigro up is a left ideal in Q K ∈ e K End( T K ), the image R e K ( L g ) remains a minimal left ideal of the s emigroup Q K ∈ e K End( T K ). This minimal left ideal is equal to the pro duct Q K ∈ e K L g K where g K = g | T K and L g K = End( T K ) ◦ g K . Because of the compactness o f L g , the op erato r R e K maps iso morphically the minimal left ideal L g onto the minimal left ideal Q K ∈ e K L g K of the s emigroup Q K ∈ e K End( T K ). Now let us lo o k at the minimal left ideal L f . By Prop os ition 2.1, the right shift r f : L g → L f , r f : h 7→ h ◦ f , is a homeomorphism. So, there is a function γ ∈ E nd λ ( P ( X )) suc h that f = γ ◦ g ◦ f . F o r every K ∈ e K co nsider the restric tio ns f K = f | T K and γ K = γ | T K , which be lo ng to the semig roup End( T K ). It follows fr om f = γ ◦ g ◦ f that f K = γ K ◦ g K ◦ f K . Since g K ∈ K (E nd( T K )), we conclude that f K also b elongs to the minimal ideal K (E nd( T K )). Then L f K = End( T K ) ◦ f K and L g K = End( T K ) ◦ g K are minimal left ideals in E nd( T K ). By Pro po sition 2.1, the right shift r f K : L g K → L f K , r f K : h 7→ h ◦ f K , is a homeomo rphism. The homeomorphisms r f K , K ∈ e K , comp ose a homeomorphism r f e K : Y K ∈ e K L g K → Y K ∈ e K L f K , r f e K : ( h K ) K ∈ e K 7→ ( h K ◦ f K ) K ∈ e K . Now co nsider the co mm utative diagr am L f R e K | L f / / Q K ∈ e K L f K L g r f O O R e K | L g / / Q K ∈ e K L g K r f e K O O Since the maps r f , r f e K , and R e K | L g are homeomorphisms, so is the map R e K | L f . Cons equent ly , the op er a tor R e K maps isomorphica lly the minimal left ideal L f = End λ ( P ( X )) ◦ f onto the minimal left ideal Q K ∈ e K L f K = Q K ∈ e K End( T K ) ◦ ( f | T K ) of the semigroup Q K ∈ e K End( T K ). 3. Assume that I ∩ b K = ∅ . In this case End I ( b T ) = End( b T ) by Pr op osition 13.3. By Pr op osition 18 .1, fo r s o me idempo ten t ˆ e ∈ End I λ ( P ( X )) the op era tor R b T e T maps is omorphically the pr inc ipa l left ideal L ˆ e = End λ ( P ( X )) ◦ ˆ e onto End I λ ( b T ) = E nd λ ( b T ). B y Lemma 18 .2, the op era tor R b T e K : End I λ ( b T ) → Q K ∈ e K End I ( T K ) = Q K ∈ e K End( T K ) is an isomorphism. So, R e K maps isomor phica lly the principal left idea l L ˆ e onto Q K ∈ e K End( T K ).  Since the function r epresentation Φ : λ ( X ) → End λ ( P ( X )), Φ : L 7→ Φ L , is a top olo gical isomorphism, the preceding theorem implies: ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 35 Corollary 18.5 . Given a [ b K ] - sele ctor e K ⊂ b K , c onsider t he c ontinuous semigr oup homomorph ism Φ e K : λ ( X ) → Y K ∈ e K End( T K ) , Φ e K : L 7→ (Φ L | T K ) K ∈ e K . If the gr oup X is twinic, then (1) Φ e K (End I I λ ( P ( X )) = Q K ∈ e K End I I ( T K ) ; (2) the homomorphism Φ e K maps isomorphic al ly e ach minimal left ide al of the semigr oup λ ( X ) onto some minimal left ide al of the semigr oup Q K ∈ e K End( T K ) . (3) If I I ∩ b K = ∅ , then for some idemp otent E ∈ λ I I ( X ) and the princip al left ide al L E = λ ( X ) ◦ E the r estriction Φ e K | L E : L E → Y K ∈ e K End( T K ) is a top olo gic al isomorphism. Corollary 18.6. If the gr ou p X is twinic, then e ach minimal left ide al of λ ( X ) is top olo gic al ly isomorph ic t o a minimal left ide al of Q K ∈ e K End( T K ) and e ach minimal left ide al of Q K ∈ e K End I I ( T K ) is top olo gic al ly isomorphic to a minimal left ide al of λ I I ( X ) . Pr o of. Corolla ry 18.5(2) implies that each minimal left ideal of λ ( X ) is topolo gically isomorphic to a minimal left ideal of Q K ∈ e K End( T K ). Now assume that L is a minimal left ideal o f the semigr o up Q K ∈ e K End I I ( T K ). It follows from Corollar y 18.5(1) that the preimage Φ − 1 e K ( L ) is a left ideal in λ I I ( X ) and hence a left ideal in λ ( X ). This left idea l contains some minimal left idea l L λ whose image Φ e K coincides with L (b eing a left ideal in L ). By Corollar y 18 .5(2), the map Φ e K | L λ : L λ → L is injective and by the compactness of L λ is a top olo gical isomorphism.  Theorem 18 . 7. L et X b e a t winic gr ou p, e K ⊂ b K b e a [ b K ] - sele ctor, and E ∈ λ ( X ) b e a minimal idemp otent. (1) The maximal sub gr oup H E = E ◦ λ ( X ) ◦ E has t he fol lowing pr op erties: (a) H E is algebr aic al ly isomorphi c to Q K ∈ e K H ( K ) ; (b) H E is top olo gic al ly isomorphic to Q K ∈ e K H ( A K ) for any twin sets A K ∈ Φ E ( T K ) , K ∈ e K ; (c) H E is a c omp act top olo gic al gr oup if and only if H ( K ) is ﬁnite for every K ∈ b K . (2) The minimal left ide al L E = λ ( X ) ◦ E has the fol lowing pr op erties: (d) L E is top olo gic al ly isomorphic to the minimal left ide al Q ∈ e K End( T K ) ◦ (Φ E | T K ) ; (e) L E is home omorphic t o Q K ∈ e K T K , which is home omorphic t o the Cantor disc ont inu um Q K ∈ e K 2 X/K ± ; (f ) L E is algebr aic al ly isomorphic to Q K ∈ e K H ( K ) × [ T K ] wher e the orbit sp ac e [ T K ] of the H ( K ) -act T K is endowe d with the left-zer o-multiplic ation; (g) L E a t op olo gic al semigr oup iﬀ L E is a semitop olo gic al semigr oup iﬀ e ach r est riction Φ E | T K , K ∈ e K , is a c ont inu ous function iﬀ the max imal sub gr oup H E and the idemp otent b and E ( L E ) of L E ar e c omp act iﬀ L E is top olo gic al ly isomorphic to Q K ∈ e K H ( K ) × [ T K ] . Pr o of. Let Φ E ∈ End λ ( P ( X )) b e the function representation of the minimal idemp otent E ∈ K ( λ ( X )). F or every K ∈ e K let f K = Φ E | T K and L f K = End( T K ) ◦ f K be the principa l left ideal in E nd( T K ), genera ted b y the function f K . By Coro l- lary 18 .5, the minimal left ideal L E is top olog ically isomorphic to a minimal left idea l of the s emigroup Q K ∈ e K End( T K ). This minimal ideal cont ains ( f K ) K ∈ e K and henc e is equal to the pro duct Q K ∈ e K L f K . This prov es the s tatement (d) of the theorem. Now a ll the o ther statements follo w from Theorems 1 4.1 and 14.3.  Theorem 18.7(b) is completed b y the follo wing theorem. Theorem 1 8.8. L et e K ⊂ b K b e a [ b K ] - sele ctor. If t he gr oup X is twinic, then for any twin sets A K ∈ T K , K ∈ e K , the minimal ide al K ( λ ( X )) of λ ( X ) c ontains a maximal sub gr oup H E , which is top olo gic al ly isomorphic to Q K ∈ e K H ( A K ) . Pr o of. In the s e migroup Q K ∈ e K K (End I I ( T K )) c hoo s e a sequence of functions ( f K ) K ∈ e K such that f K ( T K ) ⊂ ⌊ A K ⌋ for all K ∈ e K . This ca n b e do ne in the fo llowing wa y . F or every K ∈ e K ﬁrst choo se a n y minimal idemp otent g K ∈ K (End I I ( T K )). By Theorem 14.1(5), g K ( T K ) ⊂ ⌊ B K ⌋ for so me twin set B ∈ T K . Since T K is a free H ( K )-a ct, we can choo se a n equiv ar iant function ϕ : ⌊ B K ⌋ → ⌊ A K ⌋ . Then the co mpo sition f K = ϕ ◦ g K is I I - saturated and has the required pro pe r ty: f K ( T K ) ⊂ ⌊ A K ⌋ . Consider the minimal left ideal L f e K = Q K ∈ e K End( T K ) ◦ f K and let L = Φ − 1 e K ( L ) ⊂ λ ( X ) b e its preimage under the map Φ e K . By Cor ollary 18.5(1), Φ e K ( L ) = L f e K . Now let K ( L ) b e the minimal ideal o f the left ideal L . The imag e Φ e K ( K ( L )), b eing a left ideal in L f e K , c o incides with L f e K . So, we ca n ﬁnd a maximal linked s ystem L ∈ K ( L ) such that Φ e K ( L ) = ( f K ) K ∈ e K . By Theorem 18 .7(b), the maximal g roup H L is top olog ically isomorphic to Q K ∈ e K H ( A K ).  36 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Prop ositio n 18 .9. If X is a t winic gr oup, then e ach minimal left ide al of λ ( X ) is a top olo gic al semigr oup if and only if e ach maximal 2-c o gr oup K ⊂ X has ﬁnite index in X . Pr o of. Let e K ⊂ b K b e a [ b K ]- selector. If each maximal 2-c ogroup K ⊂ X has ﬁnite index in X , then the set T K is ﬁnite a nd hence the semigroup End( T K ) is ﬁnite. Consequently , Q K ∈ e K End( T K ) is a compact top ologica l semigro up and so is each minimal left idea l of this semigroup. By Cor ollary 18.5(2), each minimal left ideal o f the semigro up λ ( X ) is a top ological semigr o up. If s o me maximal 2 -cogro up in X has inﬁnite index, then Coro llary 14 .6 implies that so me minimal left ideal in Q K ∈ e K End I I ( T K ) is not a top olo gical semigroup. B y Corolla r y 18.6, so me minimal left ideal in Q K ∈ e K End I I ( T K ) is not a top ologica l semigroup.  Prop ositio n 18.10. F or a twinic gr oup X with I I ∩ b K = ∅ the fol lowing c onditions ar e e quivalent: (1) some minimal left ide al of λ ( X ) is a top olo gic al s emigr oup; (2) e ach maximal su b gr oup of λ ( X ) is a top olo gic al gro up; (3) some maximal sub gr oup of λ ( X ) is c omp act; (4) the char acteristic gr oup H ( K ) is ﬁnite for e ach maximal 2-c o gr oup K ⊂ X . Pr o of. (1) ⇒ (3) If some minimal le ft ideal of λ ( X ) is a (necessar ily compact) topo logical semigroup, then eac h maximal subgroup of this minimal ideal is a compact top ologica l group. (3) ⇒ (4) If some max imal subg r oup o f K ( λ ( X )) is co mpact, then by Theorem 18.7(c), each characteristic group H ( K ), K ∈ b K , is ﬁnite. (4) ⇒ (1 ) If each character is tic gro up H ( K ), K ∈ b K , is ﬁnite, then Pro po sition 14 .5(1) and Theorem 1 4.3 guarantee that the semigroup Q K ∈ e K End I I ( T K ) contains a minimal left ideal, which is a top olog ical semigroup. By Corolla r y 18.6, this minimal le ft ideal is to po logically isomor phic to some minimal left ideal of λ ( X ). (4) ⇒ (3) If each character istic gro up H ( K ), K ∈ b K , is ﬁnite, then Theo r em 18.7(c) guara n tees that e ach maximal subgroup of K ( λ ( X )) is a compact top olo gical group. (3) ⇒ (4). Assume that for some maximal 2-c o group K ∞ ∈ b K the characteristic gro up H ( K ∞ ) is inﬁnite. Replacing K ∞ by a co njugate cog roup, we can assume that K ∞ ∈ e K . By Theor em 8.3, the gro up H ( K ∞ ) is iso morphic to C 2 ∞ or Q 2 ∞ . In b o th cases, by Theo r ems 9.5, there is a twin se t A ∞ ∈ T K ∞ whose characteristic gr oup H ( A ∞ ) is no t a top ological group. Cho ose a minimal idempotent f e K = ( f K ) K ∈ e K ∈ Q K ∈ e K End I I ( T K ) such that f K ∞ ( T K ∞ ) ⊂ ⌊ A ⌋ . F or every K ∈ e K cho ose any t win set A K ∈ f K ( T K ) so that A K = A ∞ if K = K ∞ . By Coro llary 1 8.6, there is a minimal idempo ten t E ∈ λ I I ( X ) such that Φ e K ( E ) = f e K . By Theor em 18.7(b), the maxima l subgroup H E = λ ( X ) ◦ E ◦ λ ( X ) is top ologically isomorphic to Q K ∈ e K H ( A K ). This subgro up is no t a topolo gical g roup as it co ntains an isomor phic cop y of the right-topologic a l gro up H ( A ∞ ), which is not a top ologica l group.  Now let us write Co r ollary 18.5 and Theorem 1 8 .7 in a form, more conv enien t for calculations. F o r every g roup G ∈ { C 2 k , Q 2 k : k ∈ N ∪ {∞}} denote b y q ( X , G ) the num be r o f a ll o rbits [ K ] ∈ [ b K ] such that for some (equiv a lent ly , every) 2-cogroup K ∈ [ K ] the characteristic gro up H ( K ) is iso morphic to G . Theorem 18 . 11. F or e ach twinic gr oup X ther e is a c ar dinal m su ch that (1) e ach minimal left ide al of λ ( X ) is algebr aic al ly isomorphic to the semigr oup 2 m × Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k wher e the Cantor disc ontinu um 2 m is endowe d with the left zer o mult iplic ation; (2) If q ( X, C 2 ∞ ) = q ( X , Q 2 ∞ ) = 0 and I I ∩ b K = ∅ , t hen some minimal left ide al of λ ( X ) is top olo gic al ly isomorphic to t he c omp act top olo gic al semigr oup 2 m × Y 1 ≤ k< ∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k< ∞ Q q ( X ,Q 2 k ) 2 k . (3) e ach maximal su b gr oup of t he minimal ide al K ( λ ( X )) of λ ( X ) is algebr aic al ly isomorphic to the gr oup Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k . (4) If q ( X , C 2 ∞ ) = q ( X , Q 2 ∞ ) = 0 , then e ach maximal sub gr oup of the minimal ide al K ( λ ( X )) of λ ( X ) is top olo gic al ly isomorphi c to the c omp act t op olo gic al gr oup Y 1 ≤ k< ∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k< ∞ Q q ( X ,Q 2 k ) 2 k . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 37 Pr o of. 1. Fix an y [ b K ]- selection e K ⊂ b K . F o r every K ∈ e K put m K = | X/K ± | = { K ± x : x ∈ X } if the index of K in X is inﬁnite and m K = 2 | X/K ± | / | H ( K ) | otherwise. It follows that | [ T K ] | = | T K | | H ( K ) | = 2 m K and [ T K ] is homeomorphic to the Cantor c ub e 2 m K if the characteristic group H ( K ) is ﬁnite. Let m = P K ∈ e K m K . By Theorem 18.7(f ), any minimal left ideal L o f λ ( X ) is a lgebraica lly isomorphic to the se mig roup Q K ∈ e K H ( K ) × [ T K ], where the orbit spa ces [ T K ] are endow ed with the left zero multip lication. By Theorem 8.3, for e very K ∈ e K the characteristic g roup H ( K ) is isomorphic to C 2 k o r Q 2 k fo r some k ∈ N ∪ {∞} . According to the deﬁnition, for k ∈ { 1 , 2 } the gro up Q 2 k is isomo r phic to the quaternion g roup Q 8 . By the deﬁnition of the num ber q ( X , G ), for any gr oup G ∈ { C 2 k , Q 2 k : k ∈ N ∪ {∞}} we get q ( X , G ) = { K ∈ e K : H ( K ) ∼ = G } where ∼ = denotes the (semi)gr oup isomorphism. Now we see that L ∼ = Y K ∈ e K H ( K ) × [ T K ] ∼ = Y K ∈ e K H ( K ) × 2 m K ∼ = Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k × 2 m . 2. If q ( X , C 2 ∞ ) = q ( X, Q 2 ∞ ) = 0, then for every K ∈ e K the c haracter istic group H ( K ) is ﬁnite and the orbit space [ T K ] is a zero-dimensio nal compact Hausdorﬀ spa ce. In this c a se the spac e T K is ho meomorphic to [ T K ] × H ( K ). If K has ﬁnite index in X , then T K has cardinality 2 m K and hence is homeo mo rphic to the ﬁnite cube 2 m K . If K has inﬁnite index in X , then the space T K is ho meomorphic to the Ca n tor cub e 2 m K by Prop osition 12 .1. It follo ws from the top ological equiv alence of T K and [ T K ] × H ( K ) that [ T K ] is a retract of the Ca n tor cube T K and each p oint o f [ T K ] has character m K . Now Shc hepin’s characterization of Cantor’s cubes [20] implies that the space [ T K ] is homeo morphic to the Cantor cube 2 m K . Then the product Q K ∈ e K [ T K ] is homeo mo rphic to the Cantor cube 2 m = Q K ∈ e K 2 m K . If I I ∩ b K = ∅ , then for every K ∈ e K the endomorphism monoid End I I ( T K ) contains a contin uous minimal ide mp otent f K according to Pr op osition 1 4.5(1). By Theor em 1 4.3, the minimal left idea l L f K = End I I ( T K ) ◦ f K is top ologic a lly isomorphic to the co mpact top ologica l semigroup H ( K ) × [ T K ] where the space [ T K ] is endow ed with the left zero m ultiplication. By (the pro of of ) Cor ollary 1 8 .6, the minimal idempotent K ( λ ( X )) contains a max ima l linked s ystem E such that the minima l left idea l L E = λ ( X ) ◦ E is top ologica lly isomorphic to the minimal le ft ideal Q K ∈ e K L f K , which is top ologically isomo rphic to the compa c t topo logical semigro ups Y K ∈ e K H ( K ) × [ T K ] and Y 1 ≤ k< ∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k< ∞ Q q ( X ,Q 2 k ) 2 k × 2 m . 3. By Theo rem 18.7(b) each ma ximal subgroup H of the minimal ideal K ( λ ( X )) is top olog ically isomorphic to the right topo logical g r oup G = Q K ∈ e K H ( A K ) for some twin sets A K ∈ T K , K ∈ e K . The latter r ig ht -top ologica l group is algebraic ally isomorphic to the gr o up Y 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k ≤∞ Q q ( X ,Q 2 k ) 2 k . 4. If q ( X , C 2 ∞ ) = q ( X , Q 2 ∞ ) = 0, then all characteristic groups H ( K ), K ∈ e K , are ﬁnite a nd then the g roup G is top ologically isomo rphic to the compa c t topo logical group Y 1 ≤ k< ∞ C q ( X ,C 2 k ) 2 k × Y 3 ≤ k< ∞ Q q ( X ,Q 2 k ) 2 k .  19. The structure of the superextensions o f abelian gr oups In this section we cons ider the structure of the sup erextensio n of a belia n groups. In this ca s e some r e sults of the preceding section can b e s impliﬁed. In this section we assume that X is an ab elian gr oup. By The o rem 6.2, X is twinic and has triv ia l twinic ideal I I = {∅ } . Let us recall that for a g roup G by q ( X, G ) we denote the num ber o f orbits [ K ], K ∈ b K , such that for each K ∈ [ K ] the characteristic g r oup H ( K ) is iso morphic to the g roup G . It is clea r that q ( X , Q 2 k ) = 0 for all k ∈ N ∪ {∞} . O n the other ha nd, the num b er s q ( X , C 2 k ) can b e e a sily calculated using the following prop osition. Prop ositio n 19.1. If X is an ab elian gr oup, then for every k ∈ N ∪ {∞} t he c ar dinal q ( X , C 2 k ) is e qual to the nu mb er of sub gr oups H ⊂ X such that the quotient gr oup X/ H is isomorphic to C 2 k . If k ∈ N , then q ( X , C 2 k ) = | hom( X, C 2 k ) | − | hom( X , C 2 k − 1 ) | 2 k − 1 , wher e hom( X , C 2 k ) is the gr oup of al l homomorphisms fr om X to C 2 k . 38 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV Pr o of. Since e ach maxima l 2-cog roup K ⊂ X is no rmal, each orbit [ K ] ∈ [ b K ] consists o f a single maxima l 2- cogroup. Consequently , q ( X , H ) is equal to the num ber of maximal 2- cogro ups K ⊂ X whose characteristic g roup H ( K ) = Stab( K ) /K K = X /K K is isomorphic to H . In other words, q ( X , H ) equals to cardinality of the set b K H = { K ∈ b K : X/K K ∼ = H } , where ∼ = stands for the group isomor phism. Let G H be the set of a ll subg roups G ⊂ X such that the quotien t gr o up X/ G is isomorphic to H . The prop osition will be prov ed as so on as we c heck tha t the function f : b K H → G H , f : K 7→ K K , is bijective. T o show that f is injectiv e, take any tw o maximal 2-cogr o ups K , C with K K = f ( K ) = f ( C ) = C C . The quotient group X/K K = X/C C , b eing isomorphic to H , contains a unique element o f order 2. Since K a nd C ar e co sets o f order 2 in X/ K K = X/C C , w e conclude that K = C . T o s how that f is surjective, take a n y subgr o up G ∈ G H . The quotient g roup X/ G is iso morphic to H and thus contains a unique element K o f order 2. This element K is a ma x imal 2-cog r oup suc h that f ( K ) = K K = G . T o prove the second part of the pr op osition, observe that for a subg roup H ⊂ X the quotien t g roup X/H is is omorphic to C 2 k if a nd o nly if H coincides with the kernel o f so me epimorphism f : X → C 2 k . Obs e r ve that tw o epimor phisms f , g : X → C 2 k hav e the same kernel if and only if g = α ◦ f for some automorphism of the gr o up C 2 k . The gro up C 2 k has exactly 2 k − 1 automorphisms determined by the image of the gener ator a = e iπ 2 − k +1 of C 2 k in the 2-co group aC 2 k − 1 . A homomor phism h : X → C 2 k is an epimorphism if and only if h ( X ) 6⊂ C 2 k − 1 . Consequently , q ( X , C 2 k ) = | hom( X, C 2 k ) \ hom( X , C 2 k − 1 ) | 2 k − 1 .  Theorem 19 . 2. If X is an ab eli an gr oup, then (1) e ach maximal su b gr oup of t he minimal ide al K ( λ ( X )) is algebr aic al ly isomorphic to Q 1 ≤ k ≤∞ C q ( X ,C 2 k ) 2 k . (2) e ach minimal left ide al of λ ( X ) is home omorph ic t o the Cantor cub e (2 ω ) q ( X ,C 2 ∞ ) × Q 1 ≤ k< ∞ (2 2 k − 1 ) q ( X ,C 2 k ) and is algebr aic al ly isomorphic to the semigr oup Y 1 ≤ k ≤∞ ( C 2 k × Z k ) q ( X ,C 2 k ) wher e the cub e Z k = 2 2 k − 1 − k (e qu al to 2 ω if k = ∞ ) is endowe d with the left zer o mult iplic ation. (3) the semigr oup λ ( X ) c ontains a princip al left ide al, which is algebr aic al ly isomorphic to the semigr oup Y 1 ≤ k ≤∞ ( C 2 k ≀ Z Z k k ) q ( X ,C 2 k ) . Pr o of. Since X is ab elian, each 2 -cogro up K ∈ b K is no rmal in X a nd hence has one-element orbit [ K ] = { xK x − 1 : x ∈ X } . Then the family b K is a unique [ b K ]- selector. Since Stab ( K ) = X , the characteristic group H ( K ) = Stab( X ) /K K is equal to the quotient group X/K K a nd is abelia n. By Theor em 8 .3, H ( K ) is isomo r phic to the (quasi)cy c lic 2-g roup C 2 n for some n ∈ N ∪ {∞} . Consequently , b K = S 1 ≤ n ≤∞ b K n where b K n is the subset of b K that co nsists of all maximal 2-co groups K whose characteristic gro up H ( K ) = X/ K K is isomorphic to the group C 2 k . By the deﬁnition of the num bers q ( X , G ), we g et q ( X , C 2 n ) = | b K n | for a ll n ∈ N ∪ {∞} . 1. B y Theorem 18.7(a), each maximal subgroup of K ( λ ( X )) is algebraica lly is omorphic to Q K ∈ b K H ( K ) and the latter group is iso morphic to Y 1 ≤ n ≤∞ C | b K n | 2 n = Y 1 ≤ n ≤∞ C q ( X ,C 2 n ) 2 n . 2. B y Theorem 18.7(f ), each minima l le ft ideal of λ ( X ) is a lgebraica lly iso morphic to Q K ∈ b K ( H ( K ) × [ T K ]) where the orbit spaces [ T K ] are endow ed with the left zero m ultiplication. Let Z ∞ = 2 ω and Z n = 2 2 n − 1 − n for every n ∈ N . The cube s Z n , n ∈ N ∪ {∞} , ar e endow ed with the left zero multiplication. W e claim that | [ T K ] | = | Z n | for e ach n ∈ N ∪ {∞} . If n is ﬁnite, then | X/K ± | = 1 2 | X/K K | = 1 2 | H ( K ) | = 2 n − 1 and | [ T K ] | = | T K | | H ( K ) | = 2 X/K ± 2 n = 2 2 n − 1 − n = | Z n | . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 39 If n is inﬁnite, then the quotient gr oup H ( K ) = X/K K is isomo rphic to C 2 ∞ and then | X/K ± | = ω . By Pr op osition 12 .1, the s pa ce T K is homeomorphic to the Cantor cub e 2 ω and hence has car dinality of contin uum. Since the group H ( K ) is countable, the or bit space [ T K ] also has ca r dinality of contin uum and hence | [ T K ] | = | 2 ω | = | Z ∞ | . Now w e see that Q K ∈ b K ( H ( K ) × [ T K ]) is a lgebraica lly iso morphic to Q 1 ≤ k ≤∞ ( C 2 k × Z k ) q ( X ,C 2 k ) . 3. Since X has triv ial twinic ideal, I I ∩ b K = ∅ and b y Cor ollary 1 8.5(c) and Theor em 14.1(3), the semigro up λ ( X ) contains a principal left ideal that is algebr aically isomorphic to the s emigroup Q K ∈ b K H ( K ) ≀ [ T K ] [ T K ] , which is algebraically isomorphic to Q 1 ≤ k ≤∞ ( C 2 k ≀ Z Z k k ) q ( X ,C 2 k ) .  The following theor em characterizes the g roups X for which the algebra ic is omorphism in Theore m 19.2 a re top olog ic al. Theorem 19 . 3. F or an ab elian gr oup X the fol lowing c onditions ar e e quivalent : (1) The gr oup X admits no homomorphism onto the quasicyclic 2-gr oup C 2 ∞ . (2) Each m ax imal sub gr oup in the minimal ide al K ( λ ( X )) of λ ( X ) is top olo gic al ly iso morphic t o the c omp act top olo gic al gr oup Q k ∈ N C q ( X ,C 2 k ) 2 k . (3) Each maximal sub gr oup in K ( λ ( X )) is a t op olo gic al gr oup. (4) Some maximal sub gr oup of K ( λ ( X )) is c omp act. (5) Each minimal left ide al of λ ( X ) is top olo gic al ly isomorphic to the c omp act t op olo gic al semigr oup Y k ∈ N ( C 2 k × Z k ) q ( X ,C 2 k ) wher e the ﬁnite cub e Z k = 2 2 k − 1 − k is endowe d with the left zer o mult iplic ation. (6) λ ( X ) c ont ains a princip al left ide al, which is top olo gic al ly isomorphic to the c omp act top olo gic al semigr oup Y k ∈ N ( C 2 k ≀ Z Z k k ) q ( X ,C 2 k ) . Pr o of. Let the subfamilies b K n ⊂ b K and the Cantor cub es Z n = 2 2 n − 1 − n , n ∈ N ∪ { ∞} , b e deﬁned as in the pro o f of Theorem 19.2. Then q ( X, C 2 n ) = | b K n | . (1) ⇒ (5 , 6 ) If X a dmits no homomor phism onto C 2 ∞ , then q ( X , C 2 ∞ ) = 0 and b K ∞ = ∅ . In this case b K = S n ∈ N b K n . F or every n ∈ N and K ∈ b K n the c haracter istic group H ( K ) is isomor phic to C 2 n and the orbit space [ T K ] is homeomorphic to the cub e Z k . By Theorem 14.1(3), the endomorphism mono id End( T K ) is (topo logically) isomor phic to H ( K ) ≀ [ T K ] [ T K ] and the latter semigroup is top ologica lly isomorphic to C 2 n ≀ Z Z n n . By Theorem 18 .7(g), each minimal left ideal of λ ( X ) is topolo gically isomorphic to Y K ∈ b K H ( K ) × [ T K ] = Y n ∈ N Y K ∈ b K n H ( K ) × [ T K ] and the latter semigroup is top ologica lly isomorphic to the c o mpact top ologica l semigr oup Y k ∈ N ( C 2 k × Z k ) q ( X ,C 2 k ) . By Co rollar y 1 8.5(3) and Theor em 14.1(3), the semigro up λ ( X ) c ontains a princ ipa l left ideal that is top olo gically isomorphic to the compact topo lo gical semigroup Q K ∈ e K H ( K ) ≀ [ T K ] [ T K ] = Q n ∈ N Q K ∈ b K n H ( K ) ≀ [ T K ] [ T K ] , which is top o- logically isomor phic to the compact to p olo gical semigroup Y k ∈ N ( C 2 k ≀ Z Z k k ) q ( X ,C 2 k ) . The implication (5) ⇒ (2) ⇒ (3) ar e trivial. (3) ⇒ (1) If the group X admits a homomorphis m onto C 2 ∞ , then the family b K contains a 2-cog roup K ∞ whose characteristic group H ( K ∞ ) is is omorphic to C 2 ∞ . It follows fro m Exa mple 9 .3(2) that for some twin set A K ∞ ∈ T K ∞ the twin-generated gr oup H ( A K ∞ ) is not a top ologica l group. Now choos e a sequence ( A K ) K ∈ b K ∈ Q K ∈ b K T K of twin sets such that A K = A K ∞ if K = K ∞ . Then the r ight-topolog ic al gr oup Q K ∈ b K H ( A K ) is no t a top olog ical gro up. By Corollar y 18 .6, this right-topolo gical group is top olo gically isomo rphic to some maximal subgro up of the minimal ideal K ( λ ( X )). So , K ( λ ( X )) contains a maximal subgr oup, whic h is not a topo logical gro up. (6) ⇒ (4) If λ ( X ) contains a left idea l, which is a top ologica l semigro up, then λ ( X ) contains a minimal left ideal, which is a top ologica l semigroup. Any maximal subgro up of this minimal left ideal is a co mpact top olog ical group. 40 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV (4) ⇒ (1) I f K ( λ ( X )) contains a compact maximal subgroup, then by Theo rem 18.7(c), each characteristic g roup H ( K ), K ∈ b K , is ﬁnite and hence q ( X, C 2 ∞ ) = 0.  Finally , we shall c haracteriz e ab elian groups whose sup erextension cont ains metrizable minimal left ideals. The char- acterization inv olves the notion of the fre e rank and 2-ra nk, see [7, § 16 ] or [19, § 4.2]. Let us recall that a subset A 6∋ e of an ab elian group G with neutral elemen t e is called indep endent if for any disjoin t subsets B , C ⊂ A the s ubg roups h B i and h C i genera ted by B , C intersect by the triv ial subgroup. The ca rdinality of a maximal indep endent subse t A ⊂ G that cons ists o f element of inﬁnite order (resp. of o rder that is a pow er of 2) is called the fr e e r ank (resp. the 2-r ank ) of G and is denoted b y r 0 ( G ) (resp. r 2 ( G )). Theorem 19 . 4. F or an ab elian gr oup X the fol lowing c onditions ar e e quivalent: (1) e ach minimal left ide al of λ ( X ) is metrizable; (2) the family b K of maximal 2-c o gr ou ps is at most c ountable; (3) the gr oup X admits no epimorphism onto the gr ou p C 2 ∞ ⊕ C 2 ∞ and X has ﬁnite r anks r 0 ( X ) and r 2 ( X ) . Pr o of. (1) ⇔ (2) By Theor em 19.2(2), each minimal left idea l is homeomorphic to the cube (2 ω ) q ( X ,C 2 ∞ ) × Y 1 ≤ k< ∞ (2 2 k − 1 ) q ( X ,C 2 k ) , which is metrizable if a nd only if | b K| = P 1 ≤ k ≤∞ q ( X , C 2 k ) ≤ ℵ 0 . F o r the pro of of the equiv ale nce (2) ⇔ (3) w e need t w o lemmas. W e deﬁne a gro up G to be b K - c ount able if the family of maximal 2 -cogro ups in G is at most countable. Lemma 19. 5. Each su b gr oup and e ach quotient gr oup of a b K -c ountable gr oup is b K - c ount able. Pr o of. Assume tha t a g roup G is b K -countable. T o prov e tha t any subgroup H ⊂ G is b K -countable, observe that by Prop ositio n 7.3(2), ea ch 2 -cogro up K ⊂ H can be enlarged to a ma x imal 2-c o group ¯ K in G . The ma ximality o f K in H guarantees that K = ¯ K ∩ H . This implies that the n um ber of maximal 2- cogro ups in H do es not exceed the n um b er of maximal 2-co groups in G . T o pr ov e that any quotient gro up G/H of G by a nor mal subgro up H ⊂ G is b K - countable, obser ve that for each maximal 2- c ogroup K ⊂ G/H the preima ge q − 1 ( K ) under the quotient homomo rphism q : G → G/H is a ma ximal 2-cogr oup in G . T his implies that the num ber of maxima l 2-cogr oups in G/H do es not exceed the nu mber of ma ximal 2-cogr oups of G .  F o r a gro up G with neutral element e and a set A by ⊕ A G = { ( x α ) α ∈ A ∈ G A : |{ α ∈ A : x α 6 = e }| < ℵ 0 } we deno te the dire ct sum of | A | many copies of G . Lemma 19. 6. The gr oups ⊕ ω C 2 , ⊕ ω Z and C 2 ∞ × C 2 ∞ ar e not b K - c ount able. Pr o of. Observe that for any ab elian gro up X the num b er q ( X, C 2 ) is equal to the n um b e r of subgro ups having index 2 and is equa l to the num ber of non-trivial homo morphisms h : X → C 2 . Each (non- empt y) subset A ⊂ ω determines a (non-trivia l) homomorphism h A : ⊕ ω C 2 → C 2 , h A : ( x i ) i ∈ ω 7→ Y i ∈ A x i . F o r any distinct subsets A, B ⊂ ω the ho momorphisms h A and h B are distinct. Consequently , for the group X = ⊕ ω C 2 , the family b K o f maximal 2-cog roups has cardinality | b K| ≥ hom( X , C 2 ) = 2 ω and hence this group is not b K -countable. Since ⊕ ω C 2 is a quo tien t group of ⊕ ω Z , the latter group is not b K -countable. Finally , we show that the group X = C 2 ∞ × C 2 ∞ is no t b K -countable. It is well-kno wn (see [7, § 43]) that the quasic y clic group C 2 ∞ has uncountable automo rphism gro up Aut( C 2 ∞ ). F or any automorphism h : C 2 ∞ → C 2 ∞ its gr aph Γ h = { ( x, h ( x )) : x ∈ C 2 ∞ } is a subgr oup of X = C 2 ∞ × C 2 ∞ such that the q uotient group X/ Γ h is isomor phic to C 2 ∞ . Consequently , q ( X , C 2 ∞ ) ≥ Auth( C 2 ∞ ) > ℵ 0 and hence the group X = C 2 ∞ ⊕ C 2 ∞ is not b K - countable.  Now we are able to pr ov e the equiv alenc e (2) ⇔ (3). The implication (2) ⇒ (3 ) follows fro m Lemmas 19 .5 and 19.6. T o prov e the implication (3) ⇒ (2 ) o f Theorem 1 9.4, assume that a n ab elian group X has ﬁnite free and 2- ranks and X a dmits no homomorphism onto the g roup C 2 ∞ ⊕ C 2 ∞ . By Pr o po sition 19 .1, the cardinality o f the set b K o f maximal 2-cogr oups in X is equal to the cardina lit y of the family H o f subgroups H ⊂ X such that the quotient group X/H is isomorphic to C 2 k fo r some 1 ≤ k ≤ ∞ . So, it suﬃces to prove that |H| ≤ ℵ 0 . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 41 Consider the subgroup X od d ⊂ X consisting of the elemen ts of odd order. Since X has ﬁnite free and 2-ranks, so do e s the quotient group X/X od d . Then quotien t group Y = X/X od d is at mos t co un table (b ecause it contains no elements of o dd order and ha s ﬁnite free and 2-ranks). Let q : X → Y b e the quotient homomorphism. Let M b e the family of maximal indep endent subsets co nsisting o f elements o f inﬁnite o rder in the gr oup Y = X/ X od d . Since the fr e e rank of Y is ﬁnite, each (independent) set M ∈ M is ﬁnite and hence M is at most coun table. F o r each M ∈ M consider the free a belia n subgr o up h M i ⊂ Y gener ated by M . Let G M = Y / h M i b e the quotient group and q M : Y → G M be the q uotient homomorphism. The maxima lit y of M implies that G M is a torsion group. Since the fr e e a nd 2-ra nk s of the gr oup Y are ﬁnite, the quotient group G M has ﬁnite 2- rank. The gro up G M is the direct sum G M = O M ⊕ D M of the subgroup O M of elements o f odd orde r and the maximal 2-subgr oup D M ⊂ G M . Let p M : G M → D M = G M /O M be the quotient homomorphism. W e claim that the g roup D M has at mo st countably ma ny subg roups. Since D M is a quo tien t group of X a nd X admits no homomorphism onto the gr oup ( C 2 ∞ ) 2 the g roup D M also admits no ho momorphism ont o C 2 2 ∞ . Tw o ca ses are p ossible. 1) The group D M contains no subgro up isomor phic to C 2 ∞ . In this case the Pr ¨ ufer’s Theor em 17.2 [7] guara n tees that D M is a direct sum of cyclic 2-g roups. Since D M has ﬁnite 2-rank , it is ﬁnite, being a ﬁnite sum of cyclic 2 -groups. Then D M has ﬁnitely many subgroups. 2) The gr oup D M contains a subgroup D ⊂ M isomor phic to C 2 ∞ . Being divisible, the subgro up D is complemented in D M , whic h means that D M = D ⊕ F for so me subgroup F ⊂ D M . Since D M admits no ho momorphism o n to ( C 2 ∞ ) 2 , the subgroup F contains no subgr oup isomorphic to C 2 ∞ and hence is ﬁnite b y the pr e ceding c ase. T aking int o account that the qua sicyclic 2 -group D has coun tably many subgr oups, we conclude that the gr o up D M = D ⊕ F also has countably many subgroups. In both c a ses the family D M of subgroups of D M is at most countable. Then the family H M = { ( p M ◦ q M ◦ q ) − 1 ( H ) : H ∈ D M } also is a t most countable. It remains to chec k that H ⊂ S M ∈M H M . Fix any subgroup H ∈ H . By the deﬁnition of H , the quo tient g r oup X/ H is a 2-g roup, which implies X od d ⊂ H . Then H = q − 1 ( H Y ) where H Y = q ( H ). Let M b e a maximal indep endent subset o f H Y that co ns ists of elements of inﬁnite order . Since Y /H Y = X/H is a tors ion group, the set M is max imal in Y and hence be longs to the family M . It follows that h M i ⊂ H Y and hence H Y = q − 1 M ( H M ) wher e H M = q M ( H Y ) ⊂ G M . Since G M /H M = Y /H Y = X/H is a 2-group, the subgro up H M contains the subgro up O M of elements of o dd or der in G M . Then H M = p − 1 M ( G M ) where G M = p M ( H M ) ⊂ D M . Since G M ∈ D M , we conclude that the group H = ( p M ◦ q M ◦ q ) − 1 ( G M ) belo ngs to the family H M ⊂ H .  20. Comp act reflexions of groups In this s e ction X is an arbitrar y gro up. Till this moment our strategy in describing the minima l left ideals of the semigroups λ ( X ) consisted in ﬁnding a relatively small subfamily F ⊂ P ( X ) such that the function representation Φ F : λ ( X ) → End λ ( F ) is injectiv e on all minima l left idea ls of λ ( X ). Now we shall simplify the group X prese rving the minimal left idea ls of λ ( X ) unchanged. W e shall describ e three s uch simplifying pro ce dur es. O ne of them is the factoriza tion of X by the subgr oup Odd = \ K ∈ b K K K . Here we assume that Odd = X if the s e t b K is empt y . The following pro po sition explains the choice of the no tation for the subgr oup Odd. W e r ecall that a gr oup G is called o dd if each elemen t of G has o dd order. Prop ositio n 20.1. O dd is the lar gest normal o dd sub gr oup of X . If X is Ab elian, then Odd c oincides with t he set of al l elements having o dd or der in X . Pr o of. The normality of the subgroup Odd = T K ∈ b K K K follows from the fact that xK x − 1 ∈ b K for every K ∈ b K and x ∈ X . Next, we show that the group Odd is o dd. Assuming the converse, w e could ﬁnd an element a ∈ Odd such that the s e ts a 2 Z = { a 2 n : n ∈ Z } and a 2 Z +1 = { a 2 n +1 : n ∈ Z } are disjoint . Then the 2-c o group a 2 Z +1 of X can be enlarge d to a max imal 2-cog r oup K ∈ b K . It follows that a ∈ K ⊂ X \ K K and th us a / ∈ Odd, which is a contradiction. It r emains to prove that Odd contains any normal odd subg roup H ⊂ X . It suﬃces to c heck that for every max imal 2-cogr oup K ∈ b K the subgro up H ⊂ X lies in the group K K . Le t K ± = K ∪ K K . Since the subg roup H is normal in X , the sets K K H = H K K and K ± H = H K ± are subgr oups. W e claim that the sets K H = H K and K K H = H K K are disjoint. Assuming that K H ∩ K K H 6 = ∅ , we can ﬁnd a p oint x ∈ K such that x ∈ K K H . Since K K = xK , there a re p o int s z ∈ K and h ∈ H such that x = xz h . Then z = h − 1 ∈ K ∩ H . No w co nsider the cyclic subg r oup z 2 Z = { z 2 n : n ∈ Z } . Since z ∈ K , the subg roup z 2 Z do es not intersect the set z 2 Z +1 = { z 2 n +1 : n ∈ Z } . On the other 42 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV hand, since H is odd, there is an int eger num b er n ∈ Z with z 2 n +1 = z 0 ∈ z 2 Z +1 ∩ z 2 Z . This contradiction shows that K H and K K H a re disjoint. Consequently , the subg roup K K H has index 2 in the group K ± H and hence K H = K ± H \ K K H is a 2 -cogro up in X containing H . The max imalit y of K in K guara n tees that K = K H a nd hence H ⊂ K K .  The quotient homomorphis m q od d : X → X/ Odd g e ne r ates a contin uous semigro up homomorphism λ ( q od d ) : λ ( X ) → λ ( X/ Odd). The following theorem was proved in [4, 3.3]. Theorem 20 . 2. The homomorphism λ ( q od d ) : λ ( X ) → λ ( X/ Odd) is inje ctive on e ach minimal left ide al of λ ( X ) . Next, we deﬁne tw o compact top olo gical groups ca lled the ﬁrst a nd second pro ﬁnite reﬂe x ions of the gro up X . T o deﬁne the ﬁrst proﬁnite reﬂexion, consider the family N o f all normal subgroups of X w ith ﬁnite index in X . F or each subgroup H ∈ N consider the quotient homomorphism q H : X → X/H . The diagonal pro duct o f those ho momorphisms determines the homomorphism q : X → Q H ∈N X/H of X into the compac t topo logical group Q H ∈N X/H . The closure of the image q ( X ) in Q H ∈N X/H is denoted b y ¯ X a nd is called the pr oﬁnite r eﬂexion of X . The second proﬁnite reﬂexion ¯ X 2 is deﬁned in a similar wa y with help o f the subfamily N 2 = n \ x ∈ X xK K x − 1 : K ∈ b K , | X/K | < ℵ 0 o of N . The quo tient ho mo morphisms q H : X → X/H , H ∈ N 2 , comp ose a homomorphism q 2 : X → Q H ∈N 2 X/H . The closure of the ima g e q 2 ( X ) in Q H ∈N 2 X/H is denoted by ¯ X 2 and is called the se c ond pr oﬁnite r eﬂexion of X . Since Ker( q 2 ) = T N 2 ⊃ T K ∈ b K K K ⊃ Odd, the homomorphism q 2 : X → ¯ X 2 factorizes through the group X/ Odd in the sense that there is a unique homomo rphism q even : X/ Odd → ¯ X 2 such tha t q 2 = q even ◦ q od d . Thu s we get the following comm utative dia gram: X q odd / / q 2 " " F F F F F F F F F q   X/ Odd q even   ¯ X pr / / ¯ X 2 Applying to this diagram the functor λ of sup erextension we get the dia gram λ ( X ) λ ( q odd ) / / λ ( q 2 ) % % K K K K K K K K K K λ ( q )   λ ( X/ Odd) λ ( q even )   λ ( ¯ X ) λ (pr) / / λ ( ¯ X 2 ) In this diag ram λ ( ¯ X ) and λ ( ¯ X 2 ) ar e the sup erextensions o f the co mpact top olog ic al g r oups ¯ X and ¯ X 2 . W e reca ll that the sup e rextension λ ( K ) of a compa c t Hausdorﬀ space K is the close d subspace of the second exp onent exp(exp( K )) that consists of the maximal linked systems of closed subsets of K , see [22, § 2.1.3]. Theorem 20. 3 . If e ach maximal 2-c o gr oup K of a twinic gr oup X has ﬁn ite index in X , then the homomorphism λ ( q 2 ) : λ ( X ) → λ ( ¯ X 2 ) is inje ctive on e ach minimal left ide al of λ ( X ) . Pr o of. The injectivity of the homomorphism λ ( q 2 ) on a minimal left ideal L o f λ ( X ) will follow as so on as for any distinct maximal linked systems A , B ∈ L w e ﬁnd a subgr oup H ∈ N 2 such tha t λq H ( A ) 6 = λq H ( B ). Fix any [ b K ]-selecto r e K ⊂ b K . By Co rollar y 1 8.5, the homomor phis m Φ e T : λ ( X ) → Q K ∈ e K End λ ( T K ), Φ e T : L 7→ (Φ L | T K ) K ∈ e K is injective on the minimal left ideal L . Consequently , Φ A | T K 6 = Φ B | T K for s o me K ∈ e K a nd we can ﬁnd a set T ∈ T K such that Φ A ( T ) 6 = Φ B ( T ). Since the 2-co group K has ﬁnite index in X , the normal subgroup H = T x ∈ X xK K x − 1 has ﬁnite index in X a nd belo ngs to the family N 2 . Consider the ﬁnite quotient g roup X/H and let q H : X → X/H b e the quo tien t homomorphism. Since H ⊂ K K , the set T = K K T coincides with the preimag e q − 1 H ( T ′ ) o f some twin s et T ′ ∈ X/ H . This fact can b e used to show that λq H ( A ) 6 = λq H ( B ).  Remark 20 . 4. F or ea c h ﬁnite abe lian gr oup X the gr oup X/ Odd is a 2 -group. F or non-co mmutative gro ups it is not alwa ys true: for the gr oup X = A 4 of even p ermutations of the set 4 = { 0 , 1 , 2 , 3 } the g roup X/ Odd coincides with X , see Section 2 1.5. Also X/ Odd coincides with X for a ny s imple gro up. ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 43 21. Some examples Now we consider the sup erex tens io ns of so me concrete groups. 21.1. The i nﬁnite cyclic group Z . In order to compare the a lgebraic prop erties of the semigroups λ ( Z ) a nd β ( Z ) let us reca ll a deep result of E. Zelenyuk [23] (see also [12 , § 7.1]) who pr ov ed tha t each ﬁnite subgroup in the subsemigroup β ( Z ) ⊂ λ ( Z ) is trivial. It tur ns out that the semigroup λ ( Z ) has a totally diﬀerent prop erty . Theorem 21 . 1. (1) The semigr oup λ ( Z ) c ontains a princip al left ide al t op olo gic al ly isomorp hic t o ∞ Q k =1 C 2 k ≀ Z Z k k wher e Z k = 2 2 k − 1 − k . (2) Each minimal left ide al of λ ( Z ) is top olo gic al ly isomorph ic to 2 ω × Q ∞ k =1 C 2 k wher e the Cantor cub e 2 ω is endow e d with the left-zer o multiplic atio n. (3) e ach maximal gr oup of the minimal ide al K ( λ ( Z )) is top olo gic al ly isomorphic to ∞ Q k =1 C 2 k . (4) The semigr oup λ ( Z ) c ontains a top olo gic al ly isomorphic c opy of e ach se c ond c oun table pr oﬁnite top olo gic al semi- gr oup. Pr o of. The group Z is ab elian a nd hence has tr iv ial twinic ideal accor ding to Theore m 6.2. It is ea sy to see that q ( Z , C 2 k ) = 1 for a ll k ∈ N , while q ( Z , C 2 ∞ ) = 0. 1. B y Theor em 19.3(6), the s emigroup λ ( Z ) contains a principal left idea l that is top ologically iso morphic to Q ∞ k =1 C 2 k ≀ Z Z k k where Z k = 2 2 k − 1 − k . 2. By Theorem 19.3(5), ea c h minimal left ide a l L o f λ ( Z ) is top ologic a lly isomorphic to Q ∞ k =1 C 2 k × Z k where each cube Z k = 2 2 k − 1 − k is endow ed w ith the left zero multiplication. It is easy to see that the left zer o semigroup Q ∞ k =1 Z k is top ologically isomorphic to the Cantor cube 2 ω endow ed with the left zero multiplication. Consequently , L is top ologically isomorphic to 2 ω × Q ∞ k =1 C 2 k . 3. The preceding item implies that ea ch ma ximal group o f the minimal idea l K ( λ ( Z )) is top olo gically isomorphic to ∞ Q k =1 C 2 k . 4. The fourth item follows from the ﬁrs t item and the following well-known fact, se e [5, I.1.3].  Lemma 21.2 . Each s emigr oup S is algebr aic al ly isomorphic t o a subsemigr oup of the semigr oup A A of al l self-maps of a set A of c ar dinality | A | ≥ | S 1 | wher e S 1 is S with attache d un it. 21.2. The (quasi)cyclic 2-groups C 2 n . F or a cyclic 2-group X = C 2 n the num ber q ( X , C 2 k ) = ( 1 if k ≤ n 0 otherwise . Applying Theor em 19.3 we get: Theorem 21 . 3. F or every n ∈ N (1) The semigr oup λ ( C 2 n ) c ontains a princip al left ide al isomorphic to n Q k =1 C 2 k ≀ Z Z k k wher e Z k = 2 2 k − 1 − k . (2) Each minimal left ide al of λ ( C 2 n ) is isomorphic to Q n k =1 C 2 k × Z k wher e e ach cub e Z k = 2 2 k − 1 − k is endowe d with left-zer o mu ltiplic ation. (3) Each maximal gr oup of t he minimal ide al K ( λ ( C 2 n )) is isomorph ic to n Q k =1 C 2 k . (4) The semigr oup λ ( C 2 n ) c ontains an isomorphic c opy of e ach semigr oup S of c ar dinality | S | < 2 2 n − 1 − n . The sup erextension λ ( C 2 ∞ ) has even more interesting prop erties. Theorem 21 . 4. (1) Minimal left ide als of the semigr oup λ ( C 2 ∞ ) ar e not t op olo gic al semigr oups. (2) e ach minimal left ide al of λ ( C 2 ∞ ) is home omorphi c to the Cantor cub e 2 ω and is algebr aic al ly isomorphic to c × ( C 2 ∞ ) ω wher e t he c ar dinal c = 2 ℵ 0 is endowe d with left zer o multiplic ation; (3) the semigr oup λ ( C 2 ∞ ) c ontains a princip al left ide al, which is algebr aic al ly isomorphic t o ( C 2 ∞ ≀ c c ) ω ; (4) λ ( C 2 ∞ ) c ontains an isomorphic c opy of e ach semigr oup of c ar dinality ≤ c ; (5) e ach maximal su b gr oup of t he minimal ide al K ( λ ( C 2 ∞ )) of λ ( C 2 ∞ ) is algebr aic al ly isomorphi c t o ( C 2 ∞ ) ω ; (6) e ach maximal sub gr oup of t he minimal ide al K ( λ ( C 2 ∞ )) is top olo gic al ly isomorphic t o the c ountable pr o duct Q ∞ n =1 ( C 2 ∞ , τ n ) of quasicyclic 2-gr oups endowe d with twin-gener ate d top olo gies; 44 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV (7) for any twin-gener ate d top olo gies τ n , n ∈ N , on C 2 ∞ the right-top olo gic al gr oup Q ∞ n =1 ( C 2 ∞ , τ n ) is top olo gic al ly isomorphi c to a maximal su b gr oup of K ( λ ( C 2 ∞ )) . Pr o of. Since ea c h prop er subgro up of C 2 ∞ is ﬁnite, the family b K o f maximal 2-cog roups is coun table and hence can b e enum erated as b K = { K n : n ∈ ω } . Each maximal 2 -cogro up K ∈ b K has inﬁnite index and its characteris tic group H ( K ) is isomorphic to C 2 ∞ . 1. The equiv ale nc e (1) ⇔ (2) of Theor em 1 9.3 implies that no minimal left ideal of λ ( C 2 ∞ ) is a top ological semigroup. 2,3,5. The statements (2), (3) and (5) follow from Theor em 19.2. 4. The forth item fo llows from the third one beca use each semigr o up S o f cardina lit y | S | ≤ c e m be ds into the semigroup c c according to Lemma 21.2. 6. By Theorem 18.7(b), each max imal subgr oup G in the minimal idea l K ( λ ( C 2 ∞ )) is topo logically isomor phic to the pro duct Q K ∈ b K H ( A K ) of the structure g r oups of suitable twin subsets A K ∈ T K = T [ K ] , K ∈ b K . F or each maxima l 2-cogr oup K ∈ b K the structure gro up H ( A K ) is just C 2 ∞ endow ed with a twin-generated top o logy . 7. Now assume conv ersely tha t τ n , n ∈ N , are twin genera ted top ologies on the qua sicyclic group C 2 ∞ . F or every n ∈ N ﬁnd a twin s ubset A n ∈ T K n whose structure g roup H ( A n ) is top olog ically isomorphic to ( C 2 ∞ , τ n ). By Theorem 18.8, the pro duct Q ∞ n =1 H ( A n ) is top olog ically isomorphic to so me maximal subgro up of K ( λ ( C 2 ∞ )).  Remark 21 . 5. Theo rems 21.4(7) and 9.5 imply that among maximal subgro ups of the minimal ideal o f λ ( C 2 ∞ ) there are: • Raiko v complete topo logical groups ; • incomplete totally bounded topo logical gro ups ; • paratop olo gical g roups, which are not top ologica l groups; • semitop ologica l groups, whic h are not paratop olog ic al gro ups. 21.3. The groups of generalized quaternions Q 2 n . W e start with the quaternion gro up Q 8 = {± 1 , ± i , ± j , ± k } . It contains 3 cyclic subgr oups of order 4 corre s po nding to 4-element max imal 2-cogr oups: K 1 = Q 8 \ h i i , K 2 = Q 8 \ h j i , K 3 = Q 8 \ h k i . The characteris tic gro ups of those 2-co groups ar e isomo r phic to C 2 . The trivial subgroup of Q 8 corres p onds to the maximal 2 -cogro up K 0 = {− 1 } whose character is tic gr oup coincides with Q 8 . By Pro po sition 14.2, w e get | [ T K 0 ] | = | T K 0 | | H ( K 0 ) | = 2 | X/K ± 0 | | Q 8 | = 2 and | [ T K i ] | = 1 for i ∈ { 1 , 2 , 3 } . By Theore m 1 8.7(2), each minimal left ideal of the semigroup λ ( Q 8 ) is isomo r phic to ( Q 8 × 2) × ( C 2 × 1) 3 = 2 × Q 8 × C 3 2 . Next, given any ﬁnite num ber n ≥ 3 we c o nsider the genera lized quaternio n gr oup Q 2 n +1 . Ma ximal 2 -cogr o ups in Q 2 n +1 are of the follo wing form: K 0 = {− 1 } , K 1 = Q 2 n +1 \ C 2 n and K k,x = { 1 , x } · ( C 2 k \ C 2 k − 1 ) for 2 ≤ k ≤ n a nd x ∈ Q 2 n +1 \ C 2 n . It follows that H ( K 0 ) = Q 2 n +1 , H ( K 1 ) = C 2 and H ( K k,x ) = C 2 . Also | [ T K 0 ] | = | T K 0 | | H ( K 0 ) | = 2 | Q 2 n +1 /K ± 0 | | Q 2 n +1 | = 2 2 n 2 n +1 = 2 2 n − n − 1 , | [ T K 1 ] | = | T K 1 | | H ( K 1 ) | = 2 | Q 2 n +1 /K ± 1 | | C 2 | = 2 1 2 = 1 , and | [ T K k,x ] | = | T K k,x | | H ( K k,x ) | = 2 | Q 2 n +1 /K ± k,x | | C 2 | = 2 2 n +1 / 2 k +1 2 = 2 2 n − k − 1 . It is eas y to chec k that tw o 2-cogroups K k,x and K k,y are conjugated if and only if xy − 1 ∈ C 2 n − 1 . T aking any elemen ts x, y ∈ Q 2 n +1 \ C 2 n with xy − 1 / ∈ C 2 n , we conclude that the family e K = { K 0 , K k,x , K k,y : 2 ≤ k ≤ n } is a [ b K ]-selecto r . Applying Theo rems 18.7, 14 .1(3) and Corollar y 1 8.5(3), we get: Theorem 21 . 6. L et n ≥ 2 b e a ﬁnite numb er. Then ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 45 (1) e ach minimal left ide al of the semigr oup λ ( Q 2 n +1 ) is isomorph ic t o Q 2 n +1 × 2 2 n − n − 1 × C 2 × n Y k =2 ( C 2 × 2 2 n − k − 1 ) 2 , wher e the cub es 2 2 n − n − 1 and 2 2 n − k − 1 ar e endowe d with the left zer o mu lt iplic ation; (2) e ach maximal su b gr oup of t he minimal ide al K ( λ ( Q 2 n +1 )) is isomorph ic to Q 2 n +1 × C 2 n − 1 2 . The inﬁnite group Q 2 ∞ of generalize d quaternions has a similar structure. This gro up contains the following maximal 2-cogr oups: K 0 = {− 1 } , K 1 = Q 2 ∞ \ C 2 ∞ , and K k,x = { 1 , x } · C 2 k \ C 2 k − 1 where k ≥ 2 and x ∈ Q 2 ∞ \ C 2 ∞ . F or these 2- c ogroups we g et H ( K 0 ) = Q 2 ∞ , H ( K 1 ) = C 2 , and H ( K k,x ) = C 2 and | [ T K 0 ] | = c , | [ T K 1 ] | = 1 , and | [ T K k,x ] | = c . An y tw o 2-co groups K k,x , K k,y are conjugated. T he n for any b ∈ Q 2 ∞ \ C 2 ∞ the family e K = { K 0 , K k,b : k ∈ N } is a [ b K ]-selecto r . By a nalogy with Theor em 21.4 we can prov e: Theorem 21 . 7. F or the gr oup Q 2 ∞ (1) minimal left ide als of t he semigr oup λ ( C 2 ∞ ) ar e not top olo gic al semigr oups; (2) e ach minimal lef t ide al of the semigr oup λ ( Q 2 ∞ ) is home omorphic to the Cantor cub e and is algebr aic al ly isomor- phic to Q 2 ∞ × C ω 2 × c , wher e the c ar dinal c is endowe d with the left zer o multiplic ation; (3) the semigr oup λ ( Q 2 ∞ ) c ontains a princip al ide al isomorphic to ( Q 2 ∞ ≀ c c ) × C 2 × ( C 2 ≀ c c ) ω ; (4) λ ( Q 2 ∞ ) c ontains an isomorphic c opy of e ach semigr oup of c ar dinality ≤ c ; (5) e ach maximal sub gr oup of the minimal ide al K ( λ ( Q 2 ∞ )) is top olo gic al ly isomorphic to ( Q 2 ∞ , τ ) × C ω 2 wher e τ is a twin-gener ate d top olo gy on Q 2 ∞ ; (6) for any twin-gener ate d top olo gy τ on Q 2 ∞ the right-top olo gic al gr oup ( Q 2 ∞ , τ ) × C ω 2 is top olo gic al ly isomorphic to a maximal sub gr oup of K ( λ ( Q 2 ∞ )) . Remark 21 . 8. Theorems 21.7(6) and 9.5 imply tha t among ma x imal subgroups of the minimal idea l of λ ( Q 2 ∞ ) there are: • Raiko v complete topo logical groups , • incomplete totally bounded topo logical gro ups , • right-topo lo gical groups, which ar e not left-top olog ical groups, • semitop ologica l groups, whic h are not paratop olog ic al gro ups. 21.4. The dihedral 2 -groups D 2 n . By the dihe dr al gr oup D 2 n of even or der 2 n we understand any group with presen- tation h a, b | a n = b 2 = 1 , bab − 1 = a − 1 i . It can be re a lized a s the group of symmetries of a r egular n -gon. So, D 2 n is a s ubg roup of the or thogonal gr o up O (2). The group D 2 n contains the cyclic subgroup C n = h a i as a subgroup of index 2. The subgro up of all elements of odd order is normal in D 2 n and hence coincide with the maximal nor mal odd subgroup Odd. By Theo rem 20.2, the s up er e xtension λ ( D 2 n ) is isomorphic to the sup erextension λ ( D 2 n / Odd) of the quotient group D 2 n / Odd. The latter group is isomor phic to the dihedral group D 2 k where 2 k maximal p ow er of 2 that divides 2 n . Therefor e it suﬃces to consider the sup e r extensions of the dihedral 2-gro ups D 2 k . By the inﬁnite dihe dr al 2-gr oup w e unders ta nd the union D 2 ∞ = [ k ∈ N D 2 k ⊂ O (2) . It contains the quasicyclic 2-g roup C 2 ∞ as a nor ma l subgroup of index 2 . Now we ana lyze the structure of the sup erextension λ ( D 2 n ) for ﬁnite n ≥ 1. Maxima l 2 -cogro up in D 2 n are of the following for m: K 0 = D 2 n \ C 2 n − 1 and K k,x = { 1 , x } · ( C 2 k \ C 2 k − 1 ) 46 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV where 1 ≤ k < n and x ∈ K 0 = D 2 n \ C 2 n − 1 . The characteris tic groups of these maximal 2 -cogro ups are isomorphic to the 2-element cyclic group C 2 . Also | [ T K 0 ] | = 1 and | [ T K k,x ] | = | 2 D 2 n /K ± k,x | | H ( K ) | = 2 2 n − k − 1 for all 1 ≤ k < n and x ∈ K 0 . Let b ∈ D 2 n \ C 2 n − 1 be any elemen t and a be the generator o f the cyclic subgroup C 2 n − 1 ⊂ D 2 n . One can chec k that t wo 2-cogroups K k,x and K k,y are conjuga ted if and only if x − 1 y ∈ C 2 n − 2 . Therefore the family e K = { K 0 , K k,b , K k,ab : 1 ≤ k < n } is a [ b K ]-selecto r . Applying Theorems 18 .7, 14.1 and Cor ollary 18.5(3), we get Theorem 21 . 9. F or every n ∈ N (1) The semigr oup λ ( D 2 n ) c ontains a princip al left ide al isomorphic to C 2 × n − 1 Q k =1 ( C 2 ≀ Z Z k k ) 2 wher e Z k = 2 2 n − k − 1 . (2) Each minimal lef t id e al of λ ( D 2 n ) is isomorphic to C 2 × Q n k =1 ( C 2 × Z k ) 2 wher e cub es Z p ar e en dowe d with left-zer o multiplic ation. (3) Each maximal gr oup of t he minimal ide al K ( λ ( D 2 n +1 )) is isomorph ic to C 2 n − 1 2 . (4) The semigr oup λ ( D 2 n +1 ) c ontains an isomorph ic c opy of e ach semigr oup S of c ar dinality | S | < 2 2 n − 1 − 1 . The sup erextensio n of the inﬁnite dihedral 2-group D 2 ∞ has quite interesting pro per ties. All maximal subgroups o f the minimal ideal K ( λ ( D 2 ∞ )) are compact top olo gical g roups. On the other ha nd, in the semigroup λ ( D 2 ∞ ) ther e a re minimal left idea ls, which are (or are not) to po logical semigro ups. Theorem 21 . 10. F or the gr oup D 2 ∞ (1) e ach minimal left ide al of the semigr oup λ ( D 2 ∞ ) is home omorphi c to the Cantor cub e 2 ω and is algebr aic al ly isomorphi c to the c omp act top olo gic al semigr oup C ω 2 × 2 ω wher e the Cantor cub e 2 ω is endowe d with the left zer o multiplic ation; (2) e ach maximal sub gr oup of the minimal ide al K ( λ ( D 2 ∞ )) is top olo gic al ly isomorphic to the c omp act t op olo gic al gr oup C ω 2 ; (3) λ ( D 2 ∞ ) c ontains a minimal left ide al, which is top olo gic al ly isomorphic to the c omp act top olo gic al s emigr oup C ω 2 × 2 ω ; (4) λ ( D 2 ∞ ) c ontains a minimal left ide al, which is not a semitop olo gic al semigr oup. (5) the semigr oup λ ( D 2 ∞ ) c ontains a princip al ide al isomorphi c to C 2 × ( C 2 ≀ c c ) ω ; (6) λ ( D 2 ∞ ) c ontains an isomorphic c opy of e ach semigr oup of c ar dinality ≤ c . Pr o of. First no te that by Theorem 6.2 the tor sion group X = D 2 ∞ is twinic and has trivia l twinic ideal. Maximal 2-co group in D 2 ∞ are of the follo wing form: K 0 = D 2 ∞ \ C 2 ∞ and K k,x = { 1 , x } · ( C 2 k \ C 2 k − 1 ) where k ∈ N a nd x ∈ K 0 . The characteristic groups of these maximal 2- cogroups ar e isomorphic to the 2-element cyclic group C 2 . Cons equent ly , for a n y t win set A ∈ b T its c haracteristic group H ( A ) is top olog ically isomorphic to C 2 . Observe that | [ T K 0 ] | = 1 and | [ T K k,x ] | = 2 ω for all k ∈ N a nd x ∈ K 0 . Since the characteristic g roup H ( K k,x ) = C 2 is ﬁnite, the or bit spa ce [ T K p,x ] is a compact Hausdorﬀ space, homeo morphic to the Cantor cube 2 ω . O ne can chec k that any tw o 2-co groups K k,x and K k,y are conjugated. Therefore for any b ∈ D 2 ∞ \ C 2 ∞ the family e K = { K 0 , K k,b : k ∈ N } is a [ b K ]- selector. 1. By Theor em 18.7(e) and P rop osition 12.1, each minim al left idea l of λ ( X ) is ho meomorphic to the pro duct Q K ∈ e K T K , which is homeomorphic to 2 X/K ± 0 × Q k ∈ N 2 X/K ± k,b . The latter s pace is homeomor phic to the Cantor cube 2 ω . By Theorem 18.7(e), each minima l le ft ideal of λ ( X ) is a lgebraica lly isomor phic to Q K ∈ e K H ( K ) × [ T K ] and the latter semigroup is is o morphic to C ω 2 × 2 ω where the Ca ntor cube 2 ω is endow ed with the left zero m ultiplication. 2. T aking into account that each characteristic group H ( A ), A ∈ b T , is to p olo gically isomorphic to C 2 and applying Theorem 18.7(b), we conclude that each maximal subgroup in the minimal ideal K ( λ ( X )) is top olog ically isomor phic to the compact top olo gical group C ω 2 . ALGEBRA IN SUP EREXTENSIONS OF TWINIC GR OUPS 47 3. Since each characteristic g roup H ( K ), K ∈ b K , is ﬁnite (b eing iso morphic to C 2 ), Prop ositio n 1 8.10 implies that some minimal left ideal of λ ( X ) is a top ological semigr oup, whic h is topolog ically isomorphic to the compact top ologica l semigroup Y K ∈ e K H ( K ) × [ T K ] = H ( K 0 ) × [ T K 0 ] × Y k ∈ N ( H ( K k,b ) × [ T K k,b ]) by Theo rem 18.7(g). The latter top ological semigr oup is topolo gically isomor phic to C ω 2 × 2 ω . 4. Since the maximal 2-cogr oups K k,b , k ∈ N , hav e inﬁnite index in D 2 ∞ , Prop osition 18.9 implies that the semigroup λ ( D 2 ∞ ) contains a minimal left idea l, which is not a semitop olog ical semigroup. 5. B y Corollar y 1 8.5(3) and Theorem 14.1(3), the semigroup λ ( D 2 ∞ ) cont ains a principa l left idea l that is algebr aically isomorphic to the semigroup Q K ∈ e K ( H ( K ) ≀ [ T K ] [ T K ] ), which is isomorphic to C 2 × ( C 2 ≀ c c ) ω . 6. B y the preceding item, λ ( D 2 ∞ ) contains a subsemigro up, isomor phic to the semigroup c c of all self-ma pping of the contin uum c . By Le mma 21.2, the latter semigroup contains a n isomorphic copy of eac h semigroup of ca rdinality ≤ c .  21.5. Sup erextensio ns of ﬁnite groups o f order < 1 6 . The o rem 1 9 .3 and Prop os ition 19.1 give us a n algo rithmic wa y of calculating the minimal left idea ls of the sup erextensions of ﬁnitely-ge nerated ab elian gr oups. F or non-a belia n groups the situation is a bit more co mplica ted. In this s ection we sha ll desc r ib e the minimal left ideals of ﬁnite gro ups X of or de r | X | < 16. In fact, Theor em 20.2 helps us to re duce the problem to studying sup erex tensions of groups X / Odd. The group X/ Odd is trivial if the o rder of X is o dd. So, it suﬃces to chec k no n-ab elian groups of even o rder. If X is a 2 - group, then the subgroup Odd o f X is trivial and hence X/ Odd = X . Also the subgroup Odd is triv ia l for simple g roups. The next table describ es the structure of minimal left ideals of the sup erextensio ns of g roups X = X/ Odd of or der | X | ≤ 1 5. In this table E stands for a minimal idempotent of λ ( X ), which generates the principal left idea l λ ( X ) ◦ E and lies in the maximal subgroup H ( E ) = E ◦ λ ( X ) ◦ E . Below the cube s 2 n are consider ed as semigroups of left zeros. X | E ( λ ( X ) ◦ E ) | E ◦ λ ( X ) ◦ E λ ( X ) ◦ E C 2 1 C 2 C 2 C 4 1 C 2 × C 4 C 2 × C 4 C 2 2 1 C 3 2 C 3 2 C 3 2 1 C 7 2 C 7 2 C 2 ⊕ C 4 1 C 2 2 × C 2 4 C 3 2 × C 2 4 C 8 2 C 2 × C 4 × C 8 2 × C 2 × C 4 × C 8 D 8 2 C 5 2 2 2 × C 5 2 Q 8 2 C 3 2 × Q 8 2 × C 3 2 × Q 8 A 4 2 6 C 3 2 2 6 × C 3 2 F o r ab elian groups the entries of this table are calculated with help of The o rem 19.3 a nd Prop os ition 1 9.1. Let us illustrate this in the example of the group C 2 ⊕ C 4 . By Prop os ition 19.1, for the gro up X = C 2 ⊕ C 4 we g et • q ( X , C 2 ) = | ho m( X, C 2 ) | − | hom( X , C 1 ) | = 2 · 2 − 1 = 3; • q ( X , C 4 ) = 1 2 ( | hom( X, C 4 ) | − | ho m( X, C 2 ) | ) = 1 2 (2 · 4 − 2 · 2) = 2; • q ( X , C 2 k ) = 0 for k > 2. Then each minimal left ideal o f λ ( C 2 ⊕ C 4 ) is isomo r phic to ( C 2 × 2 2 1 − 1 − 1 ) q ( X ,C 2 ) × ( C 4 × 2 2 2 − 1 − 2 ) q ( X ,C 4 ) = ( C 2 × 2 0 ) 3 × ( C 4 × 2 0 ) 2 = C 3 2 × C 2 4 . Next, we consider the non- a be lia n g roups. In fact, the g roups Q 8 and D 8 hav e b een trea ted in Theorems 2 1.6 a nd 21.9. So, it r emains to consider the alternating gro up A 4 . This gro up has order 12, contains a normal subgroup isomorphic to C 2 × C 2 and contains no subgro up of or der 6. This implies that all 2-co groups of A 4 lie in C 2 × C 2 and co nsequently , A 4 contains 3 ma ximal 2-co groups. E ach maxima l 2-cogr oup K ⊂ A 4 contains tw o elements and ha s characteris tic group H ( K ) isomo rphic to C 2 . Since | X/K ± | = 3, Prop ositio n 14 .2 g ua rantees that | [ T K ] | = 2 | X/K ± | / | H ( K ) | = 2 3 − 1 = 2 2 . Applying Theor em 18.7, we see that each minimal left idea l of the se mig roup λ ( A 4 ) is isomo r phic to ( C 2 × 2 2 ) 3 = 2 6 × C 3 2 . 48 T ARAS BANAKH AND VOLOD YMYR GA VR YLKIV 22. Some Open Problems Problem 2 2.1. Describ e the struc tur e of (minimal left ideals) of sup erextens io ns of the simple groups A n for n ≥ 5. Problem 2 2.2. Describ e the struc tur e of (minimal left ideals) of sup erextens io ns of the ﬁnite groups of order 16. Since the free gr oup F 2 with tw o generators is not twinic, the results obtaine d in this pap er cannot b e a pplied to this group. Problem 2 2.3. What can b e said ab out the s tructure of the sup erex tension λ ( F 2 ) of the free group F 2 ? Problem 22. 4. Inv estigate the p ermanence pro pe r ties of the cla ss of twinic g roups. Is this cla s s closed under taking subgroups? pro ducts? References [1] S.I. 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Dep a r tmen t of M a them a tics , Iv a n Franko Na tional University of L viv, Un iversytetska 1 , 7 9000, Ukraine, a n d, In stytut Ma tema tyki, Uniwersytet Humanistyczno-Przyrodniczy w Kielcach,, ´ Swie ¸ tokrzyska 15, Kielce, Polan d, E-mail: tb anakh@y ahoo.com; t.o.banakh@gmail.com F acul ty of Ma thema tics and Compu ter Sciences,, V asyl Stef any k Precarp a thian Na tional University, Shevchenko str, 57, Iv ano-Frankivsk, 7 6025, Ukraine, E-mail: vga vr ylkiv@y ah oo.com

Algebra in superextensions of twinic groups

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