Algebra in superextensions of groups, I: zeros and commutativity

ALGEBRA IN SUPEREXTENSIONS OF GR OUPS, I: ZER OS AND COMMUT A TIVITY T.BANAKH, V.GA VR Y LK IV, O.NYKYFOR CHYN Abstract. Giv en a group X we study the algebraic structure of its superex- tension λ ( X ). This is a right-topological semigr oup consisting of all maxim al linked systems on X endo w ed with the op eration A ◦ B = { C ⊂ X : { x ∈ X : x − 1 C ∈ B } ∈ A} that extends the g roup operation of X . W e characterize righ t z eros of λ ( X ) as i n v ariant maximal linked systems on X and prov e that λ ( X ) has a right zero if and only if each element of X has odd order. On the othe r hand, the semigroup λ ( X ) con tains a left zero if and only if it conta i ns a zero if and only if X has odd orde r | X | ≤ 5. The semigr oup λ ( X ) i s commut ative if a nd only if | X | ≤ 4. W e finish the paper with a complete description of t he algebraic structure of the s emi groups λ ( X ) for all groups X of cardinali ty | X | ≤ 5. Contents Int r o duction 1 1. Self-linked sets in g roups 4 2. Maximal in v ariant linked systems 9 3. Right zero s in λ ( X ) 16 4. (Left) zeros of the semigroup λ ( X ) 18 5. The comm utativity of λ ( X ) 19 6. The superex tensions of finite groups 20 References 26 Introduction After the topolog ical pro o f of the Hindman theorem [H1 ] giv e n b y Galvin and Glazer 1 , to po logical methods b e c ome a standar d to ol in the mo dern com binato rics of n um b er s, see [HS], [P]. The crucial p oint is that any semig roup o pe ration ∗ defined on a discrete space X can b e extended to a r ight-topolog ical semigroup 1991 M athematics Subject Classific ation. 20M99, 54B20. 1 Unpublished, see [ HS, p.102], [H2] 1 2 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN op eration on β ( X ), the Stone- ˇ Cech compactification of X . The extension of the op eration from X to β ( X ) can b e defined by the simple for mula: (1) U ◦ V = { A ⊂ X : { x ∈ X : x − 1 A ∈ V } ∈ U } , where U , V are ultrafilters on X and x − 1 A = { y ∈ X : xy ∈ A } . Endow ed with the so-extended op era tion, the Stone- ˇ Cech compactification β ( X ) b ecomes a compact righ t- to po logical semigroup. The algebraic prop erties of this semigroup (for example, the existence of idemp otents or minimal left idea ls ) have impor tant consequences in combin atorics of n umbers, see [HS], [P]. The Stone- ˇ Cech compactification β ( X ) o f X is the subspace o f the double p ow er - set P ( P ( X )), which is a complete lattice with resp e ct to the o p erations o f union and intersection. In [G2] it was obser ved that the s emigroup op eration e xtends not only to β ( X ) but als o to the complete subla ttice G ( X ) of P ( P ( X )) generated by β ( X ). This complete sublattice co ns ists of all inc lus ion hyperspaces ov er X . By definition, a f amily F of non-empty subsets of a discr e te space X is ca lle d an inclusion hyp ersp ac e if F is monotone in the s e nse that a subset A ⊂ X belongs to F provided A con ta ins so me set B ∈ F . On the set G ( X ) there is an impor- tant transversality op eration assigning to each inclusio n h yp erspace F ∈ G ( X ) the inclusion h yp er space F ⊥ = { A ⊂ X : ∀ F ∈ F ( A ∩ F 6 = ∅ ) } . This oper ation is in volutiv e in the sense that ( F ⊥ ) ⊥ = F . It is k nown that the family G ( X ) of inclusion hyperspaces on X is clo sed in the double p ower-set P ( P ( X )) = { 0 , 1 } P ( X ) endow ed with the natural pro duct top ology . T he induced top ology o n G ( X ) c a n b e descr ibed directly: it is g enerated by the sub-bas e consisting of the sets U + = { F ∈ G ( X ) : U ∈ F } and U − = { F ∈ G ( X ) : U ∈ F ⊥ } where U runs ov er subsets of X . Endow ed with this topolo gy , G ( X ) becomes a Hausdorff s uper compact space. The latter means that each cov er of G ( X ) by the sub-basic sets has a 2-elemen t sub cov er . The extension of a binary operation ∗ from X to G ( X ) can b e defined in the same way a s for ult r afilters, i.e., b y the formula (1) applied to any t wo inclusio n hyperspaces U , V ∈ G ( X ). In [G2] it was s hown that for an ass o ciativ e bina ry op eration ∗ on X the space G ( X ) endow ed with t he extended op er ation b ecomes a compact right-topolo gical semigr oup. The algebr aic pr op erties of this semigro ups were studied in details in [G2]. ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 3 Besides the Stone- ˇ Cech compactification β ( X ), the semigro up G ( X ) contains many imp or ta nt spaces as closed subsemigroups. In particular, the space λ ( X ) = {F ∈ G ( X ) : F = F ⊥ } of max imal linked sys tems o n X is a closed subsemigr oup o f G ( X ). The spa ce λ ( X ) is well-known in General a nd Categoria l T op ology a s the sup er ex t ension of X , se e [vM], [TZ]. E ndow ed with the extended binary op eration, the sup erextensio n λ ( X ) of a se mig roup X is a sup erco mpact right-topolog ic a l semigroup containing β ( X ) as a subsemigroup. The space λ ( X ) c o nsists of max imal linked sys tems o n X . W e reca ll that a system of subsets L of X is linke d if A ∩ B 6 = ∅ for all A, B ∈ L . An inclusion hyperspace A ∈ G ( X ) is link ed if a nd only if A ⊂ A ⊥ . The family o f a ll link ed inclusion h y p er space on X is denoted by N 2 ( X ). It is a clos e d subset in G ( X ). Moreov er, if X is a se migroup, t hen N 2 ( X ) is a closed subsemigroup of G ( X ). The sup e rextension λ ( X ) consists of all maximal elemen ts o f N 2 ( X ), see [G1], [G2]. In this pap er we start a systematic inv estigation of the algebra ic structure of the semigroup λ ( X ). This progr am will b e cont inued in the forthcoming pap ers [BG2] and [BG3]. The interest to studying the semigroup λ ( X ) was motiv ated by the fact that for eac h maximal linked system L on X and each par tition X = A ∪ B of X into t wo sets A, B either A or B belongs to L . This makes p oss ible to apply maximal linked systems to Combinatorics and Ramsey Theory . In this pap er we concentrate on descr ibing ze ros and commutativit y o f the semi- group λ ( X ). In Pro po sition 3.1 we shall show that a maximal linked sys tem L ∈ λ ( X ) is a right zero of λ ( X ) if and o nly if L is inv ariant in the sense that xL ∈ L for all L ∈ L and all x ∈ X . In Theo rem 3.2 we s hall prove that a gr oup X admits an inv ariant maximal linked s ystem (equiv alen tly , λ ( X ) con ta ins a right zero) if and only if each element of X has o dd order. The situation with (left) zero s is a bit different: a max imal linked sys tem L ∈ λ ( X ) is a left zero in λ ( X ) if and only if L is a zero in λ ( X ) if and only if L is a unique inv aria nt max imal link ed system on X . The semigroup λ ( X ) has a (left) zero if and only if X is a finite group of odd orde r | X | ≤ 5 (equiv alently , X is iso mo rphic to the cy clic gr oup C 1 , C 3 or C 5 ). The semigr oup λ ( X ) ra rely is co mm uta tive: this holds if and only if the group X has finit e order | X | ≤ 4 . W e sta rt the pap er s tudying self-linked subs ets of gro ups. By definition, a subset A of a g roup X is ca lle d self-linke d if A ∩ xA 6 = ∅ fo r all x ∈ X . In Prop ositio n 1.1 we shall g ive low er and upp er b ounds for the smalles t cardinality sl ( X ) of a self-linked subset of X . W e use those bounds to characterize g roups X with sl ( X ) ≥ | X | / 2 in Theorem 1.2. 4 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN In Sec tio n 2 we apply self-linked sets to ev aluating the ca r dinality of the (rect- angular) semigro up ↔ λ ( X ) o f maximal in v ar iant linked systems o n a g r oup X . In Theorem 2.2 we show that for an infinite gro up X the cardinality of ↔ λ ( X ) equa ls 2 2 | X | . In Prop osition 2.3 and Theorem 2 .6 we calculate the cardina lit y of ↔ λ ( X ) for all finite gr oups X of order | X | ≤ 8 and also detect gro ups X with | ↔ λ ( X ) | = 1. In Sections 4 and 5 these r esults are a pplied for characteriz ing groups X who se sup e rextensions hav e zeros or are comm utative. W e finish the paper with a des c r iption of the algebr aic structure of the superex- tensions of groups X of o rder | X | ≤ 5. Now a couple of words ab out notations. F ollo wing the a lgebraic traditio n, b y C n we denote the cyclic group o f o rder n and by D 2 n the dihedr al g roup of cardinality 2 n , that is, the isometry group o f the r egular n -go n. F or a group X by e we denote the neutral elemen t o f X . F or a real n umber x we put ⌈ x ⌉ = min { n ∈ Z : n ≥ x } and ⌊ x ⌋ = max { n ∈ Z : n ≤ x } . 1. Self-linked sets in gr oups In this section we study self-linked subsets in gro ups. By definition, a subset A of a g roup G is self-linke d if A ∩ xA 6 = ∅ fo r each x ∈ G . In fact, this notion can b e defined in the more general con text of G -spaces. By a G -sp ac e we understand a set X e ndowed with a left action G × X → X of a group G . Each gro up G will be c onsidered as a G -space endow ed with the left action of G . An imp ortant example of a G -space is the homogeneous space G/H = { xH : x ∈ G } of a gro up G b y a subgr oup H ⊂ G . A subset A ⊂ X of a G -s pace X defined to b e self-linke d if A ∩ g A 6 = ∅ for all g ∈ G . Let us obser ve that a subset A ⊂ G of a group G is self-linked if and only if AA − 1 = G . F or a G -space X by sl ( X ) we denote the smalles t ca rdinality | A | of a self-linked subset A ⊂ X . Some low er and upper bounds for sl ( G ) are e s tablished in the following pro p o sition. Prop ositi on 1.1. L et G b e a finite gr oup and H b e a su b gr oup of G . Then (1) sl ( G ) ≥ (1 + p 4 | G | − 3) / 2 ; (2) sl ( G ) ≤ sl ( H ) · sl ( G/H ) ≤ sl ( H ) · ⌈ ( | G/H | + 1) / 2 ⌉ . (3) sl ( G ) < | H | + | G/H | . Pr o of. 1. T a ke any self-linked s et A ⊂ G of car dinality | A | = sl ( G ) and consider the surjectiv e map f : A × A → G , f : ( x, y ) 7→ xy − 1 . Since f ( x, y ) = xy − 1 = e for all ( x, y ) ∈ ∆ A = { ( x, y ) ∈ A 2 : x = y } , we g e t | G | = | G \ { e }| + 1 ≤ | A 2 \ ∆ A | + 1 = sl ( G ) 2 − sl ( G ) + 1, whic h just implies that sl ( G ) ≥ (1 + p 4 | G | − 3 ) / 2. ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 5 2a. Let H b e a s ubgroup of G . T ake self-linked sets A ⊂ H and B ⊂ G/H = { xH : x ∈ G } having sizes | A | = sl ( H ) and |B | = sl ( G/H ). Fix any subs et B ⊂ G such that | B | = |B | a nd { xH : x ∈ B } = B . W e claim that the set C = B A is self-linked. Given arbitr a ry x ∈ G w e should prov e that the in tersection C ∩ xC is not empty . Since B is self-linked, the int e r section B ∩ x B contains the co set bH = xb ′ H for some b, b ′ ∈ B . It follows that b − 1 xb ′ ∈ H = AA − 1 . The latter equality follows from the fa c t tha t the se t A ⊂ H is self-linked in H . Consequently , b − 1 xb ′ = a ′ a − 1 for some a, a ′ ∈ A . Then xC ∋ xb ′ a = ba ′ ∈ C a nd th us C ∩ xC 6 = ∅ . The self-link ednes s of C implies the desir e d upp er b ound sl ( G ) ≤ | C | ≤ | A | · | B | = sl ( H ) · sl ( G/H ) . 2b. Next, w e show that sl ( G/H ) ≤ ⌈ ( | G/H | + 1) / 2 ⌉ . T ake any subset A ⊂ G/H of size | A | = ⌈ ( | G/H | + 1) / 2 ⌉ and note that | A | > | G /H | / 2. Then for each x ∈ G the shift xA has s ize | x A | = | A | > | G/H | / 2. Since | A | + | xA | > | G/H | , the sets A and x A meet ea ch other. C o nsequently , A is self-linked and s l ( G/H ) ≤ | A | = ⌈ ( | G/H | + 1) / 2 ⌉ . 3. Pick a subset B ⊂ G of size | B | = | G/H | suc h that B H = G and observe that the set A = H ∪ B is self-linked and has size | A | ≤ | H | + | B | − 1 (b ecause B ∩ H is a singleton).  Theorem 1.2. F or a fi nite gr oup G (i) sl ( G ) = ⌈ ( | G | + 1) / 2 ⌉ > | G | / 2 if and only if G is isomorphic to one of the gr oups: C 1 , C 2 , C 3 , C 4 , C 2 × C 2 , C 5 , D 6 , ( C 2 ) 3 ; (ii) sl ( G ) = | G | / 2 if and only if G is isomorphic t o one of the gr oups: C 6 , C 8 , C 4 × C 2 , D 8 , Q 8 . Pr o of. I. First we e s tablish the inequality sl ( G ) < | G | / 2 for a ll g roups G not iso- morphic to the groups app ear ing in the items (i), (ii). Given such a gro up G w e should find a self-link ed subset A ⊂ G with | A | < | G | / 2 . W e consider 8 cases. 1) G co nt a ins a subgroup H of or der | H | = 3 and index | G/H | = 3. Then sl ( H ) = 2 and we ca n apply Prop o sition 1.1(2) to conclude that sl ( G ) ≤ sl ( H ) · sl ( G/H ) ≤ 2 · 2 < 9 / 2 = | G | / 2 . 2) | G | / ∈ { 9 , 12 , 15 } and G contains a subgroup H o f or der n = | H | ≥ 3 and index m = | G/H | ≥ 3. It this case n + m − 1 < nm/ 2 and sl ( G ) ≤ | H | + | G/H | − 1 = n + m − 1 < nm / 2 b y Pr op osition 1.1(3). 3) G is cyclic of or der n = | G | ≥ 9. Giv en a genera tor a of G , construct a sequence ( x i ) 2 ≤ i ≤ n/ 2 letting x 2 = a 0 , x 3 = a , x 4 = a 3 , x 5 = a 5 , and x i = x i − 1 a i 6 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN for 5 < i ≤ n/ 2. Then the s et A = { x i : 2 ≤ i ≤ n/ 2 } has size | A | < n/ 2 and is self-linked. 4) G is cyclic of o rder | G | = 7. Given a generator a of G observe that A = { e, a, a 3 } is a 3-elemen t self-linked subset and th us sl ( G ) ≤ 3 < | G | / 2. 5) G contains a c y clic subgro up H ⊂ G of pr ime order | H | ≥ 7. By the preceding t wo cases, sl ( H ) < | H | / 2 and then sl ( G ) ≤ sl ( H ) · sl ( G/H ) < | H | 2 · | G | | H | = | G | / 2 . 6) | G | > 6 and | G | / ∈ { 8 , 10 , 1 2 } . If | G | is prime or | G | = 15, then G is cyclic of or de r | G | ≥ 7 and th us has sl ( G ) < | G | / 2 by the items (3), (4). If | G | = 2 p for some pr ime num be r p , then G contains a cyclic s ubg roup of order p ≥ 7 and thus has sl ( G ) < | G | / 2 by the it e m (5 ). If | G | = 4 n for some n ≥ 4, then by Sylow’s Theorem (see [OA , p.7 4]), G c ontains a subg r oup H ⊂ G of o rder | H | = 4 and index | G/H | ≥ 4. Then sl ( G ) < | G | / 2 b y the item (2). If the ab ove conditio ns do not hold, then | G | = nm 6 = 15 for so me o dd num b ers n, m ≥ 3 and we can apply the items (1) and (2) t o conclude that sl ( G ) < | G | / 2. 7) If | G | = 8, then G is isomorphic to o ne of the g r oups: C 8 , C 2 × C 4 , ( C 2 ) 3 , D 8 , Q 8 . All those g roups app ea r in the items (i), (ii) and th us are excluded from our consideration. 8) If | G | = 10 , then G is isomorphic to C 10 or D 10 . If G is isomorphic to C 10 , then sl ( G ) < | G | / 2 b y the item (3). If G is isomorphic t o D 10 , then G contains an element a of order 5 and an element b of order 2 such tha t bab − 1 = a − 1 . Now it is easy to chec k that the 4-element set A = { e , a, b , ba 2 } is se lf-linked and hence sl ( G ) ≤ 4 < | G | / 2. 9) In this item w e consider gr oups G with | G | = 12. It is w e ll- known that ther e are fiv e non-isomorphic g roups of order 12: the cyclic group C 12 , the dir ect sum of t wo cyclic groups C 6 ⊕ C 2 , the dihedr al group D 12 , the a lternating group A 4 , and the semidirect pro duct C 3 ⋊ C 4 with presentation h a, b | a 4 = b 3 = 1 , aba − 1 = b − 1 i . If G is is omorphic to C 12 , C 6 ⊕ C 2 or A 4 , then G contains a normal 4-element subgroup H . By Sylow’s Theorem, G co ntains also an element a of or der 3. T aking int o account that a 2 / ∈ H and H a − 1 = a − 1 H , we conclude that the 5- e le men t set A = { a } ∪ H is s elf-linked and hence sl ( G ) ≤ 5 < | G | / 2 . If G is isomo rphic to C 3 ⋊ C 4 , then G co ntains a nor mal subgroup H o f order 3 a nd an element a ∈ G suc h that a 2 / ∈ H . Observe that the 5-element set A = H ∪ { a, a 2 } is self-linked. Indeed, AA − 1 ⊃ H ∪ aH ∪ a 2 H ∪ H a − 1 = G . Consequently , sl ( G ) ≤ 5 < | G | / 2. Finally , consider the case of the dihedral group D 12 . It contains an element a generating a cy c lic subgroup of o rder 6 and an element b of or der 2 such that bab − 1 = a − 1 . Consider the 5 -element set A = { e, a , a 3 , b, b a } and note that AA − 1 = { e, a, a 3 , b, b a } · { e, a 5 , a 3 , b, b a } = G . This yields the desired inequa lit y sl ( G ) ≤ 5 < 6 = | G | / 2. ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 7 Therefore w e ha ve completed the pr o of of the inequality sl ( G ) < | G | / 2 for all groups not appear ing in the items ( i),(ii) of the theorem. II. Now w e sha ll prov e the item ( i). The lower bound from Prop os ition 1.1(1) implies that sl ( G ) = ⌈ ( | G | + 1) / 2 ⌉ > | G | / 2 for all groups G wit h | G | ≤ 5. It remains to chec k that sl ( G ) > | G | / 2 if G is isomorphic to D 6 or C 3 2 . First we consider the case G = D 6 . In this case G c ontains a normal 3-element subgro up T . Assuming that sl ( G ) ≤ | G | / 2 = 3, find a self-linked 3-element subset A . Without loss of g enerality we can a ssume that the neutral elemen t e of G belongs to A (otherwise replace A by a suitable shift xA ). T aking in to account that AA − 1 = G , we conclude that A 6⊂ T and th us we can find a n element a ∈ A \ T . This element has order 2. Then AA − 1 = { e, a, b } · { e, a, b − 1 } = { e, a , b, a , e, ba , b − 1 , ba, e } 6 = G, which is a contradiction. Now as sume that G is isomorphic to C 3 2 . In this case G is the 3-dimensiona l linear space ov er the field C 2 . Assuming that s l ( A ) ≤ 4 = | G | / 2, find a 4-element self-linked subset A ⊂ G . Replacing A by a suitable shift, w e can assume that A contains a neutral element e o f G . Since AA − 1 = G , the set A contains three linearly independent po int s a, b, c . Then AA − 1 = { e, a, b, c } · { e , a, b , c } = { e , a, b , c, ab , ac, b c } 6 = G, which contradicts the c hoice of A . II I. Finally , w e prove the equalit y sl ( G ) = | G | / 2 for the groups app earing in the item (ii). If G = C 6 , then sl ( G ) ≥ 3 by Prop ositio n 1.1(1). On the other ha nd, we can chec k that for any generato r a of G the 3-element subset A = { e , a, a 3 } is self-linked in G , whic h yields sl ( G ) = 3 = | G | / 2. If | G | = 8, then sl ( G ) ≥ 4 by Pr op osition 1.1(1). If G is cyclic of or der 8 and a is a generator of G , then the set A = { e, a, a 3 , a 4 } is self-link ed and th us sl ( C 8 ) = 4. If G is is omorphic to C 4 ⊕ C 2 , then G ha s tw o commuting genera tors a, b such that a 4 = b 2 = 1 . One can chec k that the set A = { e, a, a 2 , b } is self-linked a nd th us sl ( C 4 ⊕ C 2 ) = 4. If G is isomorphic to the dihedral g roup D 8 , then G has t wo ge ne r ators a , b connected by the relations a 4 = b 2 = 1 and bab − 1 = a − 1 . One can chec k that the 4-element s ubset A = { e, a, b, ba 2 } is self-linked. 8 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN If G is is o morphic t o the group Q 8 = {± 1 , ± i, ± j, ± k } of q uaternion unit s , then we can check that the 4-element subset A = {− 1 , 1 , i, j } is self-linked and thus sl ( Q 8 ) = 4.  In the following prop osition w e complete Theo rem 1.2 calculating the v alues of the cardinal sl ( G ) for all groups G of cardinality | G | ≤ 1 3 . Prop ositi on 1.3. The numb er sl ( G ) for a gr oup G of size | G | ≤ 13 c an b e found fr om the table: G C 2 C 3 C 5 C 4 C 2 ⊕ C 2 C 6 D 6 C 8 C 2 ⊕ C 4 D 8 Q 8 C 3 2 sl ( G ) 2 2 3 3 3 3 4 4 4 4 4 5 G C 7 C 11 C 13 C 9 C 3 ⊕ C 3 C 10 D 10 C 12 C 2 ⊕ C 6 D 12 A 4 C 3 ⋊ C 4 sl ( G ) 3 4 4 4 4 4 4 4 5 5 5 5 Pr o of. F or g r oups G of or de r | G | ≤ 10 the v alue of sl ( G ) is uniquely deter mined by the low er bound sl ( G ) ≥ 1+ √ 4 | G |− 3 2 from Prop o sition 1.1(1) and the upper b ound from Theorem 1.2. It rema ins to consider the g roups G of order 11 ≤ | G | ≤ 13. 1. If G is cyclic of order 11 or 13, then take a ge ne r ator a o f G and chec k that the 4-element set A = { e , a 4 , a 5 , a 7 } is self-linked, witnessing that sl ( C 12 ) = 4. 2. If G is cyclic of order 12, then t a ke a gener ator a for G a nd check that the 4-element s ubset A = { e, a, a 3 , a 7 } is self-linked witnessing that sl ( G ) = 4. It remains to consider a ll o ther gro ups of order 12. Theo rem 1.2 gives us an upper b ound sl ( G ) ≤ 5 . So, we need to show that sl ( G ) > 4 for a ll non-cyclic groups G with | G | = 1 2. 3. If G is is o morphic to C 6 ⊕ C 2 or A 4 , then G contains a normal subgro up H isomorphic to C 2 ⊕ C 2 . Assuming that sl ( G ) = 4, we can find a 4-e le men t self-linked subset A ⊂ G . Since AA − 1 = G , we ca n find a suitable shift xA suc h that xA ∩ H contains the neutral elemen t e of G and some other elemen t a o f H . Replacing A by xA , w e can assume that e, a ∈ A . Since A 6⊂ H , there is a point b ∈ A \ H . Since the quotien t gr oup G/H has order 3 , bH ∩ H b − 1 = ∅ . Concerning the fo rth element c ∈ A \ { e, a, b } there are three p ossibilities: c ∈ H , c ∈ b − 1 H , and c ∈ bH . If c ∈ H , then bH = bH ∩ AA − 1 = b ( A ∩ H ) − 1 consists of 3 elements which is a c o ntradiction. If c ∈ b − 1 H , then H = H ∩ AA − 1 = { e , a } , which is absur d. So, c ∈ bH and thus c = bh for some h ∈ H . Since h = h − 1 , we get cb − 1 = b hb − 1 = b h − 1 b − 1 = b c − 1 . Then H = H ∩ AA − 1 = { e, a, cb − 1 , bc − 1 } has cardinality | H | = |{ e, a, cb − 1 = bc − 1 }| ≤ 3, whic h is not true. This co ntradiction completes the pro of of the inequalit y sl ( G ) > 4 for the groups C 6 ⊕ C 2 and A 4 . 4. Assume that G is isomorphic to the dihedral group D 12 . Then G contains a normal cyclic subgroup H of order 6, a nd for each b ∈ G \ H and a ∈ H we ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 9 get b 2 = e and bab − 1 = a − 1 . Assuming that sl ( D 12 ) = 4, we can find a 4- element self-link ed s ubset A ⊂ G . Let a be a generato r of the g r oup H . Since a ∈ AA − 1 = G , we can fin d t wo elemen t x, y ∈ A such that a = xy − 1 . Then the shift Ay − 1 contains e and a . Replacing A by Ay − 1 , if necessary , w e can a s sume that e, a ∈ A . Since A 6⊂ H , ther e is a n element b ∈ A \ H . Concerning the forth elemen t c ∈ A \ { e, a, b } there are t wo poss ibilities : c ∈ H and c / ∈ H . If c ∈ H , t he n the se t A H = A ∩ H = { e, a, c } contains three elements and is equal to bA − 1 H b − 1 , which implies bA − 1 H = A H b − 1 = bA − 1 H ∪ A H b − 1 = AA − 1 ∩ bH = bH . This is a contradiction, because | H | = 4 > 3 = | bA − 1 H | . Then c ∈ bH and hence H = H ∩ AA − 1 = { e, a, a − 1 , bc − 1 , cb − 1 } which is no t true because | H | = 6 > 5. 5. Assume that G is is omorphic to the semidirect pro duct C 3 ⋊ C 4 and hence has a presentation h a, b | a 4 = b 3 = 1 , aba − 1 = b − 1 i . Then the cyclic subgroup H generated by b is no r mal in G a nd the q uotient G/H is cyclic o f order 4 . Assuming that sl ( G ) = 4, ta ke any 4-elemen t self-linked s ubs e t A ⊂ G . After a suitable s hift of A , we can a ssume that e, b ∈ A . Since A 6⊂ H , there is an element c ∈ A \ H . W e claim that the four th element d ∈ A \ { e , b, c } does not belo ng to H ∪ c H ∪ c − 1 H . Otherwise, AA − 1 ⊂ H ∪ cH ∪ c − 1 H 6 = G . This implies that one o f the elements, say c b elongs to the coset a 2 H a nd the other to aH or a − 1 H . W e lose no generality a ssuming that d ∈ aH . Then c = a 2 b i , d = ab j for some i, j ∈ { − 1 , 0 , 1 } . It follows that aH = aH ∩ AA − 1 = { d, db − 1 , cd − 1 } = { ab j , ab j − 1 , a 2 b i − j a − 1 } = { ab j , ab j − 1 , ab j − i } which implies that i = − 1 a nd thus c = a 2 b − 1 . In this case w e a rrive to a contra- diction loo king at a 2 H ∩ AA − 1 = { c, cb − 1 , c − 1 , bc − 1 } = { a 2 b − 1 , a 2 b − 2 , ba 2 , b 2 a 2 } 6∋ a 2 .  Problem 1 .4. W hat is the value of sl ( G ) for other gr oups G of smal l c ar dinality? Is sl ( G ) = ⌈ (1 + p 4 | G | − 3) / 2 ⌉ for al l finite cyclic gr oups G ? 2. Ma ximal inv ariant linked systems In this section we study (max imal) inv ar iant linked sys tems o n gr oups. An inclusion hyper space A on a group X is ca lle d invariant if x A = A for all x ∈ X . The set of all inv a r iant inclusion h yp ers paces on X is denoted by ↔ G ( X ). By [G2], ↔ G ( X ) is a clo sed rectang ular subsemigroup o f G ( X ) coinciding with the minimal ideal o f G ( X ). The r e ctangularity of ↔ G ( X ) means that A ◦ B = B for all A , B ∈ ↔ G ( X ). Let ↔ N 2 ( X ) = N 2 ( X ) ∩ ↔ G ( X ) deno te th e set of all in v ariant link ed systems on X and ↔ λ ( X ) = max ↔ N 2 ( X ) b e the family of all ma ximal elements of ↔ N 2 ( X ). 10 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN Elements o f ↔ λ ( X ) are called maximal invariant linke d systems . The r e ader should be concisions of the fact that max ima l inv ar iant linked systems need no t be maximal linked! Theorem 2.1. F or every gr oup X the set ↔ λ ( X ) is a non-empty close d r e ctangular subsemigr oup of G ( X ) . Pr o of. The rectangular ity of ↔ λ ( X ) implies from t he rectangularity o f ↔ G ( X ) estab- lished in [G2, § 5] and the inclusion ↔ λ ( X ) ⊂ ↔ G ( X ). The Zorn Lemma implies that eac h in v ar iant linked system on X (in p articular, { X } ) ca n b e enlarged to a maximal inv a riant linked system o n X . This obse r v ation implies the set ↔ λ ( X ) is not empt y . Next, we show that the subsemigroup ↔ λ ( X ) is closed in G ( X ). Since the set ↔ N 2 ( X ) = N 2 ( X ) ∩ ↔ G ( X ) is clos e d in G ( X ), it suffices to show that ↔ λ ( X ) is close d in ↔ N 2 ( X ). T ake any inv ariant linked sy stem L ∈ ↔ N 2 ( X ) \ ↔ λ ( X ). Being not maximal inv ar iant, the linked sys tem L can b e enlarged to a maximal inv ariant link ed system M that c o ntains a subset B ∈ M \ L . Since M ∋ B is inv ar iant, the system { xB : x ∈ X } ⊂ M is link ed. Observe that B / ∈ L a nd B ∈ M ⊃ L implies X \ B ∈ L ⊥ and B ∈ L ⊥ . W e claim that O ( L ) = B − ∩ ( X \ B ) − ∩ ↔ N 2 ( X ) is a neighbor ho o d of L in ↔ N 2 ( X ) that misses the set ↔ λ ( X ). Indeed, for any A ∈ O ( L ), w e ge t that A is a n inv a riant link ed system such that B ∈ A ⊥ . Obser ve that for every x ∈ X and A ∈ A we get x − 1 A ∈ A b y the inv a riantness of A and hence the set B ∩ x − 1 A a nd its shift xB ∩ A b oth are no t empt y . This witnes ses tha t xB ∈ A ⊥ for e very x ∈ X . Then the max imal inv ar ia nt linked sys tem gener ated by A ∪ { xB : x ∈ X } is an inv ariant linked enlar g ement of A , whic h shows that A is no t maximal in v ar iant linked.  Next, w e shall ev aluate the cardinality of ↔ λ ( X ). Theorem 2.2. F or any infi nite gr ou p X t he semigr oup ↔ λ ( X ) has c ar dinality | ↔ λ ( X ) | = 2 2 | X | . Pr o of. The upper bo und | ↔ λ ( X ) | ≤ 2 2 | X | follows fr om the c hain of inclusions: ↔ λ ( X ) ⊂ G ( X ) ⊂ P ( P ( X )) . Now we prove that | ↔ λ ( X ) | ≥ 2 2 | X | . Let | X | = κ a nd X = { x α : α < κ } b e an injectiv e enumeration of X by ordinals < κ such that x 0 is the ne utr al element of X . F o r ev ery α < κ let B α = { x β , x − 1 β : β < α } . By tra nsfinite induct ion, c ho o se a transfinite sequence ( a α ) α<κ such that a 0 = x 0 and a α / ∈ B − 1 α B α A <α where A <α = { a β : β < α } . ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 11 Consider the set A = { a α : α < κ } . By [H S, 3 .58], the set U κ ( A ) of κ -uniform ultrafilters on A has cardina lity | U κ ( A ) | = 2 2 κ . W e recall that an ultrafilter U is κ -uniform if for ev ery s et U ∈ U and a ny subset K ⊂ U of size | K | < κ the set U \ K still b elongs to U . T o each κ -uniform ultrafilter U ∈ U κ ( A ) assign the inv a riant filter F U = T x ∈ X x U . This filter can be extended to a maximal inv ariant linked sy s tem L U . W e claim that L U 6 = L V for t wo different κ -uniform ultrafilters U , V on A . Indeed, U 6 = V yields a subset U ⊂ A suc h that U ∈ U and U / ∈ V . L e t V = A \ U . Since U , V are κ -uniform, | U | = | V | = κ . F or every α < κ consider the sets U α = { a β ∈ U : β > α } ∈ U a nd V α = { a β ∈ V : β > α } ∈ V . It is clear that F U = [ α<κ x α U α ∈ F U and F V = [ α<κ x α V α ∈ F V . Let us show that F U ∩ F V = ∅ . O therwise there would exist tw o or dinals α, β and points u ∈ U α , v ∈ V β such that x α u = x β v . It follo w s from u 6 = v that α 6 = β . W rite the p oints u, v as u = a γ and v = a δ for some γ > α and δ > β . Then we hav e the equality x α a γ = x β a δ . The inequality u 6 = v implies that γ 6 = δ . W e lose no generality ass uming that δ > γ . Then a δ = x − 1 β x α a γ ∈ B − 1 δ B δ A <δ which contradicts the c hoice of a δ . Therefore, F U ∩ F V = ∅ . T a king int o a ccount that the linked systems L U ⊃ F U ∋ F U and L V ⊃ F V ∋ F V contain disjoint sets F U , F V , we conclude that L U 6 = L V . Consequently , | ↔ λ ( X ) | ≥ |{L U : U ∈ U κ ( A ) }| = | U κ ( A ) | = 2 2 κ .  The preceding theorem implies that | ↔ λ ( G ) | = 2 c for a n y countable gr oup G . Next, w e ev a luate the cardinality of ↔ λ ( G ) for finite groups G . Given a finite gr oup G consider the in v ariant linked system L 0 = { A ⊂ X : 2 | A | > | G |} and the subset ↑L 0 = { A ∈ ↔ λ ( G ) : A ⊃ L 0 } of ↔ λ ( G ). Prop ositi on 2.3. L et G b e a finite gr oup. If s l ( G ) ≥ | G | / 2 , then ↔ λ ( G ) = ↑L 0 . 12 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN Pr o of. W e s hould prov e that each ma ximal inv aria nt linked system A ∈ ↔ λ ( G ) contains L 0 . T ake any set L ∈ L 0 . T aking int o a c count that sl ( G ) ≥ | G | / 2 and each se t A ∈ A is self-linked, w e conclude that | A | ≥ | G | / 2 and hence A in terse c ts each shift xL of L (bec ause | A | + | xL | > | G | ). Since the set L is self-linked, w e get that the inv ar iant linked system A ∪ { xL : x ∈ G } is equal to A b y the max imality of A . Conseque ntly , L ∈ A and hence L 0 ⊂ A .  In light of Prop osition 2 .3 it is imp ortant to ev aluate the ca rdinality of the set ↑L 0 . In | G | is odd, then the in v ariant linked system L 0 is maximal linked and thus ↑L 0 is a singleton. The ca se of ev en | G | is less trivial. Given an g r oup G of finite ev en order | G | , consider the family S = { A ⊂ G : AA − 1 = G, | A | = | G | / 2 } of self-link ed s ubsets A ⊂ G of cardinality | A | = | G | / 2. O n the family S consider the eq uiv alence rela tion ∼ letting A ∼ B f o r A, B ∈ S if there is x ∈ G such that A = xB or X \ A = xB . Let S / ∼ the quotien t set of S b y this equiv alence relation and s = |S / ∼ | stand for the cardinality of S/ ∼ . Prop ositi on 2.4. | ↔ λ ( G ) | ≥ |↑L 0 | = 2 s . Pr o of. First w e show that ∼ indeed is an equiv alence relation on S . So, a ssume that S 6 = ∅ . Let us show that G \ A ∈ S for every A ∈ S . Let B = G \ A . As s uming that B / ∈ S , we conclude that B ∩ xB = ∅ for some x ∈ G . Since | B | = | A | = | G | / 2 , we conclude that x B = A and G \ A = B = x − 1 A . The equalit y A ∩ x − 1 A = ∅ implies x − 1 / ∈ AA − 1 = G , which is a con tradiction. T aking into account that A = eA for every A ∈ S , we conclude that ∼ is a reflexive relation on S . If A ∼ B , then there is x ∈ X such that A = xB or G \ A = xB . This implies that B = x − 1 A or X \ B = x − 1 A , that is B ∼ A and ∼ is symmetric. It r emains to prov e that the r elation ∼ is transitive on S . So let A ∼ B ∼ C . This mea ns that there exist x, y ∈ G such that A = xB o r G \ A = xB and B = y C or G \ B = y C . It is easy to c heck that in t hese cases A = xy C or X \ A = xy C . Cho ose a s ubset T of S in ter secting each equiv alence class of ∼ at a single p oint. Observe that |T | = |S / ∼ | = s . Now for ev ery function f : T → 2 = { 0 , 1 } consider the maximal in v ar iant linked system L f = L 0 ∪ { xT : x ∈ G, T ∈ f − 1 (0) } ∪ { x ( G \ T ) : x ∈ G, T ∈ f − 1 (1) } . It can be shown that |↑L 0 | = |{L f : f ∈ 2 T }| = 2 |T | = 2 s .  ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 13 This prop osition will help us to calculate the car dinality of the set ↔ λ ( G ) for all finite groups G of order | G | ≤ 8: Theorem 2.5. The c ar dinality of ↔ λ ( G ) fo r a gr oup G of size | G | ≤ 8 c an b e found fr om the table: G C 2 C 3 C 4 C 2 ⊕ C 2 C 5 D 6 C 6 C 7 C 3 2 D 8 C 4 ⊕ C 2 C 8 Q 8 sl ( G ) 2 2 3 3 3 4 3 3 5 4 4 4 4 ↔ λ ( X ) 1 1 1 1 1 1 2 3 1 2 4 8 8 Pr o of. W e divide the pro of into 5 c a ses. 1. If s l ( G ) > | G | / 2, then L 0 is a unique maximal in v ariant linked sys tem and th us | ↔ λ ( X ) | = 1. By Theorem 1.2, sl ( G ) > | G | / 2 if and o nly if | G | ≤ 5 or G is isomorphic to D 6 or C 3 2 . 2. If sl ( G ) = | G | / 2, then | ↔ λ ( G ) | = 2 s where s = |S / ∼ | . So it rema ins to calculate the n umber s for the gro ups C 6 , D 8 , C 4 ⊕ C 2 , C 8 , and Q 8 . 2a. If G is cyc lic of order 6, then we can take an y ge ner ator a on G and by routine calculations, c heck that S = { xT , x ( G \ T ) : x ∈ G } where T = { e, a, a 3 } . It follows that s = |S / ∼ | = 1 and th us | ↔ λ ( G ) | = |↑L 0 | = 2 s = 2 . 2b. If G is cyclic of or der 8, then we can take a ny generato r a on G and b y routine verification chec k that S = { xA, G \ xA, xB , G \ xB , C, G \ xC : x ∈ G } where A = { e, a, a 2 , a 4 } , B = { e, a, a 2 , a 5 } , a nd C = { e, a , a 3 , a 5 } . It follows that s = |S / ∼ | = 3 and thus | ↔ λ ( G ) | = |↑L 0 | = 2 s = 8 . 2c. Assume that the group G is is omorphic to C 4 ⊕ C 2 and let G 2 = { x ∈ G : xx = e } b e the Bo o lean subgro up o f G . W e claim that a 4-element subset A ⊂ G is self-link ed if and only if | A ∩ G 2 | is odd. T o prov e the “if ” part of this claim, a ssume that | A ∩ G 2 | = 3. W e claim that A is self-link ed. Let A 2 = A ∩ G 2 and note that G 2 = A 2 A − 1 2 ⊂ AA − 1 bec ause | A 2 | = 3 > 2 = | G 2 | / 2. Now take a ny element a ∈ A \ G 2 and note that AA − 1 ⊃ aA − 1 2 ∪ A 2 a − 1 . Observe that b oth a A − 1 2 = aA 2 and A 2 a − 1 = a − 1 A 2 are 3-element subsets in the 4-elemen t cos et aG 2 . Those 3 -element sets a re different. Indeed, assuming that aA − 1 2 = A 2 a − 1 we would obtain that a 2 A 2 = A 2 which implies that | A 2 | = 3 is even. Co nsequently , aG 2 = aA − 1 2 ∪ A 2 a − 1 ⊂ AA − 1 and finally G = AA − 1 . 14 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN If | A ∩ G 2 | = 1, then we can take any a ∈ A \ G 2 and consider the shift Aa − 1 which has | Aa − 1 ∩ G 2 | = 3 . Then the preceding cas e implies that Aa − 1 is self-linked and so is A . T o pr ove the “only if ” part of the claim as sume that | A ∩ G 2 | is even. If | A ∩ G 2 | = 4, then A = G 2 and AA − 1 = G 2 G − 1 2 = G 2 6 = G . If | A ∩ G 2 | = 0, then A = G 2 a for any a ∈ A and hence AA − 1 = G 2 aa − 1 G − 1 2 = G 2 6 = G . If | A ∩ G 2 | = 2, then | G 2 ∩ AA − 1 | ≤ 3 and again AA − 1 6 = G . Thu s S = { A ⊂ G : | A | = 4 and | A ∩ G 2 | is odd } . Each set A ∈ S has a unique shift aA with aA ∩ G 2 = { e } . Ther e are exactly four subsets A ∈ S with A ∩ G 2 = { e } for ming tw o eq uiv alence class es with resp ect to the relation ∼ . Ther efore s = 2 and | ↔ λ ( G ) | = |↑L 0 | = 2 s = 4 . 2d. Assume that G is isomorphic to the dihedral g roup D 8 of is ometries o f the s quare. T hen G con tains an element a of or der 4 generating a normal cyclic subgroup H . The elemen t a 2 commutes with all the elements of the group G . W e claim that for each self-linked 4-element subset A ⊂ G we get | A ∩ H | = 2. Indeed, if | A ∩ H | e q uals 0 or 4, th en A = H b for some b ∈ G and then AA − 1 = Abb − 1 A − 1 = H 6 = G . If | A ∩ H | equals 1 or 3, then repla c ing A by a s uita ble shift, w e can assume that A ∩ H = { e } and hence A = { e } ∪ B for some 3-element subset B ⊂ G \ H . It follows that G \ H = AA − 1 \ H = ( B ∪ B − 1 ) = B 6 = G \ H . This contradiction s hows that | A ∩ H | = 2 . Without loss of genera lit y , we can assume that A ∩ H = { e, a 2 } (if it is not the case, replac e A by its s hift Ax − 1 where x, y ∈ A a re s uch that y x − 1 = a 2 ). Now take any element b ∈ A \ H . Since G is not commutativ e, w e get ab = ba 3 . O bserve that ba 2 / ∈ A (otherwise A = { e, b, a 2 , ba 2 } would be a subgroup of G with AA − 1 = A 6 = G ). Consequently , the 4- th element c ∈ A \ { e, a 2 , b } o f A should b e of the for m c = ba or c = ba 3 = ab . Obs erve t ha t bo th the sets A 1 = { e, a 2 , b, b a } a nd A 2 = { e, a 2 , b, a b } a re self-linked. Observe also that a 3 ( G \ A 1 ) = a 3 · { a, a 3 , ba 2 , ba 3 } = { e, a 2 , ab, b } = A 2 . Consequently , s = |S / ∼ | = 1 and | ↔ λ ( G ) | = 2 s = 2 . 2e. Finally assume that G is isomorphic to the group Q 8 = {± 1 , ± i, ± j, ± k } of quaternion units. The tw o- element subs e t H = {− 1 , 1 } is a normal s ubgroup in X . Let S ± = { A ∈ S : H ⊂ A } and o bserve that ea ch set A ∈ S has a left shift in S . T ak e any set A ∈ S ± and pic k a p oint a ∈ A \ { 1 , − 1 } . Observe that the 4 - th element b ∈ A \ { 1 , − 1 , a } of A is not equa l to − a (otherwis e, A is a subgr oup of G ). ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 15 Conv ersely , one can easily c heck that each set A = { 1 , − 1 , a, b } with a, b ∈ G \ H and a 6 = − b is self-linked. This means that S ± = {{− 1 , 1 , a, b } : a 6 = − b and a, b ∈ G \ H } and thus |S ± | = C 2 6 − 3 = 1 2 . O bserve that for each A ∈ S 2 the set − A ∈ S 2 and there ax actly tw o shifts o f X \ A that b elong to S 2 . This means that the equiv alence cla ss [ A ] ∼ of any set A ∈ S intersects S 2 in fo ur sets. Consequently , s = |S / ∼ | = |S ± | / 4 = 12 / 4 = 3 and | ↔ λ ( G ) | = |↑L 0 | = 2 s = 8 . 3. If | G | = 7 , t hen L 0 is one of three elements o f ↔ λ ( G ). The other tw o elemen ts can be found as follows. Consider the inv aria nt linked system L 1 = { A ⊂ G : | A | ≥ 5 } and observe t ha t L 1 ⊂ A for each A ∈ ↔ λ ( G ). Indeed, as suming that some A ∈ L 1 do es not belo ng to A , w e would conclude that B = G \ A ∈ A b y the max imality of A . Since | G \ B | ≤ 2 we can find x ∈ G \ B B − 1 . It follows that B , xB are t wo disjoint sets in A which is not p os s ible. Thu s L 1 ⊂ A . Observe that L 1 ⊂ A ⊂ L 0 ∪ L 3 , where L 3 = { A ⊂ G : | A | = 3 , AA − 1 = G } . Given a genera tor a of the cyclic gro up G , consider the 3 -element set T = { a, a 2 , a 4 } and note tha t T T − 1 = G a nd T − 1 ∩ T = ∅ . B y a routine calculatio n, one can chec k that L 3 = { xT , xT − 1 : x ∈ G } . Since T and T − 1 are disjoint, the inv ar iant linked system A cannot contain bo th the sets T and T − 1 . If A con tains none of the sets T , T − 1 , then A = L 0 . If A contains T , then A = ( L 0 ∪ { xT : x ∈ G } ) \ { y ( G \ T ) : y ∈ G } . If T − 1 ∈ A , then A = ( L 0 ∪ { xT − 1 : x ∈ G } ) \ { y ( G \ T − 1 ) : y ∈ G } . And those are the unique 3 maximal in v ariant sys tems in ↔ λ ( G ).  In the following t heo rem we c haracter ize groups possessing a unique maximal inv aria nt linked system. Theorem 2.6. F or a fi nite gr oup G the fol lowing c onditions ar e e quivalent: (1) | ↔ λ ( G ) | = 1 ; (2) sl ( G ) > | G | / 2 ; 16 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN (3) | G | ≤ 5 or else G is isomorphic to D 6 or C 3 2 . Pr o of. (2) ⇒ (1). If sl ( G ) > | G | / 2, then L 0 = { A ⊂ G : | A | > | G | / 2 } is a unique maximal inv ariant link ed system on G (b eca use inv ar iant link ed systems compo se of self-link e d sets). (1) ⇒ ( 2 ) Assume that sl ( G ) ≤ | G | / 2 and take a self-linked subset A ⊂ G with | A | ≤ | G | / 2. If | G | is odd, then L 0 is maxima l linked a nd then any max imal inv ari- ant linked system A containing the self-link ed set A is distinct from L 0 , w itnes sing that | ↔ λ ( G ) | > 1. If G is even, then we can enlarge A , if nece s sary , and assume tha t | A | = | G | / 2. W e claim that t he complement B = G \ A of A is self-linked to o. Assuming the conv erse , we w ould find some x / ∈ B B − 1 and conclude that B ∩ xB = ∅ , which implies that A = G \ B = xB and hence x − 1 A = B . Then the sets A and x − 1 A a re disjoint whic h cont r adicts x − 1 ∈ AA − 1 = G . Th us B B − 1 = G which implies that { xB : x ∈ G } is an inv ariant linked system. Since | G | = 2 | A | is ev en, the unions A = { xA : x ∈ G } ∪ L 0 and B = { xB : x ∈ G } ∪ L 0 are in v aria nt linked systems that ca n b e enlarged to maximal link ed s ystems ˜ A and ˜ B , resp ectively . Since t he sets A ∈ A ⊂ ˜ A and B ∈ B ⊂ ˜ B ar e disjoin t, ˜ A 6 = ˜ B are t wo distinct maximal inv aria nt s ystems on G and th us | ↔ λ ( G ) | ≥ 2 . The equiv alence (2) ⇔ (3) follows f r om Theorem 1.2(i).  3. Right zeros in λ ( X ) In this s ection we return to s tudying the sup e rextensions o f gro ups and shall detect g roups X whose superextensio ns λ ( X ) hav e righ t zeros. W e shall sho w th at for every g roup X the right zeros of λ ( X ) coincide with inv aria nt ma ximal link ed systems. W e recall that an element z of a semigr oup S is called a rig ht (resp. left ) z er o in S if xz = z (resp. z x = z ) for every x ∈ S . This is equiv alent to saying tha t the singleton { x } is a left (resp. right) idea l of S . By [G2, 5.1] a n inc lus ion hyperspac e A ∈ G ( X ) is a r ig ht zero in G ( X ) if and only if A is inv ar ia nt. This implies that the minimal ideal of the semigroup G ( X ) coincides with the set ↔ G ( X ) of in v aria nt inclusio n h yp ers paces a nd is a compac t rectangular topo logical semigroup. W e r ecall tha t a semigroup S is ca lled r e ctan- gular if xy = y for all x, y ∈ S . A similar c haracter iz a tion of r ight zeros holds also for the semigroup λ ( X ). Prop ositi on 3.1. A maximal linke d system L is a right zer o of the semigr oup λ ( X ) if and only if L is invariant. Pr o of. If L is in v ar iant, then b y prop osition 5.1 of [G2], L is a right zero in G ( X ) and consequently , a r ig ht zero in λ ( X ). ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 17 Assume conv ers e ly that L is a right zero in λ ( X ). Then for every x ∈ X we get x L = L , which m e ans that L is in v ariant.  Unlik e the s emigroup G ( X ) which alw ays con ta ins right ze ros, the s emigroup λ ( X ) contains rig ht z eros o nly for so- c alled o dd g roups. W e define a group X to b e o dd if eac h elemen t x ∈ X has odd order. W e reca ll that the or der o f an elemen t x is the s ma llest integer num b er n ≥ 1 such that x n coincides w ith the neutra l element e of X . Theorem 3.2. F or a gr oup X t he fol lowi n g c onditions ar e e qu ivalent: (1) the semigr oup λ ( X ) has a right zer o; (2) some maximal invariant linke d system on X is m ax imal linke d (which c an b e writt en as ↔ λ ( X ) ∩ λ ( X ) 6 = ∅ ); (3) e ach maximal invariant linke d system is max imal linke d (which c an b e writ- ten as ↔ λ ( X ) ⊂ λ ( X ) ); (4) for any p artition X = A ∪ B either AA − 1 = X or B B − 1 = X ; (5) e ach element of X has o dd or der. Pr o of. The equiv alence (1) ⇔ (2) follows f r om Prop osition 3.1. (2) ⇒ (4) Assume that λ ( X ) contains an in v ariant maximal link ed system A . Given an y partition X = A 1 ∪ A 2 , use the ma ximality o f A to find i ∈ { 1 , 2 } with A i ∈ A . W e claim that A i A − 1 i = X . Indeed, for every x ∈ X the in v ar iantness of A implies that xA i ∈ A and hence A i ∩ xA i 6 = ∅ , which implies x ∈ A i A − 1 i . (4) ⇒ (3 ) Assume tha t for every partition X = A ∪ B either AA − 1 = X or B B − 1 = X . W e nee d to check that each maximal in v aria nt linked sys tem L is maximal linked. In the other case, there would exis t a set A ∈ L ⊥ \ L . Since L 6∋ A is ma x imal inv ar iant linked system, some s hift xA o f A do es not in tersect A and thus x / ∈ AA − 1 . The n our assumption implies that B = X \ A has property B B − 1 = X , which means that the family { xB : x ∈ X } is link e d. W e c laim that B ∈ L ⊥ . Assuming the conv erse, we would find a s et L ∈ L w ith L ∩ B = ∅ and conclude that A ∈ L becaus e L ⊂ X \ B = A . But this contradicts the c ho ice of A ∈ L ⊥ \ L . Therefor e B ∈ L ⊥ and L ∪ { L ⊂ X : ∃ x ∈ X ( xB ⊂ L ) } is an inv a riant linked s y stem that enlarges L . Since L is a maximal inv aria nt linked system, we conc lude that B ∈ L , w hich is not p ossible b ecause B do es not intersect A ∈ L ⊥ . The obtained cont r adiction shows that L ⊥ \ L = ∅ , which mea ns that L belo ngs to λ ( X ) and th us is an inv ar iant maximal linked system. The implication (3) ⇒ (2) is trivial. 18 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN ¬ (5) ⇒ ¬ (4) Assume that X \ { e } contains a point a whose order is even or infinit y . Then the cyclic subgroup H = { a n : n ∈ Z } generated by a decomp oses int o t wo disjoint sets H 1 = { a n : n ∈ 2 Z + 1 } and H 2 = { a n : n ∈ 2 Z } such that aH 1 = H 2 . T ake a subset S ⊂ X meeting each co set H x , x ∈ X , in a single p o int. Cons ider the disjoin t s ets A 1 = H 1 S and A 2 = H 2 S and note that aA 1 = A 2 = X \ A 1 and aA 2 = X \ A 2 , which implies that a / ∈ A i A − 1 i for i ∈ { 1 , 2 } . Since A 1 ∪ A 2 = X , we get a negation of ( 4). (5) ⇒ (4) A ssume that each element of X has odd order and assume that X admits a par tition X = A ⊔ B such tha t a / ∈ AA − 1 and b / ∈ B B − 1 for some a, b ∈ X . Then aA ⊂ X \ A = B a nd bB ⊂ X \ B = A . Obs e r ve that baA ⊂ bB ⊂ A and b y induction, ( ba ) i A ⊂ A for all i > 0. Since all ele ments of X hav e finite order, ( ba ) n = e for some n ∈ N . Then ( ba ) n − 1 A ⊂ A implies A = ( ba ) n A ⊂ b aA ⊂ bB ⊂ A and hence bB = A . It follows from X = bA ⊔ bB = bA ⊔ A = B ⊔ A that bA = B . Thus x ∈ A if and only if bx ∈ B . Let H = { b n : n ∈ Z } ⊂ X b e the cyclic subgro up generated by b . By o ur assumption it is of o dd order. On the other hand, the equality bB = A = b − 1 B implies that t he intersections H ∩ A and H ∩ B hav e t he same ca rdinality b ecause b ( B ∩ H ) = A ∩ H . But this is not p o s sible b eca use of the o dd car dina lit y of H .  4. (Left) zer os of the semigroup λ ( X ) An elemen t z of a semigro up S is called a zer o in S if xz = z = z x for all x ∈ S . This is equiv alen t to saying that z is b oth a left and right zer o in S . Prop ositi on 4.1. L et X b e a gr oup. F or a m ax imal li n ke d system L ∈ λ ( X ) the fol lowing c onditions ar e e qu ivalent: (1) L is a left zer o in λ ( X ) ; (2) L is a zer o in λ ( X ) ; (3) L is a unique invariant maximal linke d system on X . Pr o of. (1) ⇒ (3) Assume that Z is a left zer o in λ ( X ). Then Z x = Z for all x ∈ X and th us Z − 1 = { Z − 1 : Z ∈ Z } ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 19 is an inv a riant maximal linked s y stem on X , which implies that the g roup X is o dd according to Theorem 3.2. Note that for every righ t zero A of λ ( X ) w e get Z = Z ◦ A = A which implies that Z is a unique right ze r o in λ ( X ) and by Prop os ition 3 .1 a unique inv aria nt ma ximal link e d system on X . (3) ⇒ (2) Assume that Z is a unique inv aria nt max imal link ed system on X . W e claim tha t Z is a left zero of λ ( X ). Indeed, for every A ∈ A and x ∈ X we get x Z ◦ A = Z ◦ A , which means tha t Z ◦ A is an inv ariant ma ximal linked sys tem. By Prop ositio n 3 .1, Z ◦ A is a right zero and hence Z ◦ A = Z b ecause Z is a unique right zero . This means that Z is a left zer o, a nd b eing a r ight zero, a zero in λ ( X ). (2) ⇒ (1) is trivial.  Theorem 4. 2. Th e sup er extension λ ( X ) of a gr oup X has a zer o if and only if X is isomorphic to C 1 , C 3 or C 5 . Pr o of. If X is a group of odd or der | X | ≤ 5, then ↔ λ ( X ) ⊂ λ ( X ) because X is o dd and | ↔ λ ( X ) | = 1 by Theo rem 2.6. This means that λ ( X ) co nt a ins a unique inv ariant maximal linked system, which is the ze r o of λ ( X ) by P rop osition 4.1. Now as s ume conv er sely tha t the semigr oup λ ( X ) has a zero element Z . By Prop ositio n 3.1 and Theorem 3 .2, X is o dd and thus ↔ λ ( X ) ⊂ λ ( X ). Since the zero Z of λ ( X ) is a unique inv ar iant maximal linked system on X , we get | ↔ λ ( X ) | ≤ 1. By Theo rem 2.6, X has order | X | ≤ 5 or is iso morphic to D 3 or C 3 2 . Since X is o dd, X m ust b e isomorphic to C 1 , C 3 or C 5 .  5. The comm ut a tivity of λ ( X ) In this section w e detect gro ups X with comm utative superextension. Theorem 5. 1. Th e sup er extension λ ( X ) of a gr oup X is c ommutative if and only if | X | ≤ 4 . Pr o of. The commut a tivity of the superextens io ns λ ( X ) of g roups X of or de r | X | ≤ 4 will be established in Sectio n 6. Now as s ume t ha t a group X has commu tative sup erextensio n λ ( X ). Then X is commutativ e. W e need t o show that | X | ≤ 4. First we show that | ↔ λ ( X ) | = 1. Assume that ↔ λ ( X ) co ntains tw o distinct ma ximal in v ar iant link ed systems A and B . T a king into accoun t that A , B ∈ ↔ λ ( X ) ⊂ ↔ G ( X ) a nd each element of ↔ G ( X ) is a righ t zero in G ( X ) (see [G2, 5.1]) w e conclude that A ◦ B = B 6 = A = B ◦ A . 20 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN Extend the linked systems systems A , B to maximal link ed systems ˜ A ⊃ A and ˜ B ⊃ B . Because of the comm uta tivit y of λ ( X ), we get A = B ◦ A ⊂ ˜ B ◦ ˜ A = ˜ A ◦ ˜ B ⊃ A ◦ B = B . This implies that the union A ∪ B 6 = A is a n inv aria nt linked system extending A , which is not p ossible b ecause of the ma ximality of A . This contradiction shows that | ↔ λ ( X ) | = 1. Applying Theorem 2.6, w e conclude that | X | ≤ 5 or X is iso morphic to C 3 2 . It remains to show that the semigroups λ ( C 5 ) and λ ( C 3 2 ) are not commutativ e. The non-commut a tivity of λ ( C 5 ) will be shown in Section 6. T o see that λ ( C 3 2 ) is not comm utative, take an y 3 generator s a, b, c of C 3 2 and consider the sets A = { e, a, b, abc } , H 1 = { e, a, b, a b } , H 2 = { e, a, bc, abc } . Observe that H 1 , H 2 are subgroups in C 3 2 . F or every i ∈ { 1 , 2 } consider the link ed system A i = h{ H 1 , H 2 } ∪ { xA : x ∈ H i }i and extend it to a maximal linked system ˜ A i on C 3 2 . W e claim that the ma ximal linked systems ˜ A 1 and ˜ A 2 do not comm ute. Indee d, ˜ A 2 ◦ ˜ A 1 ∋ [ x ∈ H 1 x ∗ ( x − 1 bA ) = bA = { e, b, ba, ac } , ˜ A 1 ◦ ˜ A 2 ∋ [ x ∈ H 2 x ∗ ( x − 1 bcA ) = bcA = { a, c, bc, abc } . It follows from bA ∩ bcA = ∅ that ˜ A 1 ◦ ˜ A 2 6 = ˜ A 2 ◦ ˜ A 1 .  6. The superextensions of finite gr o ups In this section we shall describ e the s tr ucture of the sup ere xtensions λ ( G ) of finite gro ups G of small car dinality (mor e precisely , of cardinality | G | ≤ 5). It is known that the car dinality o f λ ( G ) growth v er y quic kly as | G | tends to infinit y . The calculation of the ca rdinality of | λ ( G ) | seems to be a difficult com binator ial problem related to the still unso lved Dedekind’s problem of calculation of the num- ber M ( n ) of inclusion hyperpspa c es on an n - element subs e t, see [De]. W e were able to calculate the cardinalities o f λ ( G ) only for groups G of ca rdinality | G | ≤ 6. The results of (computer) calculations are present e d in the following table : | G | 1 2 3 4 5 6 | λ ( G ) | 1 2 4 12 81 26 46 | λ ( G ) /G | 1 1 2 3 17 447 Before describing the structure of sup erextensions of finite groups, let us make some remarks concerning the structur e of a se mig roup S containing a gro up G . In this case S can be thoug ht as a G -space endo wed with the left a ction of the group G . So w e c an consider the orbit space S/G = { Gs : s ∈ S } and the pro jection ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 21 π : S → S/ G . If G lies in the center of the semigroup S (which mea ns that the elements of G co mm ute with a ll the elements of S ), then the or bit space S/ G admits a unique semig roup oper ation turning S/ G in to a semigroup and the orbit pro jection π : S → S/G int o a semigr oup homomorphism. A subsemigroup T ⊂ S will be called a tr ansversal semigr oup if the restr iction π : T → S/ G is an isomorphis m of the semigr o ups. If S admits a transversal semigroup T , then it is a ho mo moprhic image of the product G × T under the semigro up homomor phism h : G × T → S, h : ( g , t ) 7→ g t. This helps to recover the a lgebraic structure of S from the structure of a transversal semigroup. F or a system B o f subsets of a set X b y hB i = { A ⊂ X : ∃ B ∈ B ( B ⊂ A ) } we denote the inclusion hypers pa ce generated by B . Now w e shall analyse the entries of the above table. First note that each group G o f s ize | G | ≤ 5 is ab e lia n and is isomorphic to one of the gro ups: C 1 , C 2 , C 3 , C 4 , C 2 ⊕ C 2 , C 5 . It will b e conv enient to think of the cyclic group C n as the m ultiplica tive subgr oups { z ∈ C : z n = 1 } of the complex plane. 6.1. The sem igroups λ ( C 1 ) and λ ( C 2 ) . F or the gr oups C n with n ∈ { 1 , 2 } the semigroup λ ( C n ) coincides with C n while the orbit semigroup λ ( C n ) /C n is trivial. 6.2. The semigroup λ ( C 3 ) . F or the group C 3 the semigr o up λ ( C 3 ) contains the three principa l ultrafilter s 1 , z , − z wher e z = e 2 π i/ 3 and the maximal linked in- clusion hype r space ⊲ = h{ 1 , z } , { 1 , − z } , { z , − z }i which is the zero in λ ( C 3 ). The sup e rextension λ ( C 3 ) is isomor phic to the mult iplicative semigroup C 0 3 = { z ∈ C : z 4 = z } of the co mplex plane. The latter semigr oup has zero 0 and unit 1 whic h are the unique idempotents. The transversal semigroup λ ( C 3 ) /C 3 is isomorphic to the semilattice 2 = { 0 , 1 } endow ed with the min-oper ation. 6.3. The semigroups λ ( C 4 ) and λ ( C 2 ⊕ C 2 ) . The semigr o up λ ( C 4 ) contains 12 elements while the orbit semigroup λ ( C 4 ) /C 4 contains 3 elemen ts. The semigroup λ ( C 4 ) contains a tra ns versal semigro up λ T ( G ) = { 1 , △ ,  } where 1 is the neutral elemen t of C 4 = { 1 , − 1 , i, − i } , △ = h{ 1 , i } , { 1 , − i } , { i , − i }i and  = h{ 1 , i } , { 1 , − i } , { 1 , − 1 } , { i, − i, − 1 }i . 22 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN The trans versal se migroup is isomorphic to the ex tension C 1 2 = C 2 ∪ { e } of the cyclic group C 2 by an external unit e / ∈ C 2 (such that ex = x = xe for all x ∈ C 1 2 ). The actio n of the gr oup C 4 on λ ( C 4 ) is free s o, λ ( C 4 ) is iso morphic to λ T ( C 4 ) ⊕ C 4 . The semigroup λ ( C 2 ⊕ C 2 ) has a simila r alg e braic str ucture. It cont a ins a transversal s e mig roup λ T ( C 2 ⊕ C 2 ) = { e, △ ,  } ⊂ λ ( C 2 ⊕ C 2 ) where e is the principal ultrafilter supp or ted by the neutral element (1 , 1 ) of C 2 ⊕ C 2 and the maximal linked inclusion hyper spaces △ and  ar e defined by ana logy with the case of the group C 4 : △ = h{ (1 , 1) , (1 , − 1) } , { (1 , 1) , ( − 1 , 1) } , { (1 , − 1) , ( − 1 , 1) }i and  = h{ (1 , 1) , (1 , − 1) } , { (1 , 1) , ( − 1 , 1) } , { (1 , 1) , ( − 1 , − 1) } , { (1 , − 1) , ( − 1 , 1) , ( − 1 , − 1) }i . The transversal s emigroup λ T ( C 2 ⊕ C 2 ) is isomorphic to C 1 2 and λ ( C 2 ⊕ C 2 ) is isomorphic to C 1 2 ⊕ C 2 ⊕ C 2 . W e summarize the obtained results on the algebraic s tructure o f the semigroups λ ( C 4 ) and λ ( C 2 ⊕ C 2 ) in the follo wing prop os itio n. Prop ositi on 6.1. L et G b e a gr oup of c ar dinality | G | = 4 . (1) The semigr oup λ ( G ) is isomorphi c to C 1 2 ⊕ G and thus is c ommutative; (2) λ ( G ) c ontains two idemp otents; (3) λ ( G ) has a unique pr op er ide al λ ( G ) \ G isomorphic t o the gr oup C 2 ⊕ G . 6.4. The semigroup λ ( C 5 ) . Unlik e to λ ( C 4 ), the se mig roup λ ( C 5 ) has compli- cated algebraic structure. It contains 81 elemen ts. One of them is zero Z = { A ⊂ C 5 : | A | ≥ 3 } , which is in v aria nt under any bijection of C 5 . All the other 80 elements hav e 5- element o rbits under the action o f C 5 , which implies that the orbit s emigroup λ ( C 5 ) /C 5 consists of 17 e le men ts. Let π : λ ( C 5 ) → λ ( C 5 ) /C 5 denote the orbit pro jection. It will b e con venient to think o f C 5 as the field { 0 , 1 , 2 , 3 , 4 } with the multi- plicative subgroup C ∗ 5 = { 1 , − 1 , 2 , − 2 } of in vertible elements (here − 1 and − 2 are ident ified with 4 and 3, resp ectively). Also for elements x, y , z ∈ C 5 we shall write xy z instead of { x, y , z } . The semigroup λ ( C 5 ) con tains 5 idemp otents: U = h 0 i , Z , Λ 4 = h 01 , 0 2 , 03 , 04 , 1234 i , Λ = h 02 , 03 , 123 , 0 14 , 2 34 i , 2Λ = h 04 , 01 , 124 , 023 , 143 i , ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 23 which commute and th us form an ab elian subsemigroup E ( λ ( C 5 )). Being a semi- lattice, E ( λ ( C 5 )) carr ies a natura l partial order : e ≤ f iff e ◦ f = e . The partial order Z ≤ Λ , 2Λ ≤ Λ 4 ≤ U on the set E ( λ ( C 5 )) is designed at the picture: Z r   ❅ ❅ r Λ r 2Λ Λ 4 r ❅ ❅ ❅ ❅   r U The other distinguished subset of λ ( C 5 ) is p E ( λ ( C 5 )) = {L ∈ λ ( C 5 ) : L ◦ L ∈ E ( λ ( C 5 )) } = = {L ∈ λ ( C 5 ) : L ◦ L ◦ L ◦ L = L ◦ L} . W e shall show that this set contains a point from each C 5 -orbit in λ ( C 5 ). First we show that this set has at most o ne-p oint intersection with each or bit. Indeed, if L ∈ p E ( λ ( C 5 )) and L ◦ L 6 = Z , then for ev ery a ∈ C 5 \ { 0 } , w e get ( L + a ) ◦ ( L + a ) ◦ ( L + a ) ◦ ( L + a ) = L ◦ L ◦ L ◦ L + 4 a = = L ◦ L + 4 a 6 = L ◦ L + 2 a = ( L + a ) ◦ ( L + a ) . witnessing that L + a / ∈ p λ T ( C 5 ). By a direct ca lculation o ne can c heck that the set λ T ( C 5 ) co ntains the following four maximal link ed systems: ∆ = h 02 , 03 , 23 i , Λ 3 = h 02 , 0 3 , 04 , 23 4 i , Θ = h 14 , 0 1 2 , 01 3 , 12 3 , 024 , 0 34 , 234 i , Γ = h 0 2 , 04 , 013 , 124 , 234 i . F or those systems w e get ∆ ◦ ∆ = ∆ ◦ ∆ ◦ ∆ = Λ , Λ 3 ◦ Λ 3 = Λ 3 ◦ Λ 3 ◦ Λ 3 = Λ , F ◦ Θ = F ◦ Γ = Z for every F ∈ λ ( C 5 ) \ C 5 . 24 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN All the o ther elemen ts of λ ( C 5 ) can b e found as images of ∆ , Θ , Γ , Λ 3 under the affine transformations of the field C 5 . Tho se are maps of the f o rm f a,b : x 7→ ax + b mo d 5 , where a ∈ { 1 , − 1 , 2 , − 2 } = C ∗ 5 and b ∈ C 5 . The ima ge of a maximal linked system L ∈ λ ( C 5 ) under suc h a transfo rmation will be denoted by a L + b . One can c heck that a Λ 4 = Λ 4 for eac h a ∈ C ∗ 5 while Λ = − Λ, and Θ = − Θ. Since th e linear tra nsformations of the form f a, 0 : C 5 → C 5 , a ∈ C ∗ 5 , are autho- morphisms of the gro up C 5 the induced tr ansformations λf a, 0 : λ ( C 5 ) → λ ( C 5 ) are authomor phisms of the semigroup λ ( C 5 ). This implies that those tra nsforma- tions do not mo ve the subsets E ( λ ( C 5 )) and p E ( λ ( C 5 )). Co nsequently , the set p E ( λ ( C 5 ) contains the max imal link ed systems: a ∆ , a Θ , a Λ 3 , a Γ , a ∈ Z ∗ 5 , which together with the idemp otents form a 1 7-element s ubset T 17 = E ( λ ( C 5 )) ∪  a ∆ , a Θ : a ∈ { 1 , 2 }  ∪ { a Λ 3 , a Γ : a ∈ Z ∗ 5 } that pro jects bijectiv ely onto the orbit s emigroup λ ( C 5 ) /C 5 . The set T 17 lo oks as follows (we connect an e le men t x ∈ T 17 with an idempotent e ∈ T 17 by an a r row if x ◦ x = e ): Z   ❅ ❅ Λ 2Λ Λ 4 ❅ ❅ ❅ ❅   U − Γ Θ ✲ 2Θ ✛   ✒ Γ ✂ ✂ ✂ ✍ 2Γ ❇ ❇ ❇ ▼ − 2Γ ❅ ❅ ■ − Λ 3 ✑ ✑ ✸ ∆ ✲ Λ 3 ◗ ◗ s − 2Λ 3 ◗ ◗ ❦ 2∆ ✛ 2Λ 3 ✑ ✑ ✰ The set p E ( λ ( C 5 )) includes 24 elements more and coincides with the union T 17 ∪ √ Z where √ Z = { a Θ + b, a Γ + b : a ∈ Z ∗ 5 , b ∈ C 5 } . Since each ele ment of λ ( C 5 ) can b e uniquely written as the sum L + b for some L ∈ T 17 and b ∈ C 5 , the multiplication ta ble for the semigro up λ ( C 5 ) can b e recov er ed from the Cayley ta ble for m ultiplicatio n of the elemen ts from T 17 : ALGEBRA IN SUPEREXTENSIONS OF GR OUP S, I 25 ◦ Λ 4 Λ ∆ Λ 3 − Λ 3 2Λ 2∆ 2Λ 3 − 2Λ 3 a Θ , a Γ Λ 4 Λ 4 Λ Λ Λ Λ 2Λ 2Λ 2Λ 2Λ Z Λ Λ Λ Λ Λ Λ Z Z Z Z Z ∆ ∆ Λ Λ Λ Λ 2Θ 2Θ 2Θ 2Θ Z Λ 3 Λ 3 Λ Λ Λ Λ 2Θ + 2 2Θ + 2 2Θ + 2 2Θ + 2 Z − Λ 3 − Λ 3 Λ Λ Λ Λ 2Θ − 2 2Θ − 2 2Θ − 2 2Θ − 2 Z 2Λ 2Λ Z Z Z Z 2Λ 2Λ 2Λ 2Λ Z 2∆ 2∆ Θ Θ Θ Θ 2Λ 2Λ 2Λ 2Λ Z 2Λ 3 2Λ 3 Θ − 1 Θ − 1 Θ − 1 Θ − 1 2Λ 2Λ 2Λ 2Λ Z − 2Λ 3 − 2Λ 3 Θ + 1 Θ + 1 Θ + 1 Θ + 1 2Λ 2Λ 2Λ 2Λ Z Θ Θ Θ Θ Θ Θ Z Z Z Z Z 2Θ 2Θ Z Z Z Z 2Θ 2Θ 2Θ 2Θ Z Γ Γ Θ + 1 Θ + 1 Θ + 1 Θ + 1 2Θ + 2 2Θ + 2 2Θ + 2 2Θ + 2 Z − Γ − Γ Θ − 1 Θ − 1 Θ − 1 Θ − 1 2Θ − 2 2Θ − 2 2Θ − 2 2Θ − 2 Z 2Γ 2Γ Θ − 1 Θ − 1 Θ − 1 Θ − 1 2Θ + 2 2Θ + 2 2Θ + 2 2Θ + 2 Z − 2Γ − 2Γ Θ + 1 Θ + 1 Θ + 1 Θ + 1 2Θ − 2 2Θ − 2 2Θ − 2 2Θ − 2 Z Lo oking at this ta ble w e ca n see that T 17 is not a subsemig r oup o f λ ( C 5 ) a nd hence is not a t r ansversal semigroup for λ ( C 5 ). This is not occa sional. Prop ositi on 6.2. The semigr oup λ ( C 5 ) c ont ains no tr ansversal semigr oup. Pr o of. Assume conversely that λ ( C 5 ) contains a subsemigroup T that pro jects bi- jectively on to the o rbit semigr o up λ ( C 5 ) /C 5 . Then T must include the set E ( λ ( C 5 )) of idempotents and also the subset p E ( λ ( C 5 )) \ √ Z . Conseq ue ntly , T ⊃ {U , Z , Λ , − Λ , ∆ , 2∆ , Λ 3 , − Λ 3 , 2Λ 3 , − 2Λ 3 } . Since 2Λ 3 ◦ Λ = Θ − 1 6 = Θ = 2 ∆ ◦ Λ, then there are tw o differen t p o int s in the int er section T ∩ (Θ + C 5 ) which should b e a singleton. This contradiction completes the pro of.  Analysing the Cayley table for the set T 17 we can esta blish the following prop- erties of the semigroup λ ( C 5 ). Prop ositi on 6.3. (1) The maximal linke d system Z is the zer o of λ ( Z ) . (2) λ ( C 5 ) c ont ains 5 idemp otent s: U , Z , Λ 4 , Λ , 2Λ , which c ommut e. (3) The set of c entr al elements of λ ( C 5 ) c oincides with C 5 ∪ {Z } . (4) Al l non-trivial su b gr oups of λ ( C 5 ) ar e isomorphic to C 5 . 26 T.BANAKH, V.GA VR YLKIV, O.NYKYFOR CHYN 6.5. Summary t able. The obtained res ults on the superextensio ns of groups G with | G | ≤ 5 are summed up in the follo wing table in whic h K ( λ ( G )) stands for the minimal ideal of λ ( G ). | G | | λ ( G ) | λ ( G ) | E ( λ ( G )) | K ( λ ( G )) maximal gro up 2 2 C 2 1 C 2 C 2 3 4 C 3 ∪ { ⊲ } 2 { ⊲ } C 3 4 12 C 1 2 × G 2 C 2 × G C 2 × G 5 81 T 17 · C 5 5 {Z } C 5 References [BG2] T. Banakh, V. Gavrylkiv. Algebra in superextension of groups, II: can celativity and cen ters, preprint. [BG3] T. Banakh, V. Gav r ylkiv. Algebra in superextension of groups, II I: the m i nimal ideal of λ ( G ), pr epri nt . [De] R. Dedekind, ¨ Ub er Zerle gunge n von Za hlen dur ch ihr e gr ¨ ussten gemeinsammen T eiler // In Gesammelte W erk e, Bd. 1 (1897), 103–148. [G1] V. Ga vryl ki v. The sp ac es of inclus ion hyp ersp ac e s over nonc omp act sp ac es , Matem. 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Iv a n Franko Na tional Un iversity of L viv, Ukraine E-mail addr ess : tb anakh@ya hoo.com V asyl S tef an yk Precarp a thian Na tiona l University, Iv ano-Frankivsk, Ukraine E-mail addr ess : vg avrylkiv @yahoo.c om