On countably compact 0-simple topological inverse semigroups

ON COUNT ABL Y COM P A CT 0 -SIMPLE TOPOLOGICAL INVERSE SEMIGR OUPS OLEG GUTIK AND DU ˇ SAN REPOV ˇ S Abstract. W e describ e the structure of 0-simple countably compact top ological in v erse semigroups and the structure of congruence- free coun tably compact top ological inv erse semi- groups. W e follo w the terminology of [3, 4, 8]. In this pap er all top olo gical spaces are Hausdorff. If S is a semigroup then w e denote the subset of idempotents o f S by E ( S ). A top ological space S that is alg ebraically a sem igroup with a contin uous sem igroup op eration is called a top olo gic al semigr oup . A top olo gic al inverse semigr oup is a top olo gical semigroup S that is algebraically an inv erse se migroup with con tinuous inv ersion. If Y is a subspace of a top ological space X and A ⊆ Y , then w e denote b y cl Y ( A ) the topolog ical closure of A in Y . The bicyc lic semigroup C ( p, q ) is the semigroup with the identit y 1 gene rated b y t w o ele- men ts p and q , s ub ject only to the condition pq =1. The bicy clic semigroup play s an imp or- tan t role in the algebraic theory of semigroups and in the theory of to p o logical se migroups. F or example, the w ell-know n Andersen’s result [1] states that a (0 –) simple semigroup is completely (0– ) sim ple if and only if it do es not con tain the bicyclic s emigro up. The bicyclic semigroup admits only the discrete to p o logy and a top ological semigroup S can con tain C ( p, q ) only a s an op en subset [7]. Neither stable nor Γ-compact top ological semigroups can con tain a cop y of the bicyclic semigroup [2 , 12 ]. Let S be a semigroup and I λ a no n- empt y set of cardinalit y λ . W e define the semigroup op eration ′ · ′ on the set B λ ( S )= I λ × S 1 × I λ ∪{ 0 } as follow s ( α, a, β ) · ( γ , b, δ ) = ( ( α, ab, δ ) , if β = γ , 0 , if β 6 = γ , and ( α , a, β ) · 0=0 · ( α, a, β )= 0 · 0=0, for α, β , γ , δ ∈ I λ , and a, b ∈ S 1 . The semigroup B λ ( S ) is called a Br andt λ -extension of the semigroup S [10]. F ur t hermore, if A ⊆ S then w e shall denote A αβ = { ( α, s, β ) | s ∈ A } f or α, β ∈ I λ . If a semigroup S is trivial ( i.e. if S contains only one eleme n t), then B λ ( S ) is the sem i g r oup of I λ × I λ -matrix units [4], whic h we shall denote b y B λ . By Theorem 3.9 o f [4], an in v erse semigroup T is completely 0-simple if and only if T is isomorphic to a Bra ndt λ -extension B λ ( G ) o f some group G and λ > 1. W e also note that if λ =1, then the semigroup B λ ( S ) is isomorphic to the semigroup S with adjoint zero. Gutik and P avly k [11 ] prov ed that an y con tinuous homomorphism from the infinite top ological semigroup of matrix units into a compact top ological semigroup is annihilating, and henc e the infinite top ological semigroup of matrix units do es not em b ed in to a compact top ological semigroup. They also show ed that if a top olog ical inv erse semigroup S contains a semigroup of matrix units B λ , then B λ is a closed subsemigroup of S . Susc hk ewitsc h [1 7] prov ed tha t an y finite semigroup S con tains a minimal ideal K . He also show ed that K is a completely simple semigroup and describ ed the structure of finite simple semigroups. Rees [15] generalized the Susc hke witsc h Theorem and sho w ed that if 0 This research w as supported b y the Slov enia n Research Agency gran ts P1-0292-0 1 01-04 and BI-UA/04-0 6-007. W e thank the referee and the editor for comments. Date : F ebruar y 28, 2022. 2000 Mathematics Subje ct Classific ation. 20M18 , 22A15. Key wor ds and phr ases. T opo logical inv erse semigroup, 0-simple semigroup, completely 0-simple semi- group, Stone- ˇ Cech compactification, congruence-free semigroup, bicyclic semigr oup, semig roup of matrix units. 1 2 OLEG GUTIK AND DU ˇ SAN REPOV ˇ S a semigroup S con ta ins a minimal ideal K then K is isomorphic to a Rees matrix semi- group M [ G ; I , Λ , P ] o v er a g r oup G with a regular sandwic h mat rix P . He also prov ed tha t an y completely 0- simple semigroup is isomorphic to a Rees matrix semigroup M [ G ; I , Λ , P ] o ver a 0-group G 0 with a regular sandwic h matrix P . W allace [18] prov ed the top ologi- cal analogue o f the Susc hk ewitsc h-Rees Theorem for compact t o p ological semigroups: every c omp act top olo gic al semigr oup c ontains a minimal ide al, which is top olo gic al ly is omorphic to a top olo gic al p ar agr oup . P a a lman-de-Miranda [14 ] prov ed that a n y 0-simple compact to - p ological semigroup S is completely 0- simple, the zero of S is an isolated p oin t in S and S \{ 0 } is homeomorphic to the top ological pro duct X × G × Y , where X a nd Y a r e compact top ological spaces and G is homeomorphic to the underlying space of a maximal subgroup of S , con tained in S \{ 0 } . Ow en [13] sho w ed that if S a lo cally compact completely simple top ological semigroup, then S has a structure similar to a compact simple t o p ological semi- group. Owe n also gav e an example whic h sho ws that a similar statemen t do es not hold for a lo cally compact completely 0-simple t o p ological semigroup. Gutik a nd P avly k [11] pro v ed that the subsemigroup of idempotents o f a compact 0-simple t o p ological in v erse semigroup is finite, and hence the top olog ical space o f a compact 0-simple top ological inv erse semigroup is ho meomorphic to a finite top olog ical sum of compact top ological group and a single p oint. A Hausdorff top ological space X is called c ountably c omp act if a n y o p en countable cov er of X con tains a finite sub co v er [8]. In this pap er we shall prov e that the bicyclic semigroup cannot b e em b edded into any countably compact top ological inv erse sem igroup. W e shall also describ e the structure of 0-simple countably compact topo lo gical in v erse semigroups and the structure of congruence-free coun tably compact to p ological inv erse semigroups. Theorem 1. A c ountably c om p act top olo gic al inv erse semi g r oup c annot c ontain the bicyclic semigr oup. Ther ef o r e every (0-)sim p le c ountably c omp act top olo gic al in v erse semigr oup is (0-)c ompletely simple. Pr o of. Let T be a coun tably compact top ological inv erse sem igroup and supp o se that T con tains C ( p, q ) a s a subsemigroup. Let S = cl T ( C ( p, q )). Then b y Theorem 3.1 0 .4 of [8], S is a coun tably compact space and by Prop osition I I.2 of [7], S is a top ological in v erse semigroup. Th us b y Corollary I.2 o f [7], the semigroup C ( p, q ) is a discrete subspace of S and by Theorem I.3 of [7 ], C ( p, q ) is an op en subspace of S and S \ C ( p, q ) is a n ideal in S . Therefore an y elemen t of C ( p, q ) is an isolated p oin t in the top ological space S . W e define the maps ϕ : S → E ( S ) and ψ : S → E ( S ) by the formulae ϕ ( x )= xx − 1 and ψ ( x )= x − 1 x . Since S \ C ( p , q ) is a n ideal of S , A = ϕ − 1 ( { 1 } ) ∪ ψ − 1 ( { 1 } ) ⊆ C ( p, q ), and since the maps ϕ and ψ are contin uous A is a clop en and hence countably compact infinite subset of S . But A is an op en subspace of S whose elemen ts are isolated p oin ts in S . A con tradiction. The second part of the t heorem follo ws from Theorem 2.54 o f [4].  Let S b e a class of top ological semigroups. Let λ b e a cardinal > 1, and ( S, τ ) ∈ S . Let τ B b e a top ology on B λ ( S ) suc h that ( B λ ( S ) , τ B ) ∈ S and τ B | ( α,S,α ) = τ for some α ∈ I λ . Then ( B λ ( S ) , τ B ) is called a top olo gic al Br and t λ -ex tens i on of ( S, τ ) in S [10]. Let α , β , γ , δ ∈ I λ and A b e a subspace o f S . Since the restriction ϕ γ δ αβ   A αβ : A αβ → A γ δ of the map ϕ γ δ αβ : B λ ( S ) → B λ ( S ) defined b y the form ula ϕ γ δ αβ ( s )=( γ , 1 , α ) · s · ( β , 1 , δ ) is a homeomorphism, we get the f ollo wing: Lemma 1. L et λ > 1 and B λ ( S ) b e a top olo gic al Br andt λ -e xtension of a top olo gic al semigr oup S and A a subsp ac e of S . Then the subsp ac es A αβ and A γ δ in B λ ( S ) ar e ho m e omorp h ic for al l α, β , γ , δ ∈ I λ . Theorem 2. L et S b e a 0 -simple c ountably c om p act top olo gic al inverse semigr oup. Then ther e exi s t a nonempty finite set I λ of c ar dinality λ and a c ountably c omp act top olo g ic al gr oup H such that S is top ol o gic al ly isomorphic to a top olo gic al Br andt λ -extension B λ ( H ) of H in the class of top olo gic al inverse sem igr oups. Mor e over, S is home omorphic to a finite top olo gic al sum of c ountable c omp act top o lo gic al gr oups an d a single p oint. ON COUNT ABL Y COMP ACT 0-S IMPLE TOPOLOGICAL IN VERSE SEMIGROUPS 3 Pr o of. By Theorem 1, the semigroup S is comple tely 0- simple. No w Theorem 3.9 of [4] implies that there exist a nonempty set I λ of car dina lity λ and a gr o up G suc h that S is algebraically isomorphic to B λ ( G ). Therefore for an y α ∈ I λ the subs et G αα is a subgroup of B λ ( G ) and since B λ ( G ) is a top ological inv erse semigroup, a top ological subspace G αα of B λ ( G ) with the induced multiplic ation is a top ological group. W e fix α ∈ I λ an put H = G αα . Then the top ological semigroup S is top ologically isomorphic to a top ological Brandt λ - extension B λ ( H ) of the top olo gical group H . Let e H b e the iden t ity of H . Then the subsemigroup B λ ( e H )= { 0 } ∪ { ( α, e H , β ) | α , β ∈ I λ } of B λ ( H ) is algebraically isomorphic to the semigroup of matrix units B λ . By Theorem 14 [11], B λ ( e H ) is a closed subsemigroup of B λ ( H ) and hence b y Theorem 3.10.4 of [8], B λ ( e H ) is a coun tably compact top ological space. Therefore Theorem 6 of [11] implies that B λ ( e H ) is a finite discrete subsemigroup of B λ ( H ) and hence t he set I λ is finite. W e define the maps ϕ : B λ ( H ) → B λ ( e H ) and ψ : B λ ( H ) → B λ ( e H ) by t he form ulae ϕ ( x ) = xx − 1 and ψ ( x )= x − 1 x . Since B λ ( H ) is a top ological in v erse semigroup the maps ϕ and ψ con tinuous and hence b y Lemma 4 of [11], the set H αβ = ϕ − 1 (( α, e H , β ) ) ∩ ϕ − 1 (( α, e H , β ) ) is clop en in B λ ( H ). By Lemma 1, the subs paces H αβ and H γ δ are homeomorphic for an y α, β , γ , δ ∈ I λ , and hence all o f them are homeomorphic to t he top o logical gro up H .  A T yc honoff top ological space X is called pseudo c omp act if ev ery con tin uous real- v alued function on X is b o unded. Since the top o lo gical space of T 0 -top ological group is T ychonoff and any top olo g ical sum o f T yc hono ff spaces is a Tyc honoff space, Theorem 3.10.2 0 o f [8 ] implies: Corollary 1. The top olo gic al sp ac e of a 0 -simp le c ountably c omp act top olo gic al inverse semi- gr oup is T ychon off and henc e pse udo c omp act. Let X b e a top ological space. The pair ( Y , c ), where Y is a compactum and c : X → X is a homeomorphic embedding of X into Y , suc h that cl Y c ( X )= Y , is called a c omp actific ation of the space X . Define the ordering 4 on the fa mily C ( X ) of all compactifications o f a top ological space X as follo ws: c 2 ( X ) 4 c 1 ( X ) if and o nly if there exis ts a con tin uous map f : c 1 ( X ) → c 2 ( X ) suc h that f c 1 = c 2 . The greatest elemen t of t he family C ( X ) with resp ect to the ordering 4 is called the Stone- ˇ Ce ch c o m p actific ation of the space X and it is denoted b y β X . Comfort and Ross [6] prov ed that the Stone- ˇ Cec h compactification of a pseudo compact top ological g r o up is a top ological group. The next theorem is an analogue o f the Comfort– Ross Theorem: Theorem 3. L et S b e a 0 -sim p le c ountable c omp act top olo gic al inverse semig r oup. Then the Stone- ˇ Ce ch c omp actific ation of S admits a structur e of 0 -sim p le top olo gic al inverse sem igr oup with r esp e ct to which the inclusion mapping of S into β S is a top olo gic al isomorph ism. Pr o of. By Theorem 2, S is top ologically isomorphic to a Bra ndt λ -extension of some top o- logical group H in the class of top o logical in vers e semigroups and λ<ω . No w by Lemma 1, the subspace s H αβ and H γ δ are homeomorphic in B λ ( H ), f or an y α, β , γ , δ ∈ I λ . Since a maximal subgroup in S is closed w e hav e that H αβ is a clop en subset of B λ ( H ), fo r ev ery α, β ∈ I λ . By Corollary 1, the top olog ical space B λ ( H ) is pseudo compact. Since any clop en subspace of a pseudo compact to p o logical space is pseudo compact (see [5]) the subspace H αβ is pseudocompact, for ev ery α, β ∈ I λ . Ob viously , the top ological space B λ ( H ) \{ 0 } is home- omorphic to H × I λ × I λ . Since the top ological space I λ × I λ is finite and hence compact, by Corollary 3 .10.27 of [8], the space B λ ( H ) \{ 0 } is pseudo compact. No w by Theorem 1 of [9], w e ha v e β ( H × I λ × I λ )= β H × β I λ × β I λ = β H × I λ × I λ and therefore β ( B λ ( H ))= B λ ( β H ).  Corollary 2. Every 0 -sim p le c o untable c o m p act top olo gic al inverse semigr oup is a dense subsemigr oup o f a 0 -simple c omp act top olo gic al inverse semigr oup. If S is completely simple in vers e semigroup then the semigroup S with joined zero S 0 is completely 0 - simple and hence b y Theorem 3 .9 of [4], the semigroup S 0 is isomorphic to a Brandt λ -extension B λ ( G ) of some group G . Therefore an y nonzero idemp oten t of S 0 is 4 OLEG GUTIK AND DU ˇ SAN REPOV ˇ S primitiv e. Let e and f are nonzero idemp oten ts of S 0 . Since S is an in vers e subsemigroup of S 0 w e ha v e ef = f e 6 e and ef = f e 6 f , and hence e = ef = f . Th us, the in ve rse semigroup S con ta ins the unique idempotent and hence it is a gr oup. Therefore a completely simple in ve rse semigroup is a group and Theorem 1 implies that every simple c ountable c om p ac t top olo gic al inverse semigr oup is a top olo gic al gr oup . A semigroup S is called c o n gruenc e-fr e e if it has only tw o congruences: the iden tity r elat io n and the univ ersal relation [16]. Theorem 4. L et S b e a c ongruenc e-fr e e c ountably c omp act top olo gic al inverse semigr oup with zer o . Then S is iso m orphic to a finite semigr oup of matrix units. Pr o of. Supp ose not. Since the semigroup S con tains a zero b y Theorem 2, S is top ologically isomorphic to a to p o logical Brandt λ -extension B λ ( H ) of a pseudo compact t o p ological group H in the class of top ological inv erse semigroups and λ<ω . Supp ose that the group H is not trivial. Then we define a map h : B λ ( H ) → B λ b y the form ula e h (( α , g , β ) ) = ( α , β ) and h (0)=0. Since h (( α, g , β )( γ , s, δ )) = h (( α , g s, δ ))=( α, δ ) = ( α, β )( γ , δ )= h (( α, g , β )) h (( γ , s, δ )) fo r β = γ and h (( α , g , β )( γ , s, δ )) = h (0)=0=( α , β )( γ , δ ) = h (( α, g , β )) h (( γ , s, δ )) fo r β 6 = γ , the map h is a homomorphism. This con tradicts the assumption that S is a congruence-free semigroup.  Reference s [1] O. Andersen, Ein Bericht ¨ ub er die St ru ktur abstr akter Halb grupp en , PhD Thesis, Hamburg, 1952 . [2] L. W. Anderson, R. P . Hun ter and R. J. Ko ch, Some r esults on st abili ty in semigr oups . T r ans. Amer. Math. So c. 117 (196 5 ), 521 —529. [3] J. H. Carruth, J. A. Hildebrant and R. J. K och, The The ory of T op olo gic al Semigr oups, I, II . Ma rcel Dekker, Inc., New Y o rk and Base l, 198 3 a nd 1 986. [4] A. H. Cliffo r d and G. B . 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Dep ar tment of Ma thema tics, Iv an Frank o L viv N a tional Un iversity, Universytetska 1, L viv, 79000, Ukraine E-mail addr ess : o gutik@ franko. lviv.ua, ovgut ik@yaho o.com Institute o f Ma thema tics, Phys ics and Mechanics, and F acul ty o f Educa tion, U niversity of Ljubljan a, P.O.Box 2964, Ljubljana, 1001, Sl o venia E-mail addr ess : d usan.re povs@gu est.arnes.si