Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

CHARA CTERIZING COMP A CT CLIFF ORD SEMIGR OUPS THA T EMBED INTO CONV OLUTION AND FUNCTOR-SEMIGROUPS T A RAS BANAKH, MA TIJA CENCELJ, OLENA HR YNIV, AND DU ˇ SAN REPOV ˇ S Abstract. W e study al gebraic and to p ological properties of the co nv olution semigroups of probability m easures on a topological groups and sho w that a compact Clifford topological semi group S embeds into the co nv olution semi- group P ( G ) o ve r some topological group G i f and only if S em beds into the semigroup exp( G ) of compact subsets of G if and only if S i s an inv erse semi- group and has zero-dimensional maximal semilattice. W e also show th at such a Clifford semigroup S em beds int o the functo r-semigroup F ( G ) o v er a suit- able compact top ological group G for eac h weakly normal monadic functor F in the category of compacta such that F ( G ) con tains a G -inv ariant elemen t (whic h is a n analogue of the Haar measure on G ). 1. Introduction According to [7] (and [19]) ea ch (comm utativ e) semigr oup S embeds into the global semigroup Γ( G ) ov er a suitable (ab elian) gr oup G . The global semigroup Γ( G ) over G is the set of all non-empty subsets o f G endow ed with the semigro up op eration ( A, B ) 7→ AB = { ab : a ∈ A, b ∈ B } . If G is a topo logical gr oup, then the global semigro up Γ( G ) con tains a subsemigroup exp( G ) consisting o f all non-empty co mpa ct subsets of G and carrying a natural top olo gy which makes it a top o logical s emigroup. This is the Vietor is top o logy generated b y the sub-ba se consisting of the s ets U + = { K ∈ ex p( G ) : K ⊂ U } and U − = { K ∈ exp( G ) : K ∩ U 6 = ∅ } where U runs ov er op en subsets of G . Endow e d with the Vietoris top ology the semigroup exp( G ) will b e referr ed to as the hyp ersemigr oup over G (b ecause its underlying top ologica l spac e is the hypers pace exp( G ) of G , see [1 7]). The prob- lem of detecting top ologica l semigroups embeddable int o the h yper semigro ups o ver top ologica l gro ups has bee n considered in the literature, see [7]. This problem was res olved in [6] for the cla ss of Clifford compact top o logical semigroups: such a semigroup S embeds into the hypersemigr oup over a topo- logical group if and only if the set E of idemp o tents of S is a zer o-dimensiona l commutativ e subsemigroup of S . This characteriz ation implies the r esult o f [8] that the closed int erv al [0 , 1] with the op er ation of the minim um do es not em b ed int o the hyper s emigroup ov er a topo logical group. Date : May 9, 2022 . 2010 Mathematics Subject Classific ation. 20M18, 20 M30, 22A15, 43A05, 54 B20, 54B30. Key wor ds and phr ases. Conv olution semigroup, global semi group, hypersemigroup, Cl ifford semigroup, regular semigroup, topological group, Radon measure, w eakly n ormal mona dic funct or. 1 2 T. BANAKH, M. CENCELJ, O. HR YNIV, AND D. REPOV ˇ S W e recall that a s emigroup S is Cliffor d if S is the union of its subgroups. W e s ay that a top ologica l semig roup S 1 embeds into another to po logical semigroup S 2 if there is a s emigroup homomo rphism h : S 1 → S 2 which is a top ologica l em bedding. In this pap er we shall apply the alrea dy mentioned result of [6 ] and shall char- acterize Clifford compact semigr oups embeddable into the co nv olution s e migroups P ( G ) over top o logical gr oups G . The conv olution semigroup P ( G ) co ns ists of probability Radon mea sures on G a nd carr ies the ∗ - weak to po logy gener ated by the sub-base { µ ∈ P ( G ) : µ ( U ) > a } where a ∈ R and U runs ov er op en subsets of G . A mea sure µ defined o n the σ -alg ebra of Borel subsets of G is called R adon if for every ε > 0 there is a compact subset K ⊂ G with µ ( K ) > 1 − ε . The semigroup opera tio n on P ( G ) is given by the con volution measures. W e reca ll that the c onvolution µ ∗ ν o f tw o mea sures µ, ν is the measure assigning to each b ounded contin uous function f : G → R the v alue of the integral R µ ∗ ν f = R η R µ f ( xy ) dy dx . F or mor e detail information o n the conv olution semigroups, see [1 2], [14]. The following theorem is the pr inc ipa l result of this pap er. Theorem 1.1. F or any Cliffor d c omp act t op olo gic al semigr oup S the fol lowing assertions ar e e quivalent: (1) S emb e ds into the hyp ersemigr oup exp( G ) over a top olo gic al gr oup G ; (2) S emb e ds into the c onvolution s emigr oup P ( G ) over a top olo gic al gr oup G ; (3) The set E of idemp otents of S is a zer o-dimensional c ommutative subsemi- gr oup of S . This theorem will be applied to a c ha racteriza tion of Clifford compa ct top o lo gical semigroups embedda ble into the h yp erpsemigr o ups or con volution semigroups over top ologica l groups G b elong ing to certa in v a r ieties of top olo gical gr oups. A cla s s G of top ologica l gr oups is called a variety if it is closed under a rbitrary Tyc ho nov pro ducts, and taking closed subgroups, and quotient g roups b y closed normal sub- groups. Theorem 1.2. L et G b e a non-trivial variety of t op olo gic al gr oups. F or a Cliffor d c omp act top olo gic al semigr ou p S the fol lowing assertions ar e e quivalent: (1) S emb e ds into the hyp ersemigr oup exp( G ) over a top olo gic al gr oup G ∈ G ; (2) S emb e ds into the c onvolut ion s emigr oup P ( G ) ov er a t op olo gic al gr oup G ∈ G ; (3) The set E of idemp otent s is a zer o-dimensional c ommut ative subsemigr oup of S and al l close d sub gr oups of S b elong to t he class G . In fact, the equiv alence of the first and las t statements in Theorems 1.1 and 1.2 was prov ed in Theorems 3 and 4 of [6] so it r emains to prov e the equiv alence of the assertions (1) and (2). This will b e done in Pro p osition 1.3. W e recall that a semigro up S is called r e gular if ea ch element x ∈ S is r e gu lar in the sense that xy x = x fo r some y ∈ S . An element x ∈ S is called ( u niquely ) invertible if there is a (unique) elemen t x − 1 ∈ S (ca lled the inverse of x ) such that xx − 1 x = x and x − 1 xx − 1 = x − 1 . A semigroup S is called inverse if eac h element o f S is uniquely invertible. By [9, 1.17], [15, I I.1.2] a semigroup S is inv erse if and only if it is reg ular and the set E o f idemp otents of S is a commutativ e subsemigroup of S . An in verse semigroup S is Clifford if a nd o nly if xx − 1 = x − 1 x for all x ∈ S . In this case S decomp oses into the union S = S e ∈ E H e of the maxima l subgroups H e = { x ∈ S : xx − 1 = e = x − 1 x } of S para metrized by idempotents e of S . CLIFFOR D SUBSEMIGROUPS OF FUNCTOR-SEMIGROUPS 3 W e recall that a top olo gical se migroup S is ca lled a top olo gic al inverse semigr oup if S is a n inverse semigroup and the inversion map ( · ) − 1 : S → S , ( · ) − 1 : x 7→ x − 1 is contin uous. The set E of idemp otents of a top olog ical inverse semig roup S is a clos ed c o mmut ative s ubs emigroup of S called the idemp otent semi lattic e of S . W e say that tw o idemp otents e, f ∈ E a r e inc omp ar able if their pro duct ef differs from e and f . T wo elements x, y of an inv er se semigro up S a re called c onjugate if x = z y z − 1 and y = z − 1 xz fo r s ome element z ∈ S . F o r any idemp otent e ∈ E le t ↑ e = { f ∈ E : ef = e } deno te the principal filter of e . A top ologic a l space X is called total ly disc onne ct e d if for a ny distinct po ints x, y ∈ X there is a clo sed-and- op en subset U ⊂ X containing x but not y . The following pro p osition shows that the semigroups exp( G ) and P ( G ) over a top ologica l g roup G have the same r egular s ubsemigroups (which are necessar ily top ologica l in verse semigroups). Mo reov er, regular s ubsemigroups of exp( G ) or P ( G ) hav e man y sp ecific top olo gical and alg ebraic features. Prop ositi on 1.3. L et G b e a top olo gic al gr oup. A top olo gic al r e gular semigr oup S emb e ds into P ( G ) if and only if S emb e ds into exp( G ) . If the latter happ ens, then (1) S is a top olo gic al inverse semigr oup; (2) The idemp otent semilattic e E of S has total ly disc onne cte d princip al filters ↑ e , e ∈ E ; (3) An element x ∈ S is an idemp otent if and only if x 2 x − 1 is an idemp otent; (4) Any distinct c onjugate d idemp otents of S ar e inc omp ar able. This prop osition allows one to construct many examples of top olo gical re g ular semigroups no n-embeddable into the hypersemigr oups or conv olution semigro ups ov er a top o logical groups . The first tw o as sertions of this pro po sition imply the result o f [8] to the e ffect that non-trivial semigroups of left (or right) zero s as w ell as connected to po logical semilattices do not embed into the h yp ersemigr oup exp( G ) ov er a t op ologic al group G . The last tw o assertions imply that the semig r oups exp( G ) and P ( G ) do not contain Brandt semigro ups and bicyclic semigro ups. By a Br andt semigr ou p we under stand a semig r oup o f the fo rm B ( H , I ) = I × H × I ∪ { 0 } where H is a group, I is a non-empty set, and the pr o duct ( α, h, β ) ∗ ( α ′ , h ′ , β ′ ) o f t wo no n- zero elements of B ( H , I ) is equal to ( α, hh ′ , β ′ ) if β = α ′ and 0 otherwise. A bicyclic semigr oup is a s emigroup g enerated by tw o elemen ts p, q with the rela tion q p = 1. Bra ndt s emigroups and b yciclic semigroups play an imp ortant r ole in the structure theory o f in verse semigroups, see [15]. In fact, the semigr oups exp( G ) and P ( G ) ar e sp ecial ca ses of the s o-called functor-semigr o ups intro duce d b y T ele iko and Zar ichn yi [17]. They obser ved that any weakly nor mal monadic functor F : C omp → C omp in the catego ry o f compact Hausdorff spaces lift s to the category o f compact to p o logical semig r oups, whic h means that for a ny compa ct to po logical semigro up X the spa ce F X p o s sesses a natural semigroup structure. The semigroup oper a tion ∗ o n F X can b e defined b y the following form ula a ∗ b = F p ( a ⊗ b ) fo r a, b ∈ F X where p : X × X → X is the semigroup op era tion of X and a ⊗ b ∈ F ( X × X ) is the tensor pro duct of the elements a, b ∈ F X , see [17, § 3 .4]. Therefore we actually consider in this pap er the following genera l problem: 4 T. BANAKH, M. CENCELJ, O. HR YNIV, AND D. REPOV ˇ S Problem 1. 4. Gi ven a we akly n ormal monadic fu n ctor F : C omp → C omp , fin d a char acterization of c omp act (r e gular, inverse, Cliffor d) top olo gic al semigr oups em- b e ddable into the s emigr oup F X over a c omp act top olo gic al gr oup X . Given a c omp act top olo gic al gr oup X describ e invertible elements and idemp otents of the semigr oup F X . Observe that for the functor s exp and P the answer to the first part o f this problem is given in Theorem 1.1. F unctor- semigroups induced b y the functors G of inclusion hyper spaces and λ of super extension hav e been studied in [2]–[5],[1 1]. In fact, Theorem 1 .2 also ca n b e par tly generalized to some monadic functors F (including the functors exp, P , G and λ ). Given a co mpact top olo g ical gro up G let us define an elemen t a ∈ F ( G ) to b e G -invariant if g ∗ a = a = a ∗ g for ev ery g ∈ G . Here we identify G with a subspace o f F ( G ) (which is p ossible b eca use F , b eing weakly normal, preserves singletons). A G -inv ar ia nt elemen t in F ( G ) exists for the functors exp, P , λ , and G . F or the functors exp a nd P a G -inv a riant e le ment o n F ( G ) is unique: it is G ∈ exp( G ) a nd the Haa r measure on G , r esp ectively . Theorem 1.5 . L et F : C omp → C omp b e a we akly normal monadic functor such that for every c omp act top olo gic al gr ou p G the s emigr oup F ( G ) c ont ains a G -invariant element. Each Cliffor d c omp act top olo gic al inverse semigr oup S with zer o-dimensional idemp otent semila ttic e E emb e ds into the functor-semigr oup F ( G ) over the c omp act top olo gic al gr oup G = Q e ∈ E e H e wher e e ach e H e is a non-trivial c omp act top olo gic al gr ou p c ontaining the maximal sub gr oup H e ⊂ S c orr esp onding to an idemp otent e ∈ E of S . Pr o of. By Theorem 3 of [6] (see also [13]), each Clifford compact to po logical in verse semigroup S with z ero-dimensio na l idemp otent s emilattice E embeds in to the pro d- uct Q e ∈ E H 0 e , where H 0 e stands fo r the extension of the maximal subgroup H e by a n isolated p oint 0 / ∈ H e such that x 0 = 0 x = 0 for a ll x ∈ H e . F or every idemp otent e ∈ E , fix a non-trivial co mpact top olog ical group e H e containing H e . By our hy- po thesis, the space F ( e H e ) con tains an e H e -inv ar iant element z e ∈ F ( e H e ). Then H 0 e can b e identified with the close d subsemigroup H e ∪ { z e } of F ( e H e ) and the product Q e ∈ E H 0 e can b e identified with a subsemig roup of the pro duct Q e ∈ E F ( e H e ). By [17, p.126 ], the la tter pro duct ca n b e identified with a subspa ce (actually a sub- semigroup) of F ( Q e ∈ E e H e ) = F ( G ), where G = Q e ∈ E e H e . In this way , we obtain an embedding of S into F ( G ).  As we hav e said, the functors λ of s up er extension and G of inclusio n hyperspa ces satisfy the hypothesis of Theor em 1.5. How ever, Prop ositio n 1.3 is sp ecific for the functor P and cannot b e gener alized to the functor s λ or G . Indeed, fo r the 4 - element cyclic group C 4 the se migroup λ ( C 4 ) is isomorphic to the co mmu tative inv e r se se migroup C 4 ⊕ C 1 2 , wher e C 1 2 = C 2 ∪ { 1 } is the result of attaching an external unit to the 2-element cyclic gro up C 2 , (se e [2]). On the other ha nd, the 12-element semig roup C 4 ⊕ C 1 2 cannot be embedded into exp( C 4 ) bec ause the set of reg ular elements of exp( C 4 ) consists o f 7 elements (which a re shifted s ubgroups o f C 4 ). Also the commutativ e inv erse se migroup λ ( C 4 ) ∼ = C 4 ⊕ C 1 2 can b e embedded into G ( C 4 ) (beca use λ is a submona d of G ) but cannot embed int o exp( C 4 ). CLIFFOR D SUBSEMIGROUPS OF FUNCTOR-SEMIGROUPS 5 2. Idempotents and inver tible elements of the convolution semigr oups In this section we pr ov e P r op osition 1.3. F or each top o logical group G the semigroups P ( G ) and exp( G ) a re r elated via the map o f the supp ort. W e r ecall that the supp ort of a Radon mea sure µ ∈ P ( G ) is the closed subs e t S µ = { x ∈ G : µ ( Ox ) > 0 for each neigh bo rho o d O x of x } of G . Let 2 G denote the semigroup of all non-empty clo sed subsets of G endow ed with the semig r oup ope ration A ∗ B = AB . By supp : P ( G ) → 2 G , supp : µ 7→ S µ we de no te the supp ort map. The follo wing prop osition is well-known, see (the proo f of ) Theorem 1.2 .1 in [12]. Prop ositi on 2.1. L et G b e a top olo gic al gr oup. F or any me asur es µ, ν ∈ P ( G ) the fol lowing holds: S µ ∗ ν = S µ · S ν . This me ans that the supp ort m ap supp : P ( G ) → 2 G is a semigr oup homomorphism. W e sha ll sho w that for any r e g ular elemen t µ of the conv o lution semigroup P ( G ) the supp ort S µ is compac t a nd thus b elongs to the subsemigr oup ex p( G ) of 2 G . First, we c har acterize idemp otent measures on a top o lo gical group G . A measur e µ ∈ P ( G ) is ca lled a n idemp otent me asure if µ ∗ µ = µ . In 1954 W endel [20] pro ved that each idempo tent measure on a compact topolo gical gr oup c oincides with the Haar mea s ure of some compa ct subgro up. La ter , W endel’s result was generalized to lo cally compact g roups by P ym [16] and to all topolo gical g roups b y T ortr at [18]. By the Haar me asu re on a compact topolog ical gr oup G w e understand the unique G -inv ar iant probabilit y measur e on G . It is a classic a l result that such a measure exists and is unique. Th us we hav e the f ollowing characterization of idempo tent measures on topo logical groups : Prop ositi on 2.2. A pr ob ability R adon m e asur e µ ∈ P ( G ) on a top olo gic al gr oup G is an idemp otent of the semigr oup P ( G ) if and only if µ is the Haar me asur e of some c omp act sub gr oup of G . W e shall use this propo sition to describe regular elements o f the con volution semigroups. T o this end we a pply Prop os ition 4 of [6] tha t describ es regula r ele- men ts of the h yp e rsemigro ups o ver topo logical gro ups : Prop ositi on 2.3 (Bana k h-Hryniv) . F or a c omp act subset K ∈ exp( G ) of a top o- lo gic al gr oup G the fol lowing assertions ar e e quivalent: (1) K is a r e gular element of the semigr oup exp( G ) ; (2) K is uniquely invertible in exp( G ) ; (3) K = H x for some c omp act sub gr oup H of G and some x ∈ G . A similar desc ription of r e gular elemen ts holds for the conv olution semigroup: Prop ositi on 2.4 . F or a me asur e µ ∈ P ( G ) on a t op olo gic al gr oup G the fol lowing assertions ar e e quivalent: (1) µ is a r e gular element of the semigr oup P ( G ) ; (2) µ uniquely invertible in P ( G ) ; (3) µ = λ ∗ x for some id emp otent me asur e λ ∈ P ( G ) and some element x ∈ G . 6 T. BANAKH, M. CENCELJ, O. HR YNIV, AND D. REPOV ˇ S Pr o of. Assume that µ is a regular element of P ( G ) and ν ∈ P ( G ) is a measure such that µ ∗ ν ∗ µ = µ . The measure µ ∗ ν , b eing an idempotent of P ( G ) coincides with the Haar measure λ on so me compact subgroup H of G . It follo ws that S µ · S ν = S µ ∗ ν = S λ = H and hence S µ and S ν are co mpa ct subsets of the g roup G . Since supp : P ( G ) → 2 G is a semigro up homomorphism, w e get S µ ∗ S ν ∗ S µ = S µ , which mea ns that S µ is a regula r elemen t of the semigroup exp( G ) and hence S µ = ˜ H x for some compact subgroup ˜ H and some element x ∈ G acco rding to Prop os itio n 2.3. W e claim that ˜ H = H . Indeed, H ˜ H x = S λ S µ = S µ ∗ ν S µ = S µ ∗ ν ∗ µ = S µ = ˜ H x implies that H ⊂ ˜ H . Next, for any point y ∈ S ν we g et ˜ H xy ⊂ ˜ H xS ν = S µ S ν = S λ = H ⊂ ˜ H which y ie lds xy ∈ ˜ H and finally H = ˜ H . Next, we sho w that µ = λ ∗ x , which is equiv alen t to λ = µ ∗ x − 1 . Observe that S µ ∗ x − 1 = S µ x − 1 = H xx − 1 = H . No w the equa lity µ ∗ x − 1 = λ will follow as so on as we check that the measure µ ∗ x − 1 is H -inv ariant. T a ke any p oint y ∈ H and note that y ∗ µ ∗ x − 1 = y ∗ µ ∗ ν ∗ µ ∗ x − 1 = y ∗ λ ∗ µ ∗ x − 1 = λ ∗ µ ∗ x − 1 = µ ∗ x − 1 , which means that the measure µ ∗ x − 1 on H is left-in v ariant. Since H p oss esses a unique left-inv ar ia nt pr obability measure λ , w e conclude that µ = λ ∗ x . Finally , we show that µ is uniquely inv ertible in P ( G ). It suffices to chec k that the measure ν is equal to x − 1 ∗ λ provided ν = ν ∗ µ ∗ ν . F or this just observe that S ν being a unique inverse of S µ is equal to x − 1 H . Then S x ∗ ν = xS ν = xx − 1 H . Finally , noticing that for every y ∈ H we get x ∗ ν ∗ y = x ∗ ν ∗ µ ∗ ν ∗ y = x ∗ ν ∗ λ ∗ y = x ∗ ν ∗ λ = x ∗ ν, which mea ns that x ∗ ν is a right inv ariant measure on H . Since λ is the unique right-in v ariant measur e on H we also get x ∗ ν = λ and hence ν = x − 1 ∗ λ .  Given a semigroup S we denote the set of reg ular elements of S by Reg( S ). Prop ositi on 2.5. F or any t op olo gic al gr oup G , the supp ort m ap supp : Reg( P ( G )) → Reg(exp( G )) is a home omorphism. Pr o of. The preceding pro p osition implies that the map supp : Reg( P ( G )) → Reg(exp( G )) is bijective. In or der to chec k the contin uity of this map, we must prov e that for any op en set U ⊂ G the preimages supp − 1 ( U + ) = { µ ∈ Reg( P ( G )) : supp( µ ) ⊂ U } and supp − 1 ( U − ) = { µ ∈ Reg( P ( G )) : supp( µ ) ∩ U 6 = ∅} are op en in P ( G ). The op enness of supp − 1 ( U − ) follows from the o bserv atio n that supp( µ ) ∩ U 6 = ∅ if and only if µ ( U ) > 0. T o se e that supp − 1 ( U + ) is op en, fix a ny measur e µ ∈ Reg ( P ( G )) with supp( µ ) ⊂ U . By Pro p o sition 2.4, supp( µ ) = H x for some compact s ubgroup H of G and some x ∈ G . The CLIFFOR D SUBSEMIGROUPS OF FUNCTOR-SEMIGROUPS 7 compactness of H a llows us to find an op en neighbo rho o d V of the neutral ele- men t of G such tha t H V 2 H V − 2 H V ⊂ U x − 1 . Now consider the op en neighbor- ho o d W = { ν ∈ Reg( P ( G )) : ν ( H V x ) > 1 2 } of the measure µ . W e claim that W ⊂ supp − 1 ( U + ). Indeed, g iven any measure ν ∈ W we can apply Prop o s i- tion 2.4 to find an idempo tent measur e λ a nd y ∈ G such that ν = λ ∗ y . Then 1 2 < ν ( H V x ) = λ ( H V xy − 1 ). W e claim that S λ ⊂ H V V H . Indeed, given an ar- bitrary p oint z ∈ S λ use the S λ -inv ar iance of λ to conclude that λ ( z H V xy − 1 ) = λ ( H V xy − 1 ) > 1 / 2, whic h implies that the int ersection z H V xy − 1 ∩ H V xy − 1 is non-empty whic h yields z ∈ H V xy − 1 ( H V xy − 1 ) − 1 = H V V H . The inequality λ ( H V xy − 1 ) > 1 / 2 implies that H V xy − 1 int ersects S λ and hence the set H V V H . Then y ∈ H V − 2 H H V x and S ν = S λ ∗ y ⊂ H V 2 H H V − 2 H V x ⊂ U x − 1 x = U , which implies that ν ∈ supp − 1 ( U + ). This completes the pro of o f the contin uity of the map supp : Reg ( P ( G )) → Reg(ex p( G )). The pro o f of the contin uity of the inv erse map supp − 1 : Reg(exp( G )) → Reg ( P ( G )) is even mor e inv olved. Assume that supp − 1 is disco ntin uous a t some p o int K 0 ∈ Reg(exp( G )). By Pro po sition 2.3, K 0 is a cose t of some c o mpact s ubg roup of G . After a suitable shift, w e can a ssume that K 0 is a compact subgroup of G and then µ 0 = supp − 1 ( K 0 ) is the unique Haar measur e on K 0 . Since s upp − 1 is discontin uous a t K 0 , ther e is a neighborho od O ( µ 0 ) ⊂ P ( G ) of µ 0 such that supp − 1 ( O ( K 0 )) 6⊂ O ( µ 0 ) for an y neighbor ho o d O ( K 0 ) ⊂ Reg(exp( G )) of K 0 in Reg(exp( G )). It it w ell-known that the top ology of G is g enerated by the left unifor m struc tur e, which is gener ated by b ounded left-inv ariant pseudometrics. Each b ounded left- inv ar ia nt pseudometric ρ on G induces a pseudometric ˆ ρ on P ( G ) defined by ˆ ρ ( µ 1 , µ 2 ) = inf { µ ( ρ ) : µ ∈ P ( G × G ) P pr 1 ( µ ) = µ 1 , P pr 2 ( µ ) = µ 2 } where P pr i : P ( G × G ) → P ( G ) is the map induced by the pro jection pr i : G × G → G ont o the i th co o rdinate. By [1, § 4] o r [10, 3.1 0], the top ology of the s pace P ( G ) is generated by the pseudo metrics ˆ ρ where ρ r uns ov er a ll b ounded left-inv ariant contin uous pseudometrics on G . Consequently , we can find a left-inv ariant co ntin uous pseudometric ρ on G such that the neig hborho o d O ( µ 0 ) contains the ε 0 -ball B ( µ 0 , ε 0 ) = { µ ∈ P ( G ) : ˆ ρ ( µ, µ 0 ) < ε 0 } for some ε 0 > 0. Replacing ρ by a larger left-inv a riant pseudometric, we can additionally assume that for the pseudometric space G ρ = ( G, ρ ) the map γ : G ρ × G ρ → G ρ , γ : ( x, y ) 7→ xy − 1 , is contin uous at each p oint ( x, y ) ∈ K 0 × K 0 (this follows from the fact that for each contin uo us left-in v ariant pseudometric ρ 1 on G we can find a c o ntin uous left-inv a r iant pseudo metric ρ 2 on G such that the map γ : G ρ 2 × G ρ 2 → G ρ 1 is con tinu ous at p o int s of the compact s ubset K 0 × K 0 ). The c o ntin uity and the left-in v a riance of the pseudometric ρ implies that the set G 0 = { x ∈ G : ρ ( x, 1) = 0 } is a closed subgr oup o f G . Let G ′ = { xG 0 : x ∈ G } be the left coset space of G by G 0 and q : G → G ′ , q : x 7→ xG 0 , b e the quotient pro jection. The space G ′ = G/G 0 will be consider ed as a G -s pa ce endowed with the natural left a ction o f the gro up G . The pseudo metric ρ induces a contin uo us left-in v aria nt metric ρ ′ on G ′ such that ρ ( x, y ) = ρ ′ ( q ( x ) , q ( y )) for all x, y ∈ G . So, q : ( G, ρ ) → ( G ′ , ρ ′ ) is a n is ometry . The pseudometrics ρ and ρ ′ induce the Hausdorff pseudometrics ρ H and ρ ′ H on the hyperspace s exp( G ) a nd exp( G ′ ) such that the map exp q : exp( G ) → exp( G ′ ) is an is ometry . Also these pseudometrics 8 T. BANAKH, M. CENCELJ, O. HR YNIV, AND D. REPOV ˇ S induce the pseudometrics ˆ ρ a nd ˆ ρ ′ on the spaces o f measures P ( G ), P ( G ′ ) such that the map P q : ( P ( G ) , ˆ ρ ) → ( P ( G ′ ) , ˆ ρ ′ ) is an iso metr y . The co nt inuit y of the map γ : G 2 ρ → G ρ at K 2 0 implies that ( K 0 , ρ ) is a (not necessarily separated) top ologica l g roup, K 0 ∩ G 0 is a closed normal s ubgroup of K 0 and hence K ′ 0 = q ( K 0 ) = K 0 /K 0 ∩ G 0 has the structur e of top olo gical group. Then µ ′ 0 = P q ( µ 0 ) is a Haar measure in K ′ 0 . By the choice of the neighborho od O ( µ 0 ), for ev er y n ∈ N w e can find a co mpact set K n ∈ Reg(exp( G )) suc h that the measure µ n = supp − 1 ( K n ) do es not b elong to O ( µ 0 ). Then ˆ ρ ( µ n , µ 0 ) ≥ ε 0 by the choice of the pseudometric ρ . F or every n ∈ N let µ ′ n = P q ( µ n ) ∈ P ( G ′ ), a nd K ′ n = q ( K n ) ∈ ex p( G ′ ). The conv er gence o f the sequence ( K n ) to K 0 in the pseudometric spa c e (exp( G ) , ρ H ) im- plies the conv er gence of the sequence ( K ′ n ) to K ′ 0 in the metric space (exp( G ′ ) , ρ ′ H ), which implies that the union K ′ = S n ∈ ω K ′ n is compact in the metric spa ce ( G ′ , ρ ′ ). Then the subspace P ( K ′ ) is compact in the metric spa c e ( P ( G ) , ˆ ρ ′ ) and hence the sequence ( µ ′ n ) n ∈ N contains a subsequence that con verges to s ome measure µ ′ in ( P ( G ′ ) , ˆ ρ ′ ). W e lose no generality as s uming that whole sequence ( µ ′ n ) n ∈ N con- verges to µ ′ . Since ε 0 ≤ ˆ ρ ( µ n , µ 0 ) = ˆ ρ ′ ( µ ′ n , µ ′ 0 ), w e conclude that µ ′ 6 = µ ′ 0 . W e shall derive a contradiction (with the uniqueness of a left-inv ar iant probability measure on compa ct gro ups) b y showing that µ ′ is a left-inv a riant measure on K ′ 0 , dis tinct from the Haar meas ur e µ ′ 0 . The ˆ ρ ′ -conv ergence µ ′ n → µ ′ and ρ ′ H -conv ergence s upp( µ ′ n ) = K ′ n → K ′ 0 imply that supp( µ ′ ) ⊂ K ′ 0 and thus µ ′ is a probability mea sure on the compact topo lo gical group K ′ 0 . It remains to chec k tha t the measure µ ′ is left-inv ariant. Assuming the conv er se, we ca n find a p oint a ∈ K ′ 0 such that a ∗ µ ′ 6 = µ ′ and thus ε = ˆ ρ ′ ( µ ′ , a ∗ µ ′ ) > 0. Sin ce the map γ : G ρ × G ρ → G ρ is contin uous at each point ( x, y ) ∈ K 0 × K 0 , we can find a p ositive δ < ε 4 so small that ρ ( xy , x ′ y ) < ε 4 for a ny x, y ∈ K 0 and x ′ ∈ G with ρ ( x ′ , x ) < δ . Since ρ H ( K n , K 0 ) → 0 a nd ˆ ρ ′ ( µ ′ n , µ ′ ) → 0, there is a nu mber n ∈ N and a p oint a n ∈ K n such that ρ ( a, a n ) < δ and ˆ ρ ′ ( µ ′ n , µ ′ ) ≤ ε/ 4. Consider tw o left shifts l a : G → G , l a : x 7→ ax , and l a n : G → G . The choice of δ guarantees tha t ρ K 0 ( l a , l a n ) = sup x ∈ K 0 ρ ( l a ( x ) , l a n ( x )) ≤ ε 4 . Then ˆ ρ ′ ( a ∗ µ ′ , a n ∗ µ ′ ) = ˆ ρ ′ ( P l a ( µ ′ ) , P l a n ( µ ′ )) ≤ ε 4 . The le ft shift l a n : G → G , b eing a n is ometry of the pseudometric spa ce ( G, ρ ), induces an is ometry l ′ a n : G ′ → G ′ of the metric space ( G ′ , ρ ′ ), which induces the isometry P l ′ a n : P ( G ′ ) → P ( G ′ ) of the corr esp onding spac e of measures. So , ˆ ρ ′ ( a n ∗ µ ′ , a n ∗ µ ′ n ) = ˆ ρ ′ ( P l ′ a n ( µ ′ ) , P l ′ a n ( µ ′ n )) = ˆ ρ ′ ( µ ′ , µ ′ n ) ≤ ε 4 . The compac t se t K n , b eing a regular element of the semigr oup e xp( G ) is equal to H n x n for some compact subgr oup H n ⊂ G and so me p oint x n ∈ G accor ding to Pro p osition 2.3. Then µ n = supp − 1 ( K n ) is eq ua l to λ n ∗ x n where λ n is the Haar measur e on the group H n . Since λ n is left-inv ariant, a n ∗ µ n = a n ∗ λ n ∗ x n = λ n ∗ x n = µ n and hence a n ∗ µ ′ n = µ ′ n . Now w e see that ˆ ρ ′ ( µ ′ , a ∗ µ ′ ) ≤ ˆ ρ ′ ( µ ′ , µ ′ n ) + ˆ ρ ′ ( µ ′ n , a n ∗ µ ′ n ) + ˆ ρ ′ ( a n ∗ µ ′ n , a n ∗ µ ′ ) + ˆ ρ ′ ( a n ∗ µ ′ , a ∗ µ ′ ) ≤ ≤ ε 4 + 0 + ε 4 + ε 4 < ε = ˆ ρ ′ ( µ ′ , a ∗ µ ′ ) , which is a desired co nt radiction.  CLIFFOR D SUBSEMIGROUPS OF FUNCTOR-SEMIGROUPS 9 The following corollary establishes the fir st part of P rop osition 1.3. The seco nd part of that pro p osition follows from Theorem 2 o f [6]. Corollary 2.6. L et G b e a top olo gic al gr oup. Then a top olo gic al r e gular semigr oup S c an b e emb e dde d into the h yp ersemigr oup exp( G ) i f and only if S c an b e emb e dde d into t he c onvolution semigr oup P ( G ) . Pr o of. If S ⊂ exp( G ) is a regula r subsemigroup, then S ⊂ Reg(ex p( G )) and supp − 1 ( S ) is an isomorphic co py of S in P ( G ) according to Pro p o sitions 2 .5. Con- versely , if S ⊂ P ( G ) is a reg ula r subsemigroup, then its imag e supp( S ) is a n isomorphic copy of S in exp( G ).  Ackno wledgements This research was suppo rted b y t he Slov e nian Resea r ch Agency grants P1-029 2 - 0101- 04, J1- 9643- 0101 and J1-205 7-010 1. W e thank the referee for comments a nd suggestions . References [1] T. Banakh, T op olo gy of sp ac e s of pr ob ability me asur es, II , Mat. Stud. 5 (1995), 88–106 (in Russian). [2] T. Banakh, V. Gavrylkiv, O. N ykyforc hy n, Algebr a in sup erextensions of g r oups, I: zer os a nd c ommutativity , A lgebra Discrete Math. no.3 (2008), 1–29. [3] T.Banakh, V.Gavrylkiv, Algebr a in sup erextensions of gr oups, II: c anc elativity and c enters , Algebra Discrete Math. no.4 (2008), ( 2008), 114. [4] T.Banakh, V.Ga vrylkiv, Algebr a in the sup er ext ensions of g r oups: minimal left ide als , Mat. Stud. 31:2 (2009) 142148. [5] T.Banakh, V.Ga vrylkiv, Algebr a in the sup er ext e nsions of twinic gr oups , Dissert. 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Instytut Ma tema tyki, Akademia ´ Swie ¸ tokrzyska, Kielce, Poland and Dep ar tm ent of Ma thema tics, Iv an Franko Na tional University of L viv, L viv, Ukrain e E-mail ad dr ess : tbanakh@yahoo. com Institute of Ma thema tics, Physics an d Mechanics, and F acul ty of Educa tion, Uni- versity of Ljubljana, P.O.B. 2964, Ljubljana, 1001 , Slovenia E-mail ad dr ess : matija.cencelj @guest.arnes.si Dep ar tment of Ma thema tics, Iv an Fran ko Na tional Univ ersity of L viv, L viv, Ukraine E-mail ad dr ess : olena hryniv@u kr.net F acul ty of Ma them atic s and Phy sics, and F acul ty of Educa tion, Univ er sity of Ljubl- jana, P.O.B. 2964, Lj ubljana, 1001, Slovenia E-mail ad dr ess : dusan.repovs@g uest.arnes.si