Toplogical semigroups embedded into topological groups
aa r X i v : . [ m a t h . GN ] J un TOPOLOGICAL SEMIGROUPS EMBEDDED INTOTOPOLOGICAL GROUPS
JULIO C´ESAR HERN ´ANDEZ ARZUSAUNIVERSIDAD DE CARTAGENAPROGRAMA DE MATEM ´ATICAS
Abstract.
In this paper we give conditions under which a topologi-cal semigroup can be embedded algebraically and topologically into acompact topological group. We prove that every feebly compact regu-lar first countable cancellative commutative topological semigroup withopen shifts is a topological group, as well as every connected locallycompact Hausdorff cancellative commutative topological monoid withopen shifts. Finally, we use these results to give sufficient conditions ona topological semigroup that guarantee it to have countable cellularity.
Primary: 54B30, 18B30, 54D10;Secondary: 54H10, 22A30
Key Words and Phrases: Cancellative topological semigroup, cellularity, σ -compact space, feebly compact space. INTRODUCTION
In [16] the author presents some properties that allow us to embed, topo-logically, a cancellative commutative topological semigroup into a topologi-cal group, we take advantage of this results to find conditions under whicha cancellative commutative topological semigroup has countable cellularity,as well as, when a topological semigroup is a topological group. The classof topological spaces having countable cellularity is wide, in fact, it contains(among other classes) the class of the σ -compact paratopological groups(see [17, Corollary 2.3]), the class of the sequentially compact σ -compactcancellative topological monoids (see [7, Theorem 4.8]) and the class of thesubsemigroups of precompact topological groups (see [20, Corollary 3.6]).These results have served us as motivation, thus we try to find similar re-sults in the context of topological semigroups.Compactness type conditions (compactness and sequential compactness) un-der which a topological semigroup is a topological group are given in [1,Theorem 2.5.2], [15, Theorem 2.4] and [3, Theorem 6], we use the feeblecompactness and local compactness to obtain similar results. The reflec-tion on the class of the regular spaces allows us to disregard the axioms ofseparation to obtain topological monoids with countable cellularity. . PRELIMINARIES
We denote by Z , R and N , the set of all the integer numbers, real numbersand positive integer numbers, respectively. If A is a set, | A | will denote thecardinal of A , ℵ = | N | . If X is a topological space and x ∈ X , N ( X ) x willdenote the set of all open neighborhoods of x in X or simply N x when thespace is understood.A semigroup is a set S = ∅ , endowed with an associative operation. If S alsohas neutral element, we say that S is a monoid . A mapping f : S −→ H between semigroups is a homomorphism if f ( xy ) = f ( x ) f ( y ) for all x, y ∈ S . A semitopological semigroup (monoid) consists of a semigroup (resp.monoid) S and a topology τ on S , such that for all a ∈ S , the shifts x ax and x xa (noted by l a and r a , respectively) are continuous mappings from S to itself. We say that a semitopological semigroup has open shifts , if foreach a ∈ S and for each open set U in S , we have that l a ( U ) and r a ( U )are open sets in S . A topological semigroup (monoid)( paratopological group )consists of a semigroup (resp. monoid)(resp. group) S and a topology τ on S , such that the operation of S is jointly continuous. Like [1] we do notrequire that semigroups to be Hausdorff. If S is a paratopological group andif also the mapping x x − is continuous, we say that S is a topologicalgroup. A congruence on a semigrouop S is an equivalence relation on S , ∼ , such that if x ∼ y and a ∼ b , then xa ∼ yb . If S is a semitopologicalsemigroup, then we say that ∼ is a closed congruence if ∼ is closed in S × S .If ∼ is an equivalence relation in a semigroup (monoid) S and π : S −→ S/ ∼ is the respective quotient mapping, then S/ ∼ is a semigroup (monoid) and π an homomorphism if and only if ∼ is a congruence ([5], Theorem 1).The axioms of separation T , T , T , T and regular are defined in accordancewith [4]. We denote by C i the class of the T i spaces, where i ∈ { , , , , r } .Let X be a topological space, a cellular family in X is a pairwise disjointnon empty family of non empty open sets in X . The cellularity of a space X is noted by c ( X ) and it is defined by c ( X ) = sup {| U | : U is cellular familiy in X } + ℵ . If c ( X ) = ℵ , we say that X has countable cellularity or X has the Souslinproperty .If X is a topological space and A ⊆ X , we will note by Int X ( A ) and Cl X ( A ),the interior and the closure of A in X , or simply Int ( A ) and A , respectively,when the space X is understood. An open set U in X , is called regular open in X if IntU = U . It is easy to prove the regular open ones form a base fora topology in X , X endowed with this topology, will note by X sr , which wewill call semiregularitation of X. From [10], it is well known that for each i ∈ { , , , , r } and every topo-logical space, X , there is a topological space, C i ( X ) ∈ C i , (unique up tohomeomorphism) and a continuous mapping ϕ ( X, C i ) of X onto C i ( X ), suchthat given a continuous mapping f : X −→ Y , being Y ∈ C i , there exists an nique continuous mapping g : C i ( X ) −→ Y , such that g ◦ ϕ ( X, C i ) = f .According to [2, Section 2] a SAP -compactification of a semitopologicalsemigroup S is a pair ( G, f ) consisting of a compact Hausdorff topologicalgroup G and a continuous homomorphism f : S −→ G such that for eachcontinuous homomorphism h : S −→ K , being K a compact Hausdorff topo-logical group, there is an unique continuous homomorphism h ∗ : G −→ K such that h = h ∗ ◦ f .A space X is feebly compact if each locally finite family of open sets in X is finite. By [8, Theorem 1.1.3], pseudocompactness is equivalent to feeblecompactness in the class of the Tychonoff spaces.A space X is locally compact if each x ∈ X has a compact neighborhood.The following proposition gives us some properties of C i ( S ), when S is atopological monoid with open shifts. Proposition 2.1.
Let S be a topological monoid with open shifts. Then i) C r ( S ) is a monoid, ϕ ( S, C i ) is an homomorphism (see [6, Proposition3.8] ) and C r ( S ) = C ( S sr ) ( see [7, Theorem 2.8] ). If S is alsocancellative, C r ( S ) is cancellative (see [7, Lemma 4.6] ). ii) If A is an open set in S , then C r ( A ) = ϕ ( S, C r ) ( A ) ( see [6, Corollary5.7] and [13, Lemma 4] ). iii) c ( S ) = c ( C i ( S ) , for each i ∈ { , , , , r } . iv) C ( S ) = S sr . Moreover, if S is T , C r ( S ) = S sr (see [7, Corollary2.7] and [13, Proposition 1] ). v) ϕ ( S, C i ) is open for each i ∈ { , , } (see [7, Proposition 2.1] ). vi) If S is a paratopolgical group, C r ( S ) is a paratopological group (see [19, Corollary 3.3 and Theorem 3.8 ] , [18, Theorem 2.4] ). It is easy to see that if X is a first countable topological space, then X sr isfirst countable. Since C r ( S ) = C ( S sr ) and ϕ ( S, C ) is open, whenever S is atopological monoid with open shifts, we have the following corollary. Corollary 2.2. If S is a first-countable topological monoid with open shifts,then C r ( S ) is first-countable. EMBEDDING TOPOLOGICAL SEMIGROUPS INTOTOPOLOGICAL GROUPS
Let S be a cancellative commutative semigroup, S × S is a cancellativecommutative semigroup, by defining the operation coordinate wise. Let usdefine in S × S the following relation: ( x, y ) R ( a, b ) if and only if xb = ya . It isnot to hard to prove that R is a congruence, hence if π : S × S −→ ( S × S ) / R is the respective quotient mapping, the operation induced by π makes of( S × S ) / R a semigrup. It is easy to prove that ( S × S ) / R is a group, wherethe equivalence class { π (( x, x )) : x ∈ S } is the neutral element, and theinverse of π (( a, b )) is π (( b, a )). Also, the function ι : S −→ ( S × S ) / R de-fined by ι ( x ) = π (( xa, a )), for each x ∈ S , is an algebraic monomorphism,where a is a fixed element of S ( ι does not depend of the choice of a ). Note hat π ( x, y ) = π ( xa , ya ) = π ( xa, a ) π ( a, ya ) = ι ( x )( ι ( y )) − , therefore ifwe identify each element of S with its image under ι , we have that eachelement of ( S × S ) / R can be written as xy − , where x, y ∈ S , that is to say( S × S ) / R = SS − . Let f : S −→ G be a homomorphism to an abelian topo-logical group G , there is an unique homomorphism f ∗ : ( S × S ) / R −→ G (defined as f ∗ ( π (( x, y )) = xy − ) such that f ∗ | S = f. In summary, we havethat the class of the abelian groups is a reflexive subcategory of the class ofthe cancellative commutative semigroups. Since the reflections are uniqueup to isomorphims, ( S × S ) / R is uniquely determined by S , and we willdenote by h S i and it is called group generated by S . If S is a topologi-cal semigroup, we will call h S i ∗ to h S i endowed with the quotient topologyinduced by π : S × S −→ h S i , S × S endowed with the Tychonoff producttopology. Definition 3.1.
Let S be a topological semigroup. We say that S has con-tinuous division if give x, y ∈ S and an open set V in S , containing y , thereare open sets in S , U and W , containing x and xy ( yx ), respectively; suchthat W ⊆ T { uV : u ∈ U } ( W ⊆ T { V u : u ∈ U } ). Proposition 3.2. ( [16, Theorem 1.15 and Theorem 1.19] ) If S is a cancella-tive and commutative Hausdorff topological semigroup with open shifts, then h S i ∗ is a Hausdorff topological group and the quotient mapping π : S × S −→h S i ∗ is open. Furthermore, if S has continuous division, ι : S −→ ι ( S ) is anhomeomorphism and ι ( S ) is open in h S i ∗ . Proposition 3.3.
Every open subsemigroup of a topological group has con-tinuous division.Proof.
Let S be an open subsemigroup of a topological group G . Let x, y ∈ S , since G is a group, x − xy = y . Let V be an open subset of S containing y , then V is open in G , the continuity of the operations of G , implies thatthere are open subsets in G , K and M , containing x and xy , respectively,such that K − M ⊆ V . Let us put U = K ∩ S and W = M ∩ S , then U and W are open subsets of S containing x and xy , respectively. Now, if t ∈ W and u ∈ U , then u − t ∈ K − M ⊆ V , hence t ∈ uV , therefore W ⊆ uV ,for every u ∈ U . We have proved that W ⊆ T u ∈ U uV . For yx we proceedanalogously. (cid:4) So far we just have embedded, algebraically, semigroups into groups, thefollowing proposition gives us a topological and algebraic embedding.
Proposition 3.4.
Let S be a cancellative commutative semitopological semi-group with open shifts. There exists a topology τ in h S i , such that ( h S i , τ ) is a semitopological group containing S as an open semigroup. Moreover i) S is 1-contable if and only if ( h S i , τ ) is 1-contable. ii) ( h S i , τ ) is a partopological group if and only if S is a topologicalsemigroup. ii) If S is T , ( h S i , τ ) is T . iv)) If S is Hausdorff and locally compact, ( h S i , τ ) is a locally compactHausdorff topological group.Proof. Let x be a fixed element in S , being S a cancellative commutativesemitopological semigroup with open shifts, and put B = { x − V : V ∈ N ( S ) x } . We will prove that { gU : U ∈ B , g ∈ h S i} is a base for a topologyof semitopological group in h S i , for it we will prove the conditions 1, 3 an 4given in [12, Page 93]. It is easy to prove the conditions 1 and 4, let us see3. Let V x ∈ N ( S ) x and let t ∈ x − V x , from the fact that h S i = SS − , we havethat t = ab − , where a, b ∈ S , thus ax ∈ bV x . Given that the shifts in S are open and continuous we can find W x ∈ N ( S ) x such that aW x ⊆ bV x , then t ( x − W x ) ⊆ x − V x and condition 3 holds. If τ is the topology generatedby B , then ( h S i , τ ) is a semitopological group, moreover, since S ∈ N x , S = x ( x − S ), we have that S is open in ( h S i , τ ).We will prove that { gx − V : V ∈ N ( S ) x } is a local base at g , for every g ∈ h S i .Indeed let U be an open set in ( h S i , τ ) and g ∈ U , then there is y ∈ h S i such that g ∈ yx − V ⊆ U . There are a, b, s, t ∈ S such that y = ab − and g = ts − , therefore ts − ∈ ab − x − V x ⊆ U , so that tbx ∈ asV x ⊆ bxsU ,hence there is W x open in S satisfying tbW x ⊆ asV x ⊆ bxsU , therefore g ( x − W x ) ⊆ U , this implies that { gx − V : V ∈ N Sx } is local base at g .Let us see that S is a subspace of ( h S i , τ ). Indeed, let U be an open set in S and let s ∈ U be, then xs ∈ xU , there is V ∈ N ( S ) x such that sV ⊆ xU , thisis equivalent to saying that s ( x − V ) ⊆ U , so that s ∈ ( s ( x − V )) ∩ S ⊆ U ,therefore U is open in the topology of subspace of S . Reciprocally, let U bean open set in ( h S i , τ ) and s ∈ U ∩ S , we can find W x ∈ N Sx and U s ∈ N ( S ) s ,such that s ∈ sx − W x ⊆ U , xU s ⊆ sW x ⊆ xU , then U s ⊆ U and therefore U s ⊆ U ∩ S , this proves that U ∩ S is open in S . From the fact that { gx − V : V ∈ N ( S ) x } is a local base at g , it follows that if S is 1-countable(locally compact), then ( h S i , τ ) is 1-contable (resp. locally compact) aswell.Let us suppose that S is a topological semigroup and let us prove that( h S i , τ ) is a paratopological group, it can be concluded if we prove that thecondition 2 of [12, Page 93] holds for B . Indeed, let V x ∈ N ( S ) x , since x ∈ xV x and the operation in S is jointly continuous, there exists W x ∈ N ( S ) x suchthat ( W x ) ⊆ xV x , then ( x − W x ) ⊆ V x and condition 2 holds, this provesthat ( h S i , τ ) is a paratopological group. Let us suppose that S es T and letus see ( h S i , τ ) is T , indeed let y, z ∈ h S i , z = y , then there are a, b, c, d ∈ S such that z = ab − and y = cd − , so that ad = bc , by fact that S is T , we canobtain V ad ∈ N ( S ) ad and V bc ∈ N ( S ) bc . Note that (( bd ) − V ad ) ∩ (( bd ) − V bc ) = ∅ ,also, ( bd ) − V ad ) ∈ N ( h S i ) z and ( bd ) − V bc ) ∈ N ( h S i ) y , that is to say ( h S i , τ )es T . Finally, if S is locally compact and T , ( h S i , τ ) is a semitopological roup locally compact and T , by Elii’s Theorem ( h S i , τ ) is a topologicalgroup. (cid:4) From the item ii ) of the Proposition 3.4 and the Proposition 3.3, we havethe following result. Corollary 3.5. If S is a cancellative commutative locally compact Hausdorffsemitopological semigroup with open shifts, then S has continuous division. Corollary 3.6.
Every cancellative commutative locally compact connectedHausdorff topological monoid with open shifts is a topological group.Proof.
Let S be a cancellative commutative locally compact connected Haus-dorff and let τ be the topology given in the Proposition 3.4. Let U ∈ N ( S ) e S ,since e S = e h S i and ( h S i , τ ) is a topological group, there is V ∈ N ( h S i ) e S satis-fying V − = V and V ⊆ U , then S n ∈ N V n ⊆ S n ∈ N U n . But S n ∈ N V n is anopen subgroup of ( h S i , τ ) and therefore is closed in S , the connectedness of S implies that S n ∈ N V n = S and S is a topological group. (cid:4) The following theorem tells us that every cancellative commutative locallycompact Hausdorff topological semigroup with open shifts can be embeddedas an open semigroup into the locally compact Hausdorff topological group, h S i ∗ . Theorem 3.7.
Let S be a cancellative, commutative topological semigroupwith open shifts. If S is Hausdorff and locally compact, then so is h S i ∗ .Moreover ι : S −→ ι ( S ) is an homeomorphism and ι ( S ) is open in h S i ∗ .Proof. Since S is locally compact and Hausdorff topological semigroup, sois S × S . By virtue the Proposition 3.2, π : S × S −→ h S i ∗ is open and h S i ∗ is Hausdorff, hence h S i ∗ is locally compact Hausdorff topological group.From Corollary 3.5 it follows that S has continuous division, therefore theProposition 3.2 guarantees that ι : S −→ ι ( S ) is a homeomorphism and ι ( S )is open in h S i ∗ . (cid:4) It is well known that every pseudocompact Tychonoff topological group canbe embedded as a subgroup dense into a compact topological group (see[8, Theorem 2.3.2]). The following theorem presents an analogue result incancellative commutative topological semigroups, where also of the pseudo-compactness, it is required the local compactness.
Theorem 3.8. If S is a cancellative commutative locally compact pseudo-compact Hausdorff topological semigroups with open shifts, then S is an opendense subsemigroup of h S i ∗ and h S i ∗ is a compact topological group.Proof. Since S is a locally compact pseudocompact space, [4, Theorem3.10.26] implies that S × S is pseudocompact. From the fact that h S i ∗ is a continuos image of S × S , we have that h S i ∗ is pseudompact, thereforethe Cˇ e ch-Stone compactification, β h S i ∗ , is a topological group containinga h S i ∗ as dense subgroup. [4, Theorem 3.3.9] guarantees that h S i ∗ is an pen subgroup of βS , therefore it is also closed. By the density of h S i ∗ , h S i ∗ = β h S i ∗ , that it to say, h S i ∗ is a compact topological group. Since h S i ∗ is compact, S is open in h S i ∗ and h S i ∗ = SS − , there are s , s ..., s n in S such that h S i ∗ = S ni =1 Ss − i . Cl h S i ∗ ( S ) is a compact Hausdorff cancella-tive semigroup, then by [1, Theorem 2.5.2], Cl h S i ∗ ( S ) is a topological group,therefore s − i ∈ Cl h S i ∗ ( S ) for every i ∈ { , , ..., n } , hence ( Cl h S i ∗ ( S )) s − i = Cl h S i ∗ ( S ) for every i ∈ { , , ..., n } . Since each shift in h S i ∗ is a homeomor-phism, we have that h S i ∗ = Cl h S i ∗ ( S ni =1 r s − i ( S )) = S ni =1 Cl h S i ∗ ( r s − i ( S )) = S ni =1 r s − i ( Cl h S i ∗ ( S )) = S ni =1 ( Cl h S i ∗ ( S )) s − i = S ni =1 Cl h S i ∗ ( S ) = Cl h S i ∗ ( S ),that it to say, S is dense in h S i ∗ . (cid:4) We obtain the following corollary.
Corollary 3.9.
The closure of any subsemigorup of a cancellative commu-tative locally compact pseudocompact Hausdorff topological semigroup withopen shifts can be embedded as a dense open subsemigroup into a compactHausdorff topological group.Proof.
Let S be a cancellative commutative locally compact pseudocom-pact Hausdorff topological semigroup with open shifts and let K be a sub-semigorup of S . By Theorem 3.8, S is an open subsemigroup of h S i ∗ , since Cl S ( K ) = Cl h S i ∗ ( K ) ∩ S , we have that Cl S ( K ) is open in Cl h S i ∗ ( K ). Now, K is dense in Cl h S i ∗ ( K ) and K ⊆ Cl S ( K ) ⊆ Cl h S i ∗ ( K ), this proves that Cl S ( K ) is dense en Cl h S i ∗ ( K ), but Cl h S i ∗ ( K ) is a compact Hausdorff can-cellative topological semigroup, so that it is a topological group by [1, The-orem 2.5.2], and so we have finished the proof. (cid:4) Proposition 3.10. If S is a cancellative commutative locally compact pseu-docompact Hausdorff topological semigroup with open shifts, then ( h S i ∗ , ι ) coincides with the SAP -compactification of S .Proof. Let S be a cancellative commutative locally compact pseudocompactHausdorff topological semigroup with open shifts and f : S −→ G a continu-ous homomorphism, being G a Hausdorff compact topological group. Since S is commutative, f ( S ) is a commutative subsemigroup of G , so that f ( S )is a compact commutative cancellative topological semigroup, which is atopological group by [1, Theorem 2.5.2]. Let us define f ∗ : h S i ∗ −→ f ( S ) by f ∗ ( xy − ) = f ( x )( f ( y )) − , f ∗ is a continuous homomorphism and moreover f ∗ ◦ ι = f , this ends the proof. (cid:4) It is known that each pseudocompact Tychonoff paratopological group is atopological group (see [14, Theorem 2.6]). The following theorem gives usa similar result in cancellative commutative topological monoids with openshifts, but instead of group structure we have required the first axiom ofcountability.
Theorem 3.11.
Let S be a cancellative commutative feebly compact topolog-ical monoid with open shifts satisfying the first axiom of countability. Then r ( S ) is a compact metrizable topological group. Moreover, the followingstatements hold: i) If S is T , S is a paratopological group. ii) If S is regular, S is a compact metrizable topological group.Proof. Let S be a commutative cancellative topological monoid with openshifts and put G = h S i . From Proposition 3.4 we have that there is a topo-logy τ , such that ( G, τ ) is a paratopological group containing S as an openmonoid. It follows from Proposition 2.1 that C r ( G ) is a regular paratopo-logical group containing C r ( S ), So [11, Corollary 5] implies that C r ( G ) isTychonoff, therefore so is C r ( S ). Since C r ( S ) is feebly compact and Ty-chonoff, it is pseudocompact. Then C r ( S ) is a pseuducompact subspace ofthe regular first-countable paratopological group C r ( G ), following [9, Corol-lary 4.18], we have that C r ( S ) is metrizable and compact. By Proposition2.1 i), C r ( S ) is cancelative, therefore [1, Theorem 2.5.2] implies that C r ( S )is a topological group. Now, if S is T , C r ( S ) = S sr , but S and S sr coin-cide algebraically, thus S is a paratopological group. If S is regular, then S = C r ( S ), this ends the proof. (cid:4) By [8, Example 2.7.10], there is a feebly compact Hausdorff 2-countableparatopological group that fails to be a compact topological group, thereforethe regularity in ii ) Theorem 3.11 cannot be weakened to the Hausdorffseparation property. Example 3.12.
Let ω be the first non countable ordinal, the space [0 , ω ) of ordinal numbers strictly less than ω with its order topology is regularfirst-countable feebly compact space, but [0 , ω ) is not compact. Then we cansee the importance of algebraic structure in the Theorem 3.11. CELLULARITY OF TOPOLOGICAL SEMIGROUPS
Finally, we present some results about the cellularity of topological semi-groups.
Theorem 4.1.
Let S be a cancellative, commutative Hausdorff locally com-pact σ -compact topological semigroup with open shifts. Then S has countablecellularity.Proof. Since S × S is locally compact and σ -compact, from Proposition3.2 we have that h S i ∗ is locally compact, Hausdorff, σ -compact topologicalgroup, which has countable celluarity following [17, Corollary 2.3]. Giventhat S is open in h S i ∗ , then c ( S ) = c ( h S i ∗ ) = ℵ . (cid:4) In the next corollary we give an analogous result to that of the proposition4.1, but without considering axioms of separation.
Corollary 4.2.
Every σ -compact locally compact cancellative commutativetopological monoid with open shifts has countable cellularity. roof. Let S be a σ -compact locally compact cancellative commutative topo-logical monoid with open shifts. By Proposition 2.1 ϕ ( S, C ) is open, therefore C ( S ) is locally compact Hausdorff topological semigroup with open shifts,this implies that C ( S ) is regular, so that C ( S ) = C r ( S ). Since C r ( S ) is can-cellative, we can apply the Theorem 4.1 and the Proposition 2.1 to concludethat c ( S ) = c ( C r ( S )) = ℵ . (cid:4) Corollary 4.3.
Every subsemigroup of a commutative cancellative locallycompact pseudocompact Hausdorff topological semigroup with open shifts hascountable cellularity.Proof.
Let S be a commutative cancellative locally compact pseudocompactHausdorff topological semigroup with open shifts and let K be a subsemi-group of S . By Corollary 3.9, there exists a compact Hausdorff topologicalgroup, G , containing Cl S ( K ) as an open semitopological group, therefore c ( K ) ≤ c ( Cl S ( K )) ≤ c ( G ) = ℵ . (cid:4) Since the compact topological groups has countable cellularity and c ( S ) = c ( C r ( S )) for every topological monoid with open shifts, the Theorem 3.11implies the following corollary. Corollary 4.4.
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