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RIGHT SIMPLE SUBSEMIGROUPS AND RIGHT SUBGROUPS OF COMPACT CONVERGENCE SEMIGROUPS
PHOEBE HO and SHING S. SO (Received 1 July 1999)
Abstract.Clifford and Preston (1961) showed several important characterizations of right groups. It was shown in Roy and So (1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the clo- sure of a right simple subsemigroup of a compact semigroup is always a right subgroup.
In this paper, it is shown that such results can be generalized in convergence semigroups.
In the discussion of maximal right simple subsemigroups and maximal right subgroups of semigroups, generalization of the results that no two maximal right simple subsemigroups and maximal right subgroups of a convergence semigroup intersect, is also established.
Keywords and phrases. Convergence semigroups, right simple semigroups, maximal right simple subsemigroups, left cancellative semigroups, right zero semigroups, right groups, maximal right subgroups.
2000 Mathematics Subject Classification. Primary 22A15.
1. Introduction.Discussion of convergence spaces, compactification of convergence spaces, and compact convergence semigroups can be found in [2, 3, 4, 5]; however, a brief summary of essential results will be repeated here.
Definition1.1. Aconvergence semigroupis a convergence spaceStogether with a continuous functionm:S×S→S such thatS is Hausdorff andmis associative.
The following notations are useful in the discussion of convergence semigroups:
(i) Fora,b∈S,ab=m(a,b).
(ii) ForA,B⊆S,AB=m(A×B)= {ab|a∈Aandb∈B}. In particular,A{b}will be denotedAb.
(iii) Ᏺ×Ᏻis the filter onS×Swith{F×G|F∈ᏲandG∈Ᏻ}as its base.
(iv) Ᏺ·Ᏻis the filter onS withm(Ᏺ×Ᏻ)as its base.
Lemma1.2. IfᏲandᏳare filters on a convergence semigroupS such thatᏲ→x andᏳ→y, thenᏲ·Ᏻ→xy.
Lemma1.3. IfS is a compact convergence semigroup, thenScontains an idempo- tent.
2. Main results. Let S be a semigroup. Then S is left cancellative provided that zx=zy impliesx=y for allx,y,z∈S; S is right simpleif it contains no proper right ideal oraS=S for alla∈S;Sis aright groupifS is both left cancellative and right simple;S is aright zero semigroupifxy=yfor allx,y∈S.
In [1], Clifford and Preston showed that a semigroupS is a right group if and only ifSis right simple and contains an idempotent.
Using this result and Lemma 1.3, the next four results in compact convergence semi- groups can be obtained in exactly the same way as in the topological setting.
Theorem2.1. LetSbe a compact convergence semigroup. Then the following state- ments are equivalent.
(i) Sis right simple.
(ii) Sis a right group.
(iii) Sis left cancellative.
Corollary2.2. Every compact convergence simple semigroup is a group.
Corollary2.3. Every compact convergence cancellative semigroup is a group.
Corollary2.4. Every closed right simple or closed left cancellative subsemigroup of a compact convergence semigroup is a right group.
The example in [6] indicates that right subgroups of compact topological semi- groups are closely related to their right simple subsemigroups, but not left cancella- tive subsemigroups. Thus the following discussion focuses only on the relationship between right simple subsemigroups and right subgroups of compact convergence semigroups.
In [6], it is shown that the closure of a right simple subgroup of a compact topological semigroup is always a right group. The next two theorems show that similar results can be obtained in compact convergence semigroups.
Theorem2.5. If S is a compact convergence semigroup andR is a right simple subsemigroup. ThenClSR, the closure ofR, is also a right simple subsemigroup ofS.
Proof. Leta,b∈ClSR. There exist filtersᏲandᏳsuch thatR∈Ᏺ∩Ᏻ,Ᏺ→a, and Ᏻ→b.
SinceRis right simple, forF∈ᏲandG∈Ᏻ, letXFG= {x∈R:g=xf , f∈F, g∈G}
and letχbe the filter withᏮas base whereᏮ= {XFG:F∈Ᏺ, G∈Ᏻ}. Thenχ·Ᏺis the filter onSwithm(χ×Ᏺ)as its base.
SinceSis compact, there exists an ultrafilterᐅ≥χsuch thatᐅ→ywherey∈ClSR.
Thusχ·Ᏺ≤ᐅ·Ᏺandᐅ·Ᏺ→ya. On the other hand, for eachF∈Ᏺ, G⊂XFG·F for allG∈Ᏻ. It follows thatXFG·F∈Ᏻandχ·Ᏺ≤Ᏻ. Letᐁbe an ultrafilter containing χ·Ᏺ. Thenᐁ→yaandᐁ→b. Thusb=yaand it follows that ClSRis a right simple subsemigroup ofS.
Definition2.6. LetRbe a right simple subsemigroup of a semigroupS. ThenR is called amaximal right simple subsemigroupofSif and only ifR=Sand no proper right simple subsemigroup ofS properly containingR.
Definition2.7. LetRbe a right subgroup of a semigroupS. ThenRis called a maximal right subgroupofS if and only ifR=S and no proper right subgroup ofS properly containingR.
SupposeS is a compact convergence semigroup andR is a right simple subsemi- group ofS. Let={T|R⊂T ,T is a proper right simple subsemigroup ofS}. Partially
orderby set inclusion. By the Hausdorff maximal principle, there is a maximal chain Ꮿof. LetM= ∪Ꮿ.
Letx,y∈M. Thenx∈Tandy∈T∗for someT ,T∗∈Ꮿ. LetT=max{T ,T∗}. Then x,y∈Tand Tbeing right simple impliesy∈xT⊂xM. Therefore,M is a right simple subsemigroup ofS.
SupposeM∗ is a proper right simple subsemigroup ofsuch thatM⊂M∗. Then M∗∈ᏯsoᏯ⊂Ꮿ∪ᏹ∗, which contracts the fact thatᏯthe maximal chain of. There- fore,Mis the maximal right simple subsemigroup ofS containingR. SinceSis com- pact, by Theorems 2.1 and 2.5,Mis a compact maximal right subgroup ofScontaining R. Therefore, the following theorem is proved.
Theorem2.8. IfRis a right simple subsemigroup of a compact convergence semi- groupS, then eitherS is a right group or Ris contained in a unique maximal right subgroupMofS such thatMis compact.
Similarly, the following corollaries concerning convergence semigroups can be ob- tained.
Corollary2.9. Every right simple subsemigroupRof a compact convergence semi- groupS, withR=S, is contained in a unique maximal right subsemigroupMsuch that Mis closed.
Corollary2.10. Every right subgroupRof a compact convergence semigroupS, withR=S, is contained in a unique maximal right subgroupMsuch thatMis closed.
Clifford and Preston [1] showed that a semigroupS is a right group if and only ifS is isomorphic to the direct product ofG×EwhereGis a group andEis a right zero semigroup, denoted bySG×E. In fact,Eis the set of all idempotent ofSandG=Se for somee∈E. This result suggests a different way of analyzing compact right simple convergence semigroups.
LetSbe a compact right simple or left cancellative convergence semigroup. It follows from Theorem 2.1 thatS is a right group. SinceS is compact andG=Sg for some g∈E,Gis compact. SinceSis a right group,ef=ffore,f∈E. ThusEis a right zero semigroup. SinceEis a closed subset ofS, Eis compact.
LetZbe a right zero subsemigroup of a compact semigroupS. By Theorem 2.5, ClSZ is a subsemigroup ofS.
Letx,y∈ClSZ. Then there exist filtersᏲ andᏳsuch thatZ∈Ᏺ∩Ᏻ,Ᏺ→x, and Ᏻ→y.
ConsiderᏴ= {Z∩Ᏺ:Ᏺ∈Ᏺ} and = {Z∩Ᏻ:Ᏻ∈Ᏻ}. ThenᏴ and are filter bases of some filterᏴ∗ and∗, respectively. Note thatᏴ∗and∗containᏲandᏳ, respectively. ThusᏲ·Ᏻ≤Ᏼ∗·∗→xy.
On the other hand, forH∈Ᏼ∗andK∈∗, there existsF∈Ᏺsuch that(Z∩F)(Z∩ G)=Z∩F⊂HK. ThusᏴ∗·∗≤ᏲandᏲ→xy. It follows fromxy=ythat ClSZis a right zero subsemigroups.
The next two lemmas follow from the above discussion.
Lemma2.11. LetSbe a compact right simple or left cancellative convergence semi- group. ThenS G×E whereG is a compact group andE is a compact right zero semigroup.
Lemma2.12. LetZbe a right zero subsemigroup of a compact convergence semi- groupS. ThenClSZis also a right zero subsemigroup ofS.
Using Hausdorff’s maximal principle and Lemma 2.12, the following lemma can be easily obtained.
Lemma2.13. LetS be a compact convergence semigroup and Z be a right zero subsemigroup ofS. ThenZis contained in a maximal right zero subsemigroup ofS.
The next theorem can be proved in the same way as in the topological setting.
Theorem2.14. LetS be a compact convergence semigroup andRbe a right sub- group of S such thatRGR×ER. Then there existGM, the maximal subgroup of S containingGR, andEM, the maximal right zero subgroup ofScontainingER, such that GM×EM is isomorphic to a maximal right subgroupMofScontainingR.
Proof. SinceRGR×ER, there exists a unique maximal subgroupGM ofS con- tainingGR and a unique maximal right zero subsemigroupEM of S containing ER
by Lemma 2.13. LetM be the isomorphic image ofGM×EM inS. ThenM is a right subgroup ofS.
SupposeM∗is a maximal right subgroup containingR. ThenM∗G∗M×EM∗. By the maximality ofM∗,M⊆M∗. SinceR⊂M∗,GR⊂GM∗andER⊂EM∗. By the maximality ofGM andEM, GM∗⊆GM andEM∗⊆EM. By Lemma 2.16,M∗⊆MsoM∗=M.
It is a well-known result that no two maximal subgroups of a semigroup intersect and the following are its generalizations. In Theorem 2.15, we generalize it for max- imal right subgroups and the proof can be found in [6]. Theorem 2.17is a partial generalization of Theorem 2.15 in commutative semigroups.
Theorem2.15. No two maximal right subgroups of a semigroup intersect.
Lemma2.16. IfM1andM2are distinct maximal right simple subsemigroups ofS such thatM1M2=M2M1, then eitherM1∩M2= ∅orM1·M2=S.
Proof. Suppose thatM1∩M2= ∅andM1·M2=S. Leta,b∈M1·M2. Thenab∈ M1·M2sinceM1M2=M2M1. ThusM1M2is a subsemigroup ofS.
In fact, the following argument shows thatM1·M2is a right simple subsemigroup ofS containing bothM1andM2. Fora,b∈M1·M2,a=a1a2andb=b1b2for some a1,b1∈M1,a2,b2∈M2. Then
a=a1a2
= b1b1∗
a2for someb1∗∈M1sinceM1is simple
=b1 a∗2b1∗∗
for someb1∗∗∈M1anda∗2∈M2sinceM1M2=M2M1
=b1 b2a∗∗2
b∗∗1 for somea∗∗2 ∈M2sinceM2is simple
= b1b2
mwherem=a∗∗2 b∗∗1 ∈M2M1=M1M2
=bmfor somem∈M1·M2.
(2.1)
Therefore,M1·M2is right simple. SinceM1∩M2= ∅, andM1,M2are distinct,M1⊂ M1·M2. By the maximality ofM1and the fact thatM1=M2,M1·M2=S, which is a contradiction.
The following theorem concerning maximal simple subsemigroups of commutative semigroups follows immediately from Lemma 2.16.
Theorem2.17. No two maximal simple subsemigroups of a commutative semi- group intersect.
Theorem2.18. LetᏰbe a partition of a maximal right simple subsemigroupMof a compact convergence semigroupS such that Dis a monoid for eachD∈Ᏸ. Then D=Mefor somee∈E(M)for eachD∈Ᏸ.
Proof. SinceMis a maximal right simple subsemigroup ofSandSis compact,M is a maximal right subgroup ofSandMcan be written as union of the decomposition {Me:e∈E(M)}. For eachD∈Ᏸ, leteDbe the identity ofD. TheneD∈Mefor some e∈E(M). In fact,e=eD·e=e·eD=eDsinceeis the identity ofMe. It follows that D⊂MeD.
SupposeMeD⊂D. Then there existsD∗∈Ᏸsuch thatD∗∩(MeD−D)= ∅. LeteD∗
be the identity ofD∗. Then it follows from the discussion aboveD∗⊂MeD∗ which impliesD∗∩(MeD−D)⊂D∗∩MeD ⊂MeD∗∩MeD. This contradicts the fact that {Me:e∈E(M)}is a decomposition ofM. ThereforeD=MeD=Mefor eachD∈Ᏸ.
Definition2.19. LetRbe a right group. For each idempotenteofR, letfe:R→R be defined byfe(x)=(ex)−1. ThenRis called aconvergence right group if and only iffeis continuous for every idempotenteinR.
Theorem2.20. LetS be a compact pseudotopological semigroup.
(i) IfS is right simple, thenSis a right convergence group.
(ii) Either every maximal right simple subsemigroup ofSis closed and hence a con- vergence right subgroup ofS orSitself is a convergence right group.
Proof. (i) By Theorem 2.1,S is a right group. For each idempotenteofS, letfe: S→Sbe defined byfe(x)=(ex)−1and letᏲbe a filter such thatᏲ→x. Thenfe(Ᏺ) is a filter inSe. Letᐁbe an ultrafilter such thatᐁ≥fe(Ᏺ). SinceSeis compact,ᐁ→y for someyinSe. Since for everyU∈ᐁ,U∩f (Ᏺ)= ∅for allF∈Ᏺ. Thene∈UF for allU∈ᐁandF∈Ᏺwhereeis the identity of the groupSe. NowᐁᏲ→y,ᐁᏲ=e, and˙
˙
e→eimplyyx=e. Thusy=(ex)−1. SinceS is pseudotopological,f (Ᏺ)→(ex)−1. Thus the result follows.
(ii) SupposeMis a maximal right simple subsemigroup ofS such thatM=ClSM.
Then ClsM =S by the maximality ofM and Theorem 2.5. The result follows from Theorem 2.1 and part (i) of this theorem.
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Ho: Central Missouri State University, Warrensburg, Missouri, USA So: Central Missouri State University, Warrensburg, Missouri, USA E-mail address:[email protected]
