Right simple subsemigroups and right subgroups of compact convergence semigroups

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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.

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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 subsemisemi-groups, 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.

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order᏿by set inclusion. By the Hausdorﬀ maximal principle, there is a maximal chain

Ꮿof᏿. LetM= ∪Ꮿ.

Letx,y∈M. Thenx∈Tandy∈T∗_{for some}_{T ,T}∗_∈_Ꮿ_{. Let}_T₌_max{_{T ,T}∗_{}. Then} x,y∈T_and _T_{being right simple implies}_y_∈_xT_⊂_xM_{. Therefore,}_M _{is a right} simple subsemigroup ofS.

SupposeM∗ _{is a proper right simple subsemigroup of}_᏿_{such that}_M_⊂_M∗_{. Then} M∗_∈_Ꮿ_so_Ꮿ_⊂_Ꮿ_∪_ᏹ∗_{, which contracts the fact that}_Ꮿ_{the 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 that}_Ᏼ∗_and_᏷∗_contain_Ᏺ_and_Ᏻ_, respectively. ThusᏲ·Ᏻ≤Ᏼ∗_·_᏷∗_→_xy_.

On the other hand, forH∈Ᏼ∗_and_K_∈_᏷∗_{, there exists}_F_∈_Ᏺ_{such that}_(Z_∩_F)(Z_∩ G)=Z∩F⊂HK. ThusᏴ∗_·_᏷∗_≤_Ᏺ_and_Ᏺ_→_xy_{. It follows from}_xy₌_y_{that Cl}_S_Z_is a right zero subsemigroups.

The next two lemmas follow from the above discussion.

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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 containing}_R_{. Then}_M∗_G∗

M×EM∗. By the

maximality ofM∗_,_M_⊆_M∗_{. Since}_R_⊂_M∗_,_G_R_⊂_G_M_∗_and_E_R_⊂_E_M_∗_{. By the maximality} ofGM andEM, GM∗⊆G_M andE_M∗⊆E_M. 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 =b1a∗2b1∗∗

for someb∗∗

1 ∈M1anda∗2∈M2sinceM1M2=M2M1 =b1b2a∗∗2 b1∗∗for somea∗∗2 ∈M2sinceM2is simple

=b1b2mwherem=a2∗∗b∗∗1 ∈M2M1=M1M2 =bmfor somem∈M1·M2.

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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 above}_D∗_⊂_Me

D∗ which impliesD∗_∩_(Me_D₋_D)_⊂_D∗_∩_Me_D _⊂_Me_D_∗_∩_Me_D_{. 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_{. Since}_S _{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.

References

[1] A. H. Cliﬀord and G. B. Preston,The Algebraic Theory of Semigroups. Vol. I, American Mathematical Society, Providence, R.I., 1961. MR 24#A2627. Zbl 111.03403. [2] P. Ho, P. Plummer, and S. So,Some results on compact convergence semigroups deﬁned

by ﬁlters, Internat. J. Math. Math. Sci.21(1998), no. 1, 153–158. CMP 1 486 971. Zbl 888.54038.

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[4] D. C. Kent and G. D. Richardson,Compactiﬁcations of convergence spaces, Internat. J. Math. Math. Sci.2(1979), no. 3, 345–368. MR 80i:54004. Zbl 418.54005.

[5] G. D. Richardson and D. C. Kent,The star compactiﬁcation, Internat. J. Math. Math. Sci.4

(1981), no. 3, 451–472. MR 82g:54038. Zbl 479.54002.

[6] K. Roy and S. S. So,Right simple subsemigroups and right subgroups of a compact semi-group, J. Inst. Math. Comput. Sci. Math. Ser.11(1998), no. 2, 121–125. MR 99i:22008. Zbl 990.21715.

Ho: Central Missouri State University, Warrensburg, Missouri, USA