The Translational Hull of Strongly Inverse Wrpp Semigroups

QIU Shu-ming

(Elementary Educational College,Jiangxi Normal University,Nanchang 330022,China)

The Translational Hull of Strongly Inverse Wrpp Semigroups

QIU Shu-ming

(Elementary Educational College,Jiangxi Normal University,Nanchang 330022,China)

In this paper,we obtain some characterizations of the translational hull of strongly inverse wrpp semigroups.And we prove that the translational hull of a strongly inverse wrpp semigroup is still of the same type.

translational hull;wrpp semigroups;strongly inverse wrpp semigroups

§1.Introduction

We call a mapping λ from a semigroup S into itself a left translation of S if λ(ab)=(λa)b for all a,b∈S.Similarly,we call a mapping ρ from S into itself a right translation of S if (ab)ρ=a(bρ)for all a,b∈S.A left translation λ and a right translation ρ of S are said to be linked if a(λb)=(aρ)b for all a,b∈S.In this case,we call the pair(λ,ρ)a bitranslation of S.The set Λ(S)of all left translations(and also the set P(S)of all right translations)of the semigroup S forms a semigroup under the composition of mappings.By the translational hull of S,we mean a subsemigroup Ω(S)consisting of all bitranslations(λ,ρ)of S in the direct product Λ(S)×P(S).The concept of translational hull of semigroups and rings was first introduced by Petrich in 1970(see[9]).Moreover,the translational hull of a semigroup plays an important role in the general theory of semigroups.The translational hull of an inverse semigroup was first studied by Ault in 1973.Later on,Fountain and Lawson[2]further studied the translational hull of adequate semigroups.Guo and Y Q Guo[3]researched the translational hull of a stronglyright type-A semigroup.Recently,Guo and Shum[4]investigated the translational hull of a type-A semigroup.In particular,the result obtained by Ault[1]was substantially generalized and extended.

Tang[11]introduced Green’s∗∗-relations L∗∗and R∗∗,which are common generalizations of usual Green’s relations L,R and Green’s∗-relations L∗,R∗.Recently,Liu and Guo[8]researched the natural partial orders on wrpp semigroups.A semigroup S is called a w rpp semigroup if for any a∈S,we have

(i)Each L∗∗-class of S contains at least one idempotent of S;

According to Hu[5],a[wlpp;wrpp]wpp semigroup S is called an inverse[wlpp;wrpp]wpp semigroup if the set of idempotents E(S)is a semilattice under the multiplication of S.Similar to the definition of strongly rpp semigroups,a wrpp semigroup S is called a strongly wrpp semigroup if for any a∈S,there exists a unique idempotent e such that aL∗∗e and a=ea. Thus,we naturally call an inverse wrpp semigroup S a strongly inverse wrpp semigroup if S is a strongly wrpp semigroup.

In this paper,we obtain some characterizations of the translational hull of strongly inverse wrpp semigroups.Furthermore,we show that the translational hull of a strongly inverse wrpp semigroup is still the same type.It seems that it is similar to[6],but factually our results are a generalization of Hu.Meanwhile,we easily obtain that the translational hull of a strongly right adequate semigroup is still of the same type[10].

§2.Preliminaries

Throughout this paper we use the notations and terminologies of Tang[11]and Howie[7].It is easy to see that each L∗∗-class contains precisely an idempotent in an inverse wrpp semigroup. For convenience,we shall denote by a∗the typical idempotent in the L∗∗-class of S containing a.

For a semigroup S,we define the relations on S as follows

For all a,b∈S,

Lemma 2.1[11](1)L∗∗is a right congruence and L⊆L∗⊆L∗∗;

(2)R∗∗is a left congruence and R⊆R∗⊆R∗∗.

By the definition of strongly inverse wrpp semigroups,we can easily obtain the following results.

Lemma 2.2If S is a strongly inverse wrpp semigroup,then

(1)Every L∗∗-class of S containing a has an unique idempotent a∗of S and aa∗=a=a∗a;

(2)For any a,b∈S,aL∗∗b if and only if a∗=b∗.

Lemma 2.3Let S be a strongly inverse wrpp semigroup.If λ,λ′(ρ,ρ′)are left(right) translations of S whose restrictions to the set of idempotents of S are equal,then λ=λ′(ρ=ρ′).

ProofLet a be an element of S and e be an idempotent in the L∗∗-class of a.Then ae=a=ea(since S is strongly)and so

Hence,ρ=ρ′.Similarly,we have λ=λ′.

Lemma 2.4Let S be a strongly inverse wrpp semigroup and(λi,ρi)∈Ω(S)with i=1,2. Then the following statements are equivalent

(1)(λ1,ρ1)=(λ2,ρ2);

(2)ρ1=ρ2;

(3)λ1=λ2.

ProofNote that(1)⇔(2)is dual to(1)⇔(3),it suffices to verify(1)⇔(2).Since (1)⇒(2)is clear,we need only to show that(2)⇒(1).Let ρ1=ρ2.For all e∈E(S),we have eρ1=eρ2.Therefore,

Similarly,λ2e=(λ2e)∗(λ1e).Thus,λ1e=(λ1e)∗(λ2e)∗(λ1e).Since S is an inverse wrpp semigroup,

And because S is strongly,so(λ2e)∗=(λ1e)∗.Fatherly,

Consequently,λ1=λ2.

Lemma 2.5Let a,b be elements of a strongly inverse wrpp semigroup S.Then the following conditions hold in S

(i)(ab)∗=(a∗b)∗;

(ii)(ae)∗=a∗e,for all e∈E(S).

Proof(i)Since L∗∗is a right congruence on S,abL∗∗a∗b.By Lemma 2.2,we have (ab)∗=(a∗b)∗.

(ii)We have(ae)∗=(a∗e)∗from(i),thus(ae)∗=a∗e.

§3.Main Results

Throughout this section,we always use S to denote a strongly inverse wrpp semigroup with a semilattice of idempotents E.Let(λ,ρ)∈Ω(S).Then we define the mappings λ∗and ρ∗which map S into itself by

for all a∈S.

For the mappings λ∗and ρ∗,we have the following lemmas.

Lemma 3.1For any e∈E,we have

(i)λ∗e=eρ∗∈E;

(ii)λ∗e=(λe)∗.

Proof(i)Since the set of all idempotents E of the semigroup S forms a semilattice,all idempotents of S commute.Hence,λ∗e=(λe)∗e=e(λe)∗=eρ∗.It is a routine calculation that λ∗e,eρ∗∈E by(ii).

(ii)Since L∗∗is a right congruence on S,we see that λ∗e=(λe)∗eL∗∗λe·e=λe.Now,by Lemma 2.2,we have λ∗e=(λe)∗,as required.

Lemma 3.2(λ∗,ρ∗)∈Ω(S).

ProofWe first show that λ∗is a left translation of S.For any a,b∈S,by Lemma 2.2 and Lemma 2.5,we have

We now proceed to show that ρ∗is a right translation of S.For any a,b∈S,we first observe that ab=(ab)·b∗,so(ab)∗=(ab)∗b∗by Lemma 2.5.Now,we have

In fact,the pair(λ∗,ρ∗)is clearly linked,because we have

for all a,b∈S.Thereby,the pair(λ∗,ρ∗)is an element of the translational hull Ω(S)of S.

Lemma 3.3Let S be a strongly inverse wrpp semigroup and(λ,ρ)be an element of Ω(S). Then(λ,ρ)=(λ,ρ)(λ∗,ρ∗)=(λ∗,ρ∗)(λ,ρ).

ProofFor all e∈E(S),we have

This implies that λλ∗=λ by Lemma 2.3.By Lemma 3.2,we know that(λ∗,ρ∗)∈Ω(S). Hence(λ,ρ)(λ∗,ρ∗)=(λλ∗,ρρ∗).And by Lemma 2.4,we have ρρ∗=ρ.This shows that (λ,ρ)=(λ,ρ)(λ∗,ρ∗).On the other hand,by Lemma 3.1 and λλ∗=λ,we have

that is,λ∗λ=λ.And again by Lemma 2.4,we have(λ,ρ)=(λ∗,ρ∗)(λ,ρ).

Lemma 3.4(λ∗,ρ∗)∈E(Ω(S)).

ProofFor all e∈E(S),in view of the fact that E(S)is a semilattice,by Lemma 3.1,we get

which implies that(ρ∗)2=ρ∗by Lemma 2.3.It then follows from Lemma 2.4 that(λ∗,ρ∗)= (λ∗,ρ∗)2,that is(λ∗,ρ∗)∈E(Ω(S)).

Lemma 3.5Let S be a strongly inverse wrpp semigroup and(λ,ρ)∈Ω(S),then(λ,ρ)L∗∗(λ∗,ρ∗).

ProofIt suffices to show that

for all(λi,ρi)∈Ω(S)with i=1,2.Since R is a left congruence and by Lemma 3.4,we can know that if(λ∗,ρ∗)(λ1,ρ1)R(λ∗,ρ∗)(λ2,ρ2),then

On the other hand,we need to show that

Let(λ,ρ)(λ1,ρ1)R(λ,ρ)(λ2,ρ2)for(λi,ρi)∈Ω(S)with i=1,2.Then there exist(λ3,ρ3),(λ4, ρ4)∈Ω(S)such that(λ,ρ)(λ1,ρ1)=(λ,ρ)(λ2,ρ2)(λ3,ρ3)and(λ,ρ)(λ2,ρ2)=(λ,ρ)(λ1,ρ1)(λ4, ρ4).Thus,we have ρρ1=ρρ2ρ3and ρρ2=ρρ1ρ4.We easily know that fρρ1=fρρ2ρ3for all f∈E(S),which implies that

So(fρ)∗ρ1R(fρ)∗ρ2ρ3.Since f is arbitrary,we regard(λe)∗as f,that is(fρ)∗=((λe)∗ρ)∗for any e∈E(S).Thereby we can obtain that((λe)∗ρ)∗ρ1=((λe)∗ρ)∗ρ2ρ3x for some x∈S1. Noticing that((λe)∗ρ)e=(λe)∗(λe)=λe,we have((λe)∗ρ)∗e=(λe)∗by Lemma 2.5(2). Thereby we get where ρxis the inner right translation on S determined by x and by Lemma 2.3,ρ∗ρ1= ρ∗ρ2ρ3ρx.Similarly,we have ρ∗ρ2=ρ∗ρ1ρ4ρyfor some y∈S1.Thus,by Lemma 2.4,we can obtain that

This completes the proof of the equation(B).Consequently,the equation(A)is proved.

By Lemma 3.3,Lemma 3.4 and Lemma 3.5,we can obtain the following corollary

Corollary 3.6The translational hull of a strongly inverse wrpp semigroup is a wrpp semigroup.

Lemma 3.7Let Φ(S)={(λ,ρ)∈Ω(S)|λE∪Eρ⊆E},then E(Ω(S))=Φ(S).

ProofLet(λ,ρ)∈Φ(S),then for all e∈E,eρ∈E.Since E is a semilattice,

And by Lemma 2.3,ρ2=ρ.Similarly,λ2=λ.By Lemma 2.4,we have(λ,ρ)=(λ,ρ)2,that is,(λ,ρ)∈E(Ω(S)).

Conversely,if(λ,ρ)∈E(Ω(S))and(λ,ρ)L∗∗(λ∗,ρ∗)by Lemma 3.5,then(λ∗,ρ∗)=(λ,ρ)(λ∗, ρ∗).Again by Lemma 3.3,we get(λ,ρ)=(λ,ρ)(λ∗,ρ∗).Therefore(λ,ρ)=(λ∗,ρ∗),that is, λ∗=λ and ρ∗=ρ.By Lemma 3.1 λE∪Eρ⊆E.Consequently,E(Ω(S))=Φ(S).

Lemma 3.8The elements of E(Ω(S))commute with each other.

ProofLet(λ1,ρ1),(λ2,ρ2)∈E(Ω(S)).By Lemma 3.7,for any e∈E(S),we have

This fact implies that ρ1ρ2=ρ2ρ1.By Lemma 2.4,we have(λ1,ρ1)(λ2,ρ2)=(λ2,ρ2)(λ1,ρ1).

By using the above Lemma 3.2→Lemma 3.5,Lemma 3.7 and Lemma 3.8,we can easily verify that for any(λ,ρ)∈Ω(S)there exists a unique idempotent(λ∗,ρ∗)such that(λ,ρ)L∗∗(λ∗,ρ∗) and(λ,ρ)(λ∗,ρ∗)=(λ,ρ)=(λ∗,ρ∗)(λ,ρ).Thus,Ω(S)is indeed a strongly wrpp semigroup. Again by Lemma 3.8 and the definition of a strongly inverse wrpp semigroup,we can obtain our main theorem:

Themorem 3.9The translational hull of a strongly inverse wrpp semigroup is still a strongly inverse wrpp semigroup.

X M Ren and K P Shum[10]introduced the definition of strongly right adequate semigroups. Since a strongly right adequate semigroup is a strongly inverse wrpp semigroup,we have the following corollary.

Corollary 3.10[10]The translational hull of a strongly right adequate semigroup is still of the same type.

[1]AULT J E.The translational hull of an inverse semigroup[J].Glasgow Math,1973,14:56-64.

[2]FOUNTAIN J B,LAWSON M.The translational hull of an adequate semigroup[J].Semigroup Forum,1985, 32:79-86.

[3]GUO Xiao-jiang,GUO Yu-qi.The translational hull of a strongly right type a semigroup[J].Science in China(series A),2000,43:6-12.

[4]GUO Xiao-jiang,SHUM K P.On the translational hulls of type a semigroups[J].J Algebra,2003,269(1): 240-249.

[5]HU Zhi-bin,GUO Xiao-jiang.Inverse wpp semigroups[J].Chin Quart J of Math,2011,26(4):521-525.

[6]HU Zhi-bin,GUO Xiao-jiang.The translational hulls of inverse wpp semigroups[J].International Mathematical Forum,2009,28(4):1397-1404.

[7]HOWIE J M.An Introduction to Semigroup Theory[M].London:Academic Press,1976.

[8]LIU Hai-jun,GUO Xiao-jiang.The natural partial orders on wrpp semigroups[J].Chin Quart J of Math, 2012,27(1):139-144.

[9]PETRICH M.The translational hull in semigroups and rings[J].Semigroup Forum,1970,1:283-360.

[10]REN Xue-min,SHUM K P.The translational hull of a strongly right or left adequate semigroup[J].Vietnam Journal of Mathematics,2006,34(4):441-447.

[11]TANG Xiang-dong.On a theorem of C-wrpp semigroups[J].Commun Algebra,1997,25:1499-1504.

tion:20M10

:A

1002–0462(2017)01–0059–07

date:2015-11-09

Supported by the National Natural Science Foundation of China(11361027);Supported by the Science Foundation of Education Department of Jiangxi Province(GJJ11388);Supported by the Youth Growth Fund of Jiangxi Normal University

Biography:Qiu Shu-ming(1983-),male,native of Nanchang,Jiangxi,M.S.D.,a lecturer of Jiangxi Normal University,engages in mathematics theory.

CLC number:O152.7