IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation57606110.1155/2008/576061576061Research ArticleA Note on Locally Inverse Semigroup AlgebrasGuoXiaojiang1GoichotFrancois1Department of MathematicsJiangxi Normal UniversityNanchangJiangxi 330022China050220082008912200729120082008Copyright © 2008This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let R be a commutative ring and S a finite locally inverse semigroup. It is proved that the semigroup algebra R[S] is isomorphic to the direct product of Munn algebras (R[GJ],mJ,nJ;PJ) with JS/𝒥, where mJ is the number of -classes in J, nJ the number of -classes in J, and GJ a maximum subgroup of J. As applications, we obtain the sufficient and necessary conditions for the semigroup algebra of a finite locally inverse semigroup to be semisimple.

1. Main Results

A regular semigroup S is called a locally inverse semigroup if for all idempotent eS, the local submonoid eSe is an inverse semigroup under the multiplication of S. Inverse semigroups are locally inverse semigroups. Inverse semigroup algebras are a class of semigroup algebras which is widely investigated. One of fundamentally important results is that a finite inverse semigroup algebra is the direct product of full matrix algebras over group algebras of the maximum subgroups of this finite inverse semigroup. Consider that all local submonoids of a locally inverse semigroup are inverse semigroups, it is a very natural problem whether a finite locally inverse semigroup algebra has a similar representation to inverse semigroup algebras. This is the main topic of this note.

Let 𝒜 be an R-algebra. Let m and n be positive integers, and let P be a fixed n×m matrix over 𝒜. Let :=(𝒜;m,n;P) be the vector space of all m×n matrices over 𝒜. Define a product in by AB=APB(A,B), where APB is the usual matrix product of A, P, and B. Then is an algebra over R. Following , we call the Munn m×n  matrix algebra over  𝒜  with sandwich matrix  P.

By a semisimple semigroup, we mean a semigroup each of whose principal factor is either a completely 0-simple semigroup or a completely simple semigroup. It is well known that a finite regular semigroup is semisimple. The Rees theorem tells us that any completely 0-simple semigroup (completely simple semigroup) is isomorphic to some Rees matrix semigroup 0(G,I,Λ;P) ((G,I,Λ;P)), and vice versa (for Rees matrix semigroups, refer to ). In what follows, by the phrase “Let S=JS/𝒥0(GJ;IJ,ΛJ;PJ) be a finite regular semigroup,” we mean that S is a finite regular semigroup in which the principal factor of S determined by the 𝒥-class J is isomorphic to the Rees matrix semigroup 0(GJ;IJ,ΛJ;PJ) or (GJ;IJ,ΛJ;PJ) for any JS/𝒥.

The following is the main result of this paper.

Theorem 1.1.

Let S=JS/𝒥0(GJ,IJ,ΛJ;PJ) be a finite locally inverse semigroup. Then the semigroup algebra R[S] is isomorphic to the direct product of (R[GJ];|IJ|,|ΛJ|;PJ) with JS/𝒥.

Based on Theorem 1.1 and [1, Lemma 5.17, page 162, and Lemma 5.18, page 163], the following corollary is straightforward.

Corollary 1.2.

Let S=JS/𝒥0(GJ,IJ,ΛJ;PJ) be a finite locally inverse semigroup. Then the semigroup algebra R[S] has an identity if and only if |IJ|=|ΛJ| and PJ is invertible in the full matrix algebra M|IJ|(R[GJ]) for all JS/𝒥.

Reference [1, Lemma 5.18, page 163] told us that (R[GJ],mJ,nJ;PJ) is isomorphic to the full matrix algebra MnJ(R[GJ]) if (R[GJ],mJ,nJ;PJ) has an identity. Now, we have the following.

Corollary 1.3.

Let S=JS/𝒥0(GJ,IJ,ΛJ;PJ) be a finite locally inverse semigroup. If R[S] has an identity, then R[S] is isomorphic to the direct product of the full matrix algebras M|IJ|(R[GJ]) with JS/𝒥.

The following corollary is a consequence of Corollary 1.3.

Corollary 1.4.

Let S=JS/𝒥0(GJ,IJ,ΛJ;PJ) be a finite locally inverse semigroup. Then the semigroup algebra R[S] is semisimple if and only if for all JS/𝒥,

|IJ|=|ΛJ|;

PJ is invertible in the full matrix algebra M|IJ|(R[GJ]);

R[GJ] is semisimple.

2. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>

For our purpose, we have the Möbius inversion theorem .

Lemma 2.1.

Let (P,) be a locally finite partially ordered set (i.e., intervals are finite) in which each principal ideal has a maximum and G be an Abelian group. Suppose that f:PG is a function and define g:PG by g(x)=yxf(y). Then f(x)=yxg(y)μ(x,y), where μ is a Möbius function.

Now assume that S is a regular semigroup and a,bS. Define abthereexiste,fE(S)suchthata=eb=bf. Then is a partial order on S. Following , we call the natural partial order on S. Equivalently, ab if and only if for every (for some) fE(Rb)(fE(Lb)), there exists eE(Ra)(eE(La)) such that ef and a=eb(a=be). Moreover, Nambooripad [3, 4] proved that S is a locally inverse semigroup if and only if the natural partial order is compatible with respect to the multiplication of S.

Lemma 2.2.

Let S be a locally inverse semigroup and a,bS. Then for any uab, there exist xa and yb such that u=xy, xRu and yLu.

Proof.

For any eE(Ra), we have ea=a and eab=ab. Let z be an inverse of ab. Clearly, abzE(Rab). Note that eabz=abz. It is easy to check that abzeE(S),abzee, and abzabze. Hence abzeab and there exists gE(S) such that u=gab and gabze(e). Thus gaa. On the other hand, since is a left congruence and since abzeab, we have u=gabgabze=g; while since ae, we have gage=g. These imply that uga. Dually, we have hE(S) such that u=abh, bhb and ubh. Since u=gab=abh=uh=(ga)(bh), we know that ga and bh are the required elements x and y.

Define a multiplication on S0=S{0} by xy={xyifx0,y0,andy,xyJx;0otherwise, where xy is the product of x and y in S. By the arguments of [4, page 9], (S0,) is a semigroup. We denote by S the semigroup (S0,). For any JS/𝒥, we denote J0=J{0}. It is easy to check that (J0,) is a subsemigroup of S, which is isomorphic to the principal factor of S determined by J. We will denote the semigroup (J0,) by J. By the definition of , it is easy to see that in the semigroup S,

JxJxJx for all xS;

JxJy=0 for all x,yS such that xJy.

Thus R0[S] is the direct sum of the contracted semigroup algebras R0[J] with JS/𝒥. Note that J is isomorphic to some principal factor of S. We observe that J is a completely 0-simple semigroup since S is a semisimple semigroup, and thus J is isomorphic to some Rees matrix semigroup 0(GJ,IJ,ΛJ;PJ). By a result of , R0[0(GJ,IJ,ΛJ;PJ)] is isomorphic to (R[GJ],|IJ|,|ΛJ|;PJ). Consequently, to verify Theorem 1.1, we need only to prove that R[S] is isomorphic to R0[S].

For the convenience of description, we introduce the semigroup S¯. Put S¯={x¯xS}{0}. Define a multiplication on S¯ as follows: x¯y¯=xy¯, where we will identify 0¯ with 0. It is easy to see that S¯ is isomorphic to S. Hence the contracted semigroup algebra R0[S¯] is isomorphic to the contracted semigroup algebra R0[S]. For JS/𝒥, we denote J¯={x¯xJ}{0}. It is easy to check that (J¯,) is a subsemigroup of S¯ isomorphic to the semigroup J. So, for any J,KS/𝒥, we have J¯K¯{J¯ifK=J,=0otherwise.

For Theorem 1.1, it remains to prove the following lemma.

Lemma 2.3.

R[S]R0[S¯].

Proof.

We consider the mapping φ:R[S]R0[S¯] given on the basis by φ(s)=tst¯(sS). Clearly, φ is well defined. Of course, φ and ¯ may be regarded as the mappings of the ordered set (S,) into the additive group of R0[S¯]. Now, by applying the Möbius inversion theorem to the mappings φ and ¯, we have s¯=tsφ(t)μ(t,s)=φ(tstμ(t,s)), where μ is the Möbius function for (S,). Hence φ is surjective.

We will prove that φ is injective. For α0=xSpx0xR[S], we denote by supp(α0) the set {xSpx00} and by M(α0) the set of maximal elements in the set supp(α0) with respect to the partial order . In recurrence, we define αn=αn1xM(αn1)pxn1x, where αn=xsupp(αn)pxnx. Let βn=xsupp(βn)qxnx with n=0,1,2,. If φ(αn)=φ(βn), then by the definition of φ, xM(αn)pxx¯+Γαn=φ(αn)=φ(βn)=yM(βn)qyny¯+Γβn, where Γαn=xM(αn)yS,y<xpyny¯ and Γβn=xM(βn)yS,y<xqyny¯, and hence xM(αn)pxnx¯=xM(βn)qxnx¯, thus M(αn)=M(βn) and pxn=qxn for any xM(αn). This can imply the following.

Fact 2.4.

If φ(αn)=φ(βn), then M(αn)=M(βn) and by the definition of φ, φ(αn+1)=φ(βn+1).

By the definition of φ, the following facts are immediate.

Fact 2.5.

αn=βn if and only if M(αn)=M(βn) and αn+1=βn+1.

Fact 2.6.

If φ(αn)=φ(βn) and M(αn)=supp(αn),M(βn)=supp(βn), then αn=βn.

Note that |supp(α0)|< and supp(αn+1)supp(αn). We thus have a smallest integer k such that M(αk)=supp(αk). Clearly, αk+1=0. This means that k is the smallest integer t such that αt+1=0. Similarly, there exists the smallest integer l such that βl+1=0 and M(βl)=supp(βl). Now, assume φ(α0)=φ(β0). By using Fact 2.4 repeatedly, φ(α1)=φ(β1),φ(α2)=φ(β2),,φ(αk+1)=φ(βk+1). But φ(αk+1)=0, we have φ(βk+1)=0 and by the definition of φ, βk+1=0. Thus k+1l+1 by the minimality of l, and kl. Similarly, lk. Therefore k=l. Since φ(αk)=φ(βk), by Fact 2.6, we have αk=βk since M(αk)=supp(αk) and M(βl)=supp(βl). Again by the hypothesis φ(α0)=φ(β0), and by Fact 2.4, M(α0)=M(β0); and by (2.6), M(α1)=M(β1),M(α2)=M(β2),,M(αk)=M(βk). By Fact 2.5, M(αk1)=M(βk1); and αk=βk imply αk1=βk1; moreover, by using Fact 2.5 repeatedly, αk2=βk2,,α1=β1 and α0=β0. We have now proved that φ is injective.

Finally, for any s,tS, by (2.4), we have s¯*t¯={st¯ifs,tJst,0otherwise, and by Lemma 2.2, φ(s)*φ(t)=(xsx¯)*(yty¯)=xJst,xsyJst,ytx¯*y¯=xJst,xsyJst,ytxy¯. Moreover, by Lemma 2.2, we have φ(st)=ustu¯=xJst,xsyJst,ytxy¯=xs,xJstyt,yJstx¯*y¯=φ(s)*φ(t). Thus φ is a homomorphism of R[S] into R0[S¯]. Consequently, φ is an isomorphism of R[S] onto R0[S¯].

Acknowledgment

The research is supported by the NSF of Jiangxi Province, the SF of Education Department of Jiangxi Province, and the SF of Jiangxi Normal University.

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