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 ℳ(R[GJ],mJ,nJ;PJ) with
J∈S/𝒥, where
mJ is the number of ℛ-classes in J, nJ the number of ℒ-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 e∈S, 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 ℳ:=ℳ(𝒜;m,n;P) be the vector
space of all m×n matrices over 𝒜. Define a product ∘ in ℳ by
A∘B=APB(A,B∈ℳ),
where APB is the usual
matrix product of A, P, and B. Then ℳ is an algebra
over R. Following [1], we call ℳ the Munn m×nmatrix algebra
over𝒜with sandwich
matrixP.

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 ℳ0(G,I,Λ;P) (ℳ(G,I,Λ;P)), and vice
versa (for Rees matrix semigroups, refer to [1]). In what follows, by the
phrase “Let S=⋃J∈S/𝒥ℳ0(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 ℳ0(GJ;IJ,ΛJ;PJ) or ℳ(GJ;IJ,ΛJ;PJ) for any J∈S/𝒥.

The following is the main result of this paper.

Theorem 1.1.

Let S=⋃J∈S/𝒥ℳ0(GJ,IJ,ΛJ;PJ) be a finite
locally inverse semigroup. Then the semigroup algebra R[S]
is isomorphic
to the direct product of ℳ(R[GJ];|IJ|,|ΛJ|;PJ) with J∈S/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=⋃J∈S/𝒥ℳ0(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 J∈S/𝒥.

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

Corollary 1.3.

Let S=⋃J∈S/𝒥ℳ0(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 J∈S/𝒥.

The following corollary is a consequence of Corollary
1.3.

Corollary 1.4.

Let S=⋃J∈S/𝒥ℳ0(GJ,IJ,ΛJ;PJ) be a finite
locally inverse semigroup. Then the semigroup algebra R[S]
is semisimple
if and only if for all J∈S/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 [2].

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:P→G
is a function
and define g:P→G by g(x)=∑y≤xf(y). Then f(x)=∑y≤xg(y)μ(x,y), where μ
is a Möbius
function.

Now assume that S is a regular
semigroup and a,b∈S. Define
a≤b⟺thereexiste,f∈E(S)suchthata=eb=bf.
Then ≤ is a partial
order on S. Following [3], we call ≤ the natural
partial order on S. Equivalently, a≤b if and only if
for every (for some) f∈E(Rb)(f∈E(Lb)), there exists e∈E(Ra)(e∈E(La)) such that e≤f 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,b∈S. Then for any u≤ab, there exist x≤a and y≤b such that u=xy, x∈Ru and y∈Lu.

Proof.

For any e∈E(Ra), we have ea=a and eab=ab. Let z be an inverse
of ab. Clearly, abz∈E(Rab). Note that eabz=abz. It is easy to check that abze∈E(S),abze≤e, and abzℛabze. Hence abzeℛab and there exists g∈E(S) such that u=gab and g≤abze(≤e). Thus ga≤a. On the other hand, since ℛ is a left
congruence and since abzeℛab, we have u=gabℛgabze=g; while since aℛe, we have gaℛge=g. These imply that uℛga. Dually, we have h∈E(S) such that u=abh, bh≤b and uℒbh. 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
x⊗y={xyifx≠0,y≠0,andy,xy∈Jx;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 J∈S/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⊗,

Jx⊗⊗Jx⊗⊆Jx⊗ for all x∈S;

Jx⊗⊗Jy⊗=0 for all x,y∈S such that x∉Jy.

Thus R0[S⊗] is the direct
sum of the contracted semigroup algebras R0[J⊗] with J∈S/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 ℳ0(GJ,IJ,ΛJ;PJ). By a result of [1], R0[ℳ0(GJ,IJ,ΛJ;PJ)] is isomorphic
to ℳ(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¯∣x∈S}∪{0}. Define a multiplication on S¯ as follows:
x¯∗y¯=x⊗y¯,
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 J∈S/𝒥, we denote J¯={x¯∣x∈J}∪{0}. It is easy to check that (J¯,∗) is a subsemigroup
of S¯ isomorphic to
the semigroup J⊗. So, for any J,K∈S/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)=∑t≤st¯(s∈S). 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¯=∑t≤sφ(t)μ(t,s)=φ(∑t≤stμ(t,s)),
where μ is the Möbius
function for (S,≤). Hence φ is surjective.

We will prove that φ is injective.
For α0=∑x∈Spx0x∈R[S], we denote by supp(α0) the set {x∈S∣px0≠0} 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=αn−1−∑x∈M(αn−1)pxn−1x, where αn=∑x∈supp(αn)pxnx. Let βn=∑x∈supp(βn)qxnx with n=0,1,2,…. If φ(αn)=φ(βn), then by the definition of φ, ∑x∈M(αn)pxx¯+Γαn=φ(αn)=φ(βn)=∑y∈M(βn)qyny¯+Γβn, where Γαn=∑x∈M(αn)∑y∈S,y<xpyny¯ and Γβn=∑x∈M(βn)∑y∈S,y<xqyny¯, and hence ∑x∈M(αn)pxnx¯=∑x∈M(βn)qxnx¯, thus M(αn)=M(βn) and pxn=qxn for any x∈M(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+1≥l+1 by the
minimality of l, and k≥l. Similarly, l≥k. 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(αk−1)=M(βk−1); and αk=βk imply αk−1=βk−1; moreover, by using Fact 2.5 repeatedly, αk−2=βk−2,…,α1=β1 and α0=β0. We have now proved that φ is injective.

Finally, for any s,t∈S, by (2.4), we have
s¯*t¯={st¯ifs,t∈Jst,0otherwise,
and by Lemma 2.2,
φ(s)*φ(t)=(∑x≤sx¯)*(∑y≤ty¯)=∑x∈Jst,x≤s∑y∈Jst,y≤tx¯*y¯=∑x∈Jst,x≤s∑y∈Jst,y≤txy¯.
Moreover, by Lemma 2.2, we have
φ(st)=∑u≤stu¯=∑x∈Jst,x≤s∑y∈Jst,y≤txy¯=∑x≤s,x∈Jst∑y≤t,y∈Jstx¯*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.

CliffordA. H.PrestonG. B.SteinbergB.Möbius functions and semigroup representation theoryNambooripadK. S. S.The natural partial order on a regular semigroupNambooripadK. S. S.Structure of regular semigroups. I