Research Article | Open Access

Xiaojiang Guo, "A Note on Locally Inverse Semigroup Algebras", *International Journal of Mathematics and Mathematical Sciences*, vol. 2008, Article ID 576061, 5 pages, 2008. https://doi.org/10.1155/2008/576061

# A Note on Locally Inverse Semigroup Algebras

**Academic Editor:**Francois Goichot

#### Abstract

Let be a commutative ring and a finite locally inverse semigroup. It is proved that the semigroup algebra is isomorphic to the direct product of Munn algebras with , where is the number of -classes in , the number of -classes in , and a maximum subgroup of . 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 is called a *locally
inverse semigroup* if for all idempotent , the local submonoid is an inverse
semigroup under the multiplication of . 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 be an -algebra. Let and be positive
integers, and let be a fixed matrix over . Let be the vector
space of all matrices over . Define a product in by
where is the usual
matrix product of , , and . Then is an algebra
over . Following [1], we call the Munn *matrix algebra
over* *with sandwich
matrix* .

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 -simple
semigroup (completely simple semigroup) is isomorphic to some Rees matrix
semigroup (), and vice
versa (for Rees matrix semigroups, refer to [1]). In what follows, by the
phrase “Let be a finite
regular semigroup,” we mean that is a finite
regular semigroup in which the principal factor of determined by
the -class is isomorphic
to the Rees matrix semigroup or for any .

The following is the main result of this paper.

Theorem 1.1. *Let be a finite
locally inverse semigroup. Then the semigroup algebra
is isomorphic
to the direct product of with . *

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 be a finite
locally inverse semigroup. Then the semigroup algebra
has an identity
if and only if and is invertible
in the full matrix algebra for all .*

Reference [1, Lemma 5.18, page 163] told us that is isomorphic to the full matrix algebra if has an identity. Now, we have the following.

Corollary 1.3. *Let be a finite
locally inverse semigroup. If
has an
identity, then
is isomorphic
to the direct product of the full matrix algebras with . *

The following corollary is a consequence of Corollary 1.3.

Corollary 1.4. *Let be a finite
locally inverse semigroup. Then the semigroup algebra
is semisimple
if and only if for all *

(1)*;*(2)* is invertible
in the full matrix algebra ;*(3)* is
semisimple.*

#### 2. Proof of Theorem 1.1

For our purpose, we have the Möbius inversion theorem [2].

Lemma 2.1. *Let ()
be a locally
finite partially ordered set (i.e., intervals are finite) in which each
principal ideal has a maximum and
be an Abelian
group. Suppose that
is a function
and define by Then where
is a Möbius
function. *

Now assume that is a regular
semigroup and . Define
Then is a partial
order on . Following [3], we call the *natural
partial order* on . Equivalently, if and only if
for every (for some) , there exists such that and . Moreover, Nambooripad [3, 4] proved that is a locally
inverse semigroup if and only if the natural partial order is compatible
with respect to the multiplication of .

Lemma 2.2. *Let
be a locally
inverse semigroup and . Then for any , there exist and such that , and .*

*Proof. *For any , we have and . Let be an inverse
of . Clearly, . Note that . It is easy to check that , and . Hence and there exists such that and . Thus . On the other hand, since is a left
congruence and since , we have ; while since , we have . These imply that . Dually, we have such that , and . Since , we know that and are the
required elements and .

Define a multiplication on by where is the product of and in . By the arguments of [4, page 9], is a semigroup. We denote by the semigroup . For any , we denote . It is easy to check that is a subsemigroup of , which is isomorphic to the principal factor of determined by . We will denote the semigroup by . By the definition of , it is easy to see that in the semigroup ,

(i) for all ;(ii) for all such that . Thus is the direct sum of the contracted semigroup algebras with . Note that is isomorphic to some principal factor of . We observe that is a completely -simple semigroup since is a semisimple semigroup, and thus is isomorphic to some Rees matrix semigroup . By a result of [1], is isomorphic to . Consequently, to verify Theorem 1.1, we need only to prove that is isomorphic to .

For the convenience of description, we introduce the semigroup . Put . Define a multiplication on as follows: where we will identify with . It is easy to see that is isomorphic to . Hence the contracted semigroup algebra is isomorphic to the contracted semigroup algebra . For , we denote . It is easy to check that is a subsemigroup of isomorphic to the semigroup . So, for any , we have

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

Lemma 2.3. *. *

*Proof. *We consider the mapping given on the
basis by . Clearly, is well
defined. Of course, and may be regarded
as the mappings of the ordered set () into the
additive group of . Now, by applying the Möbius inversion theorem to the
mappings and , we have
where is the Möbius
function for (). Hence is surjective.

We will prove that is injective.
For , we denote by the set and by the set of
maximal elements in the set with respect to
the partial order . In recurrence, we define , where . Let with . If , then by the definition of , , where and , and hence , thus and for any . This can imply the following.*Fact 2.4. *If , then and by the
definition of , .

By the
definition of , the following facts are immediate.*Fact 2.5. * if and only if and .*Fact 2.6. *If and , then .

Note that and . We thus have a smallest integer such that . Clearly, . This means that is the smallest
integer such that . Similarly, there exists the smallest integer such that and . Now, assume . By using Fact 2.4 repeatedly,
But , we have and by the
definition of , . Thus by the
minimality of , and . Similarly, . Therefore . Since , by Fact 2.6, we have since and . Again by the hypothesis , and by Fact 2.4, ; and by (2.6), . By Fact 2.5, and imply ; moreover, by using Fact 2.5 repeatedly, and . We have now proved that is injective.

Finally, for any , by (2.4), we have
and by Lemma 2.2,
Moreover, by Lemma 2.2, we have
Thus is a
homomorphism of into . Consequently, is an
isomorphism of onto .

#### 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.

#### References

- A. H. Clifford and G. B. Preston, in
*The Algebraic Theory of Semigroups*, vol. 1 of*Mathematical Surveys, no. 7*, American Mathematical Society, Providence, RI, USA, 1961. View at: Zentralblatt MATH | MathSciNet - B. Steinberg, “Möbius functions and semigroup representation theory,”
*Journal of Combinatorial Theory*, vol. 113, no. 5, pp. 866–881, 2006. View at: Publisher Site | Google Scholar | MathSciNet - K. S. S. Nambooripad, “The natural partial order on a regular semigroup,”
*Proceedings of the Edinburgh Mathematical Society*, vol. 23, no. 3, pp. 249–260, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. S. S. Nambooripad, “Structure of regular semigroups. I,”
*Memoirs of the American Mathematical Society*, vol. 22, no. 224, p. vii+119, 1979. View at: Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2008 Xiaojiang Guo. This 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.