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
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 , 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 ). 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 .
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
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 .
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 , 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 , 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 .
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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