International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 576061, 5 pages
doi:10.1155/2008/576061
Research Article
A Note on Locally Inverse Semigroup Algebras
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
Received 9 December 2007; Accepted 29 January 2008
Academic Editor: Francois Goichot
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.
Abstract
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.