Guo, Xiaojiang A note on locally inverse semigroup algebras. (English) Zbl 1145.20035 Int. J. Math. Math. Sci. 2008, Article ID 576061, 5 p. (2008). Summary: 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 \(\mathcal M(R[G_J],m_J,n_J;P_J)\) with \(J\in S/\mathcal J\), where \(m_J\) is the number of \(\mathcal R\)-classes in \(J\), \(n_J\) the number of \(\mathcal L\)-classes in \(J\), and \(G_J\) a maximum subgroup of \(J\). As applications, we obtain sufficient and necessary conditions for the semigroup algebra of a finite locally inverse semigroup to be semisimple. Cited in 5 Documents MSC: 20M25 Semigroup rings, multiplicative semigroups of rings 16S36 Ordinary and skew polynomial rings and semigroup rings 20M18 Inverse semigroups Keywords:finite locally inverse semigroups; semisimple semigroup algebras; direct products of Munn algebras PDFBibTeX XMLCite \textit{X. Guo}, Int. J. Math. Math. Sci. 2008, Article ID 576061, 5 p. (2008; Zbl 1145.20035) Full Text: DOI EuDML References: [1] 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. · Zbl 0111.03403 [2] B. Steinberg, “Möbius functions and semigroup representation theory,” Journal of Combinatorial Theory, vol. 113, no. 5, pp. 866-881, 2006. · Zbl 1148.20049 [3] 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. · Zbl 0459.20054 [4] K. S. S. Nambooripad, “Structure of regular semigroups. I,” Memoirs of the American Mathematical Society, vol. 22, no. 224, p. vii+119, 1979. · Zbl 0457.20051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.