ℛ-unipotent semigroup algebras

ABSTRACT

Let S be an ℛ-unipotent semigroup such that the idempotent set E(S) is locally ℒ-finite, and let R be a commutative ring with identity. In this paper, we construct a set of orthogonal idempotents $\left\{\widetilde{e_Y}\right|Y\in {(S∕\mathcal{ℒ})}^{\ast }\}$ in R₀[S], then prove that $\overline{\mathcal{ℬ}}=\left\{\stackrel{̃}{e_{L_{a^{†}}}}a\stackrel{̃}{e_{L_{a^{\ast }}}}\right|a\in S^{\ast }\}$ is an R-basis of R₀[S], and that the set $\overline{S}=\overline{\mathcal{ℬ}}\cup \left\{0\right\}$ is a 0-direct union of ℛ-unipotent completely 0-simple semigroups. As an application, we show that R₀[S] is π-semisimple if and only if S is an inverse semigroup and for each maximal subgroup G of S, the group algebra R[G] is π-semisimple.

KEYWORDS:

• ℛ-unipotent semigroups
• π-semisimple
• left regular bands
• orthogonal idempotents
• semigroup algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G30
• 17C17
• 17C20
• 17C27
• 20M25

Acknowledgments

The author would like to thank the anonymous referees for their valuable comments and suggestions.