A note on locally inverse semigroup algebras.
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.
scientific article; zbMATH DE number 5313282