On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents

Abstract

Let \(I\) be a finite set (without \(0\)) and \(J\) a subset of \(I\times I\)  without diagonal elements. Let \(S(I,J)\) denotes the semigroup generated by \(e_0=0\) and \(e_i\), \(i\in I\), with the following  relations: \(e_i^2=e_i\) for any \(i\in I\), \(e_ie_j=0\) for any  \((i,j)\in J\). In this paper we prove that, for any finite semigroup \(S=S(I,J)\) and any its matrix representation \(M\) over a field \(k\), each matrix of the form \(\sum_{i \in I}\alpha_i M(e_i)\) with \(\alpha_i\in k\) is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra.