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 × I without diagonal elements. Let S(I, J) denotes the semigroup generated by e₀ = 0 and ei, i ∈ I, with the following relations: e²i = ei for any i ∈ I, eiej = 0 for any (i, j) ∈ 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 ∑i∈IαiM(ei) with αi ∈ 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.