On tame semigroups generated by idempotents with partial null multiplication

Abstract

Let I be a finite set without 0 and J a subset in I×I without diagonal elements (i,i). We define S(I,J) to be the semigroup with generators ei, where i∈I∪0, and the following relations: e0=0; e2i=ei for any i∈I; eiej=0 for any (i,j)∈J. In this paper we study finite-dimensional representations of such semigroups over a field k. In particular, we describe all finite semigroups S(I,J) of tame representation type.