On tame semigroups generated by idempotents with partial null multiplication

Abstract

Let \(I\) be a finite set without \(0\) and \(J\) a subset in \(I\times I\)  without  diagonal elements \((i,i)\). We define  \(S(I,J)\) to be the semigroup with generators \(e_i\), where \(i\in I\cup 0\), and the following relations: \(e_0=0\);  \(e_i^2=e_i\) for any \(i\in I\); \(e_ie_j=0\) for any  \((i,j)\in J\). 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.