On tame semigroups generated by idempotents with partial null multiplication

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.