Matrix Semigroups

Abstract

Let $S$ be a semigroup of matrices over a field such that a power of each element lies in a subgroup (i.e., each element has a Drazin inverse within the semigroup). The main theorem of this paper is that there exist ideals ${I_0}, \ldots ,{I_t}$ of $S$ such that ${I_0} \subseteq \cdots \subseteq {I_t} = S$, ${I_0}$ is completely simple, and each Rees factor semigroup ${I_k}/{I_{k - 1}}$, $k = 1, \ldots ,t$, is either completely $0$-simple or else a nilpotent semigroup. The basic technique is to study the Zariski closure of $S$, which is a linear algebraic semigroup.