Matrix semigroups

Abstract

Let S 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 , … , I t {I_0}, \ldots ,{I_t} of S S such that I 0 ⊆ ⋯ ⊆ I t = S {I_0} \subseteq \cdots \subseteq {I_t} = S , I 0 {I_0} is completely simple, and each Rees factor semigroup I k / I k − 1 {I_k}/{I_{k - 1}} , k = 1 , … , t k = 1, \ldots ,t , is either completely 0 0 -simple or else a nilpotent semigroup. The basic technique is to study the Zariski closure of S S , which is a linear algebraic semigroup.