Rees Theorem and Quotients in Linear Algebraic Semigroups

Abstract

Let S be an irreducible linear algebraic semigroup over an algebraically closed field k. We analyze the Rees theorem for a regular $\mathcal{J}$-class J of S. We define the support of J to be $\mathbb{X} = J/\mathcal{H}$. We show that $\mathbb{X}$ is a quasi-projective variety that is isomorphic to the direct product of two geometric quotients of algebraic group actions. When J is completely simple, $\mathbb{X}$ is an affine variety, while in reductive monoids, $\mathbb{X}$ is always a projective variety. We define the support of S to be that of the maximum regular $\mathcal{J}$-class of S. We study closed irreducible regular subsemigroups S of the full linear monoid M _n (k) with projective supports $\mathbb{X}$. We determine the possible $\mathbb{X}$ and for a given $\mathbb{X}$, all the possible S. Along the way, we pose some open problems, the chief among which is the conjecture that any irreducible regular linear algebraic semigroup S with zero has projective support. We prove this in the simplest case of when S is a completely 0-simple semigroup.