On Linear Algebraic Semigroups

Abstract

Abstract.Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of K " along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals /",...,/, such that S = /, 3 • • • D I0, I0 is the completely simple kernel of S and each Rees factor semigroup 4//*_i is either nil or completely 0-simple (k =■ 1,...,/). We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set E(S).

Various Forms of Generating Subsemigroups in Algebraic Monoids

“…So dim N = 1. By [13,Theorem 2.10], there is a onedimensional irreducible algebraic submonoid S of M with e as its zero. Thus S ∈ 1 e M .…”

“…Putcha [12,Theorem 2.13] proved that if an irreducible algebraic monoid M is one dimensional with M = G then M = G ∪ 0 . Conversely, it is easy to show that if M = G ∪ 0 then M is one dimensional.…”