On Linear Algebraic Semigroups

Abstract

Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of ${K^n}$ along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals ${I_0}, \ldots , {I_t}$ such that $S = {I_t} \supseteq \cdots \supseteq {I_0}$, ${I_0}$ is the completely simple kernel of S and each Rees factor semigroup ${I_k}/{I_{k - 1}}$ is either nil or completely 0-simple $(k = 1, \ldots , t)$. 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 $E(S)$. (1) If $E(S)$ is a subsemigroup, then S is a semilattice of nil extensions of rectangular groups. (2) If all the subgroups of S are abelian and if for all $a \in S$, there exists $e \in E(S)$ such that $ea = ae = a$, then S is a semilattice of nil extensions of completely simple semigroups. (3) If all subgroups of S are abelian and if S is regular, then S is a subdirect product of completely simple and completely 0-simple semigroups. (4) S has only trivial subgroups if and only if S is a nil extension of a rectangular band.