On Algebraic Semigroups and Monoids, II

TL;DR

This paper explores the structure of idempotents in algebraic semigroups and monoids, revealing finiteness, geometric properties, and their relation to convex cones and conjugacy classes.

Contribution

It establishes new structural results about the finiteness and geometric nature of idempotents in algebraic semigroups and monoids, linking them to toric varieties and convex cones.

Findings

$E(S)$ is finite and reduced when $S$ is commutative.

In irreducible cases, $E(S)$ lies in a minimal irreducible subsemigroup that is an affine toric variety.

For irreducible monoids, $E(S)$ is smooth with components as conjugacy classes.

Abstract

Consider an algebraic semigroup $S$ and its closed subscheme of idempotents, $E(S)$. When $S$ is commutative, we show that $E(S)$ is finite and reduced; if in addition $S$ is irreducible, then $E(S)$ is contained in a smallest closed irreducible subsemigroup of $S$, and this subsemigroup is an affine toric variety. It follows that $E(S)$ (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when $S$ is an irreducible algebraic monoid, we show that $E(S)$ is smooth, and its connected components are conjugacy classes of the unit group.