  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.
