Categorically closed unipotent semigroups

Abstract

Let $\mathcal C$ be a class of $T_1$ topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup $X$ is $\mathcal C$-$closed$ if $X$ is closed in any topological semigroup $Y\in\mathcal C$ that contains $X$ as a discrete subsemigroup; $X$ is $injectively$ $\mathcal C$-$closed$ if for any (injective) homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. A semigroup $X$ is $unipotent$ if it contains a unique idempotent. We prove that a unipotent commutative semigroup $X$ is (injectively) $\mathcal C$-closed if and only if $X$ is bounded, nonsingular (and group-finite). This characterization implies that for every injectively $\mathcal C$-closed unipotent semigroup $X$, the center $Z(X)$ is injectively $\mathcal C$-closed.