Categorically Closed Unipotent Semigroups

Abstract

Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup Y∈C that contains X as a discrete subsemigroup; X is injectively C-closed if for any injective homomorphism h:X→Y to a topological semigroup Y∈C the image h[X] is closed in Y. A semigroup X is unipotent if it contains a unique idempotent. It is proven that a unipotent commutative semigroup X is (injectively) C-closed if and only if X is bounded and nonsingular (and group-finite). This characterization implies that for every injectively C-closed unipotent semigroup X, the center Z(X) is injectively C-closed.