Injectively closed commutative semigroups

Abstract

Let $\mathcal{C}$ be a class of topological semigroups. A semigroup $X$ is $injectively$ $\mathcal{C}$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal{C}$ containing $X$ as a subsemigroup. Let ${\mathsf{T}}_2\mathsf{S}$ (resp. ${\mathsf{T}}_{\mathsf{z}}\mathsf{S}$) be the class of Hausdorff (and zero-dimensional) topological semigroups. We prove that a commutative semigroup $X$ is injectively ${\mathsf{T}}_2\mathsf{S}$-closed if and only if $X$ is injectively ${\mathsf{T}}_{\mathsf{z}}\mathsf{S}$-closed if and only if $X$ is bounded, chain-finite, group-finite, nonsingular and not Clifford-singular.

40 pages