Absolutely closed semigroups

Abstract

Let $\mathcal{C}$ be a class of topological semigroups. A semigroup $X$ is called $absolutely$ $\mathcal{C}$-$closed$ if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in \mathcal{C}$, the image $h[X]$ is closed in $Y$. Let ${\mathsf{T}}_1\mathsf{S}$, ${\mathsf{T}}_2\mathsf{S}$, and ${\mathsf{T}}_{\mathsf{z}}\mathsf{S}$ be the classes of $T_1$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup $X$ is absolutely ${\mathsf{T}}_{\mathsf{z}}\mathsf{S}$-closed if and only if $X$ is absolutely ${\mathsf{T}}_2\mathsf{S}$-closed if and only if $X$ is chain-finite, bounded, group-finite and Clifford+finite. On the other hand, a commutative semigroup $X$ is absolutely ${\mathsf{T}}_1\mathsf{S}$-closed if and only if $X$ is finite. Also, for a given absolutely $\mathcal{C}$-closed semigroup $X$ we detect absolutely $\mathcal{C}$-closed subsemigroups in the center of $X$.

22 pages