Absolutely closed semigroups

TL;DR

This paper characterizes absolutely closed topological semigroups across various classes, establishing conditions for commutative semigroups to be absolutely closed in T1, Hausdorff, and Tychonoff topologies, and explores subsemigroup properties.

Contribution

It provides a complete characterization of absolutely closed commutative topological semigroups in different classes, linking algebraic properties with topological closure conditions.

Findings

Absolutely TzS-closed iff absolutely T2S-closed for commutative semigroups.

Such semigroups are chain-finite, bounded, group-finite, and Clifford+finite.

Finite semigroups are the only absolutely T1S-closed commutative 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 $\mathsf{T_{\!1}S}$, $\mathsf{T_{\!2}S}$, and $\mathsf{T_{\!z}S}$ be the classes of $T_1$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup $X$ is absolutely $\mathsf{T_{\!z}S}$-closed if and only if $X$ is absolutely $\mathsf{T_{\!2}S}$-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 $\mathsf{T_{\!1}S}$-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 $X$.