Injectively closed commutative semigroups

TL;DR

This paper characterizes when commutative semigroups are injectively closed in classes of Hausdorff and zero-dimensional topological semigroups, linking algebraic properties to topological closure conditions.

Contribution

It provides a complete algebraic characterization of injectively closed commutative semigroups within certain topological classes.

Findings

Injectively T2S-closed and TzS-closed semigroups are characterized by algebraic properties.

Characterization involves boundedness, chain-finiteness, group-finiteness, nonsingularity, and absence of Clifford-singularity.

Equivalence of T2S-closed and TzS-closed conditions for 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 $\mathsf{T_{\!2}S}$ (resp. $\mathsf{T_{\!z}S}$) be the class of Hausdorff (and zero-dimensional) topological semigroups. We prove that a commutative semigroup $X$ is injectively $\mathsf{T_{\!2}S}$-closed if and only if $X$ is injectively $\mathsf{T_{\!z}S}$-closed if and only if $X$ is bounded, chain-finite, group-finite, nonsingular and not Clifford-singular.