Injectively and absolutely $T_1S$-closed semigroups

TL;DR

This paper characterizes injectively and absolutely $T_1S$-closed commutative semigroups, showing their structural properties and implications for their centers within topological semigroup theory.

Contribution

It provides a complete characterization of injectively $T_1S$-closed semigroups and explores properties of their centers, advancing understanding in topological semigroup closure concepts.

Findings

Injectively $T_1S$-closed semigroups are bounded, nonsingular, and Clifford-finite.

Every injectively $T_1S$-closed semigroup has an injectively $T_1S$-closed center.

Absolutely $T_1S$-closed semigroups have finite centers.

Abstract

A semigroup $X$ is $absolutely$ (resp. $injectively$) $T_1S$-$closed$ if for any (injective) homomorphism $h:X\to Y$ to a $T_1$ topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. We prove that a commutative semigroup $X$ is injectively $T_1S$-closed if and only if $X$ is bounded, nonsingular and Clifford-finite. Using this characterization, we prove that (1) every injectively $T_1S$-closed semigroup has injectively $T_1S$-closed center, and (2) every absolutely $T_1S$-closed semigroup has finite center.