Subgroups of categorically closed semigroups

TL;DR

This paper investigates various subgroups and idempotents in categorically closed topological semigroups, establishing boundedness and finiteness properties under different closure conditions within the class of Tychonoff zero-dimensional semigroups.

Contribution

It introduces new subgroup boundedness and finiteness results for categorically closed semigroups, extending understanding of their algebraic and topological structure.

Findings

Subgroups of the center are bounded in ideally TzS-closed semigroups.

Subgroups of the ideal center are bounded in TzS-closed semigroups.

Subgroups of the center are finite in TzS-discrete or injectively TzS-closed semigroups.

Abstract

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called (1) $\mathcal C$-$closed$ if $X$ is closed in every topological semigroup $Y\in\mathcal C$ containing $X$ as a discrete subsemigroup, (2) $ideally$ $\mathcal C$-$closed$ if for any ideal $I$ in $X$ the quotient semigroup $X/I$ is $\mathcal C$-closed; (3) $absolutely$ $\mathcal C$-$closed$ if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$, (4) $injectively$ $\mathcal C$-$closed$ (resp. $\mathcal C$-$discrete$) if for any injective homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed (resp. discrete) in $Y$. Let $\mathsf{T_{\!z}S}$ be the class of Tychonoff zero-dimensional topological semigroups. For a semigroup $X$ let $V\!E(X)$ be the set of all viable idempotents of $X$, i.e., idempotents $e$ such that the complement $X\setminus\frac{H_e}e$ of the set $\frac{H_e}e=\{x\in X:xe=ex\in H_e\}$ is an ideal in $X$. We prove the following results: (i) for any ideally $\mathsf{T_{\!z}S}$-closed semigroup $X$ each subgroup of the center $Z(X)=\{z\in X:\forall x\in X\;\;(xz=zx)\}$ is bounded; (ii) for any $\mathsf{T_{\!z}S}$-closed semigroup $X$, each subgroup of the ideal center $I\!Z(X)=\{z\in Z(X):zX\subseteq Z(X)\}$ is bounded; (iii) for any $\mathsf{T_{\!z}S}$-discrete or injectively $\mathsf{T_{\!z}S}$-closed semigroup $X$, every subgroup of $Z(X)$ is finite, (iv) for any viable idempotent $e$ in an ideally (and absolutely) $\mathsf{T_{\!z}S}$-closed semigroup $X$, the maximal subgroup $H_e$ is ideally (and absolutely) $\mathsf{T_{\!z}S}$-closed and has bounded (and finite) center $Z(H_e)$.