Characterizing categorically closed commutative semigroups

TL;DR

This paper characterizes when commutative semigroups are categorically closed within a class of topological semigroups, revealing conditions related to periodicity, chain-finiteness, and subgroup properties.

Contribution

It provides a complete characterization of $ ext{C}$-closed and projectively $ ext{C}$-closed commutative semigroups based on algebraic and topological conditions.

Findings

A semigroup is $ ext{C}$-closed if it admits a homomorphism to a chain-finite semilattice with $ ext{C}$-closed fibers.

A commutative semigroup is $ ext{C}$-closed iff it is periodic, chain-finite, with bounded subgroups, and certain product conditions.

A commutative semigroup is projectively $ ext{C}$-closed iff it is chain-finite, with bounded subgroups, and the union of subgroups has finite complement.

Abstract

Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal C$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X/_\approx$ is $\mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin\{x,y\}$. We prove that a semigroup $X$ is $\mathcal C$-closed if it admits a homomorphism $h:X\to E$ to a chain-finite semilattice $E$ such that for every $e\in E$ the semigroup $h^{-1}(e)$ is $\mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $\mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $A\subseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $\mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $X\setminus H(X)$.