Categorically closed countable semigroups

TL;DR

This paper explores the relationship between categorical closedness and topologizability in countable semigroups, introducing concepts like polyboundedness and establishing conditions for topologizability and automatic continuity.

Contribution

It establishes new characterizations of topologizability and closedness in countable semigroups, introduces the notion of polybounded semigroups, and links these to automatic continuity and group structure.

Findings

Countable semigroups with finite-to-one shifts are topologizability characterized by nontopologizability.

Polyboundedness is equivalent to certain closedness properties in T_1 and zero-dimensional semigroups.

Every cancellative polybounded semigroup is a group.

Abstract

In this paper we establish a connection between categorical closedness and topologizability of semigroups. In particular, for a class $\mathsf T_{\!1}\mathsf S$ of $T_1$ topological semigroups we prove that a countable semigroup $X$ with finite-to-one shifts is injectively $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf{T_{\!1}S}$-nontopologizable in the sense that every $T_1$ semigroup topology on $X$ is discrete. Moreover, a countable cancellative semigroup $X$ is absolutely $\mathsf T_{\!1}\mathsf S$-closed if and only if every homomorphic image of $X$ is $\mathsf T_{\!1}\mathsf S$-nontopologizable. Also, we introduce and investigate a notion of a polybounded semigroup. It is proved that a countable semigroup $X$ with finite-to-one shifts is polybounded if and only if $X$ is $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf T_{\!z}\mathsf S$-closed, where $\mathsf T_{\!z}\mathsf S$ is a class of zero-dimensional Tychonoff topological semigroups. We show that polyboundedness provides an automatic continuity of the inversion in $T_1$ paratopological groups and prove that every cancellative polybounded semigroup is a group.