Pseudocompact and precompact subsemigroups of topological groups

TL;DR

This paper explores conditions under which subsemigroups of topological groups are actually subgroups, focusing on properties like pseudocompactness and precompactness, and establishing new criteria for these structures.

Contribution

It introduces new sufficient conditions for subsemigroups to be subgroups, specifically relating to pseudocompactness and precompactness in topological groups.

Findings

Closed precompact subsemigroups are subgroups.

Open pseudocompact monoids are subgroups.

Provides criteria extending previous compactness conditions.

Abstract

In this paper we give sufficient conditions under which a subsemigroup of a topological group is a subgroup, adding to the results given in \cite{Kosh, can, axioms, forum, Hof, cc, locally} where conditions exist (such as locally compactness, compactness, feeble compactness and sequential compactness) for a semigroup to be a group. In our work we proved that closed precompact subsemigroups of topological groups are semigroups, just like open pseudocompact monoids of topological groups.