The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

TL;DR

This paper investigates conditions in countably compact topological semigroups that ensure the continuity of inversion, the closedness of maximal subgroups, and the structural properties of the semigroup's Clifford part.

Contribution

It establishes criteria under which maximal subgroups are closed, the Clifford part is closed, and the inversion and projection maps are continuous in countably compact topological semigroups.

Findings

Maximal subgroups are closed under certain conditions.

The Clifford part of the semigroup can be closed.

Inversion and projection maps are continuous in these settings.

Abstract

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$, $\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous.