Joint continuity in semitopological monoids and semilattices

TL;DR

This paper explores the continuity properties of actions and operations in semitopological monoids, semilattices, and pseudocompact spaces, extending existing results and introducing new concepts like weak $q_D$-spaces.

Contribution

It generalizes Lawson's results to pseudocompact spaces and introduces the concept of weak $q_D$-spaces, linking them to Grothendieck pairs and analyzing continuity in semitopological structures.

Findings

Subgroups of pseudocompact semitopological monoids are topological groups.

Continuity of multiplication and inversion in semitopological semigroups is established.

Semitopological semilattices' operation continuity is studied.

Abstract

In this paper we study the separately continuous actions of semitopological monoids on pseudocompact spaces. The main aim of this paper is to generalize Lawson's results to some class of pseudocompact spaces. Also, we introduce a concept of a weak $q_D$-space and prove that a pseudocompact space and a weak $q_D$-space form a Grothendieck pair. As an application of the main result, we investigate the continuity of multiplication and taking inverses in subgroups of semitopological semigroups. In particular, we get that if $(S,\bullet)$ is a Tychonoff pseudocompact semitopological monoid with a quasicontinuous multiplication $\bullet$ and $G$ is a subgroup of $S$, then $G$ is a topological group. Also, we study the continuity of operations in semitopological semilattices.