On semitopological simple inverse $\omega$-semigroups with compact maximal subgroups

TL;DR

This paper characterizes the structure of simple inverse Hausdorff semitopological d-semigroups with compact maximal subgroups, showing they are topologically isomorphic to Bruck--Reilly extensions of finite semilattices of compact groups.

Contribution

It provides a structural description of semitopological d-semigroups with compact maximal subgroups, linking them to Bruck--Reilly extensions and classifying their topologies.

Findings

Such semigroups are topologically isomorphic to Bruck--Reilly extensions of finite semilattices of compact groups.

Every Hausdorff locally compact shift-continuous topology on these semigroups is either compact or has an isolated zero.

The structure theorem applies to semigroups with adjoined zero, describing their topological properties.

Abstract

We describe the structure of ($0$-)simple inverse Hausdorff semitopological $\omega$-semigroups with compact maximal subgroups. In particular, we show that if $S$ is a simple inverse Hausdorff semitopological $\omega$-semigroup with compact maximal subgroups, then $S$ is topologically isomorphic to the Bruck--Reilly extension $\left(\textbf{BR}(T,\theta),\tau_{\textbf{BR}}^{\oplus}\right)$ of a finite semilattice $T=\left[E;G_\alpha,\varphi_{\alpha,\beta}\right]$ of compact groups $G_\alpha$ in the class of topological inverse semigroups, where $\tau_{\textbf{BR}}^{\oplus}$ is the sum direct topology on $\textbf{BR}(T,\theta)$. Also we prove that every Hausdorff locally compact shift-continuous topology on the simple inverse Hausdorff semitopological $\omega$-semigroups with compact maximal subgroups with adjoined zero is either compact or the zero is an isolated point.