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

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 $\left(\text{BR}(T,\theta ),{\tau }_{\text{BR}}^{\oplus }\right)$ of a finite semilattice $T=\left[E;G_{\alpha },{\phi }_{\alpha ,\beta }\right]$ of compact groups $G_{\alpha }$ in the class of topological inverse semigroups, where ${\tau }_{\text{BR}}^{\oplus }$ is the sum direct topology on $\text{BR}(T,\theta )$. 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.

14 pages