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

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