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 $\left(\textbf{BR}(T,\theta),\tau_{\textbf{BR}}^{\oplus}\right)$ 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.