On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups

We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological Reilly semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ with adjoined zero and with a non-annihilating homomorphism $\theta\colon \mathbb{Z}_{+}\to \mathbb{Z}_{+}$: $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^0$ in a semitopological semigroup.