Title: On locally compact semitopological $0$-bisimple inverse $ω$-semigroups,
Abstract: 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.