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

## TL;DR

This paper characterizes the structure of certain locally compact semitopological inverse semigroups, showing they are either compact or a topological sum of classes, with specific results for Reilly semigroups with monothetic subgroups.

## Contribution

It provides a detailed structural description and a dichotomy result for locally compact semitopological $0$-bisimple inverse $ ext{omega}$-semigroups, including Reilly semigroups with monothetic subgroups.

## Key findings

- Such semigroups are either compact or topologically sum of $ ext{H}$-classes.
- Reilly semigroups with non-annihilating homomorphism are either compact or discrete.
- The paper discusses the closure and remainder of the semigroup in larger 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.