On locally compact semitopological $0$-bisimple inverse $\omega$-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 $\mathcal{H}$-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 $\left(\text{B}({\mathbb{Z}}_{+},\theta {)}^0,\tau \right)$ with adjoined zero and with a non-annihilating homomorphism $\theta :{\mathbb{Z}}_{+}\to {\mathbb{Z}}_{+}$: $\left(\text{B}({\mathbb{Z}}_{+},\theta {)}^0,\tau \right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\text{B}({\mathbb{Z}}_{+},\theta {)}^0$ in a semitopological semigroup.

26 pages

References in corpus (3)

• On countably compact 0-simple topological inverse semigroups
• On a complete topological inverse polycyclic monoid
• On monoids of monotone injective partial self-maps of integers with cofinite domains and images

Cited by in corpus (1)

• On a semitopological semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ when a family $\mathcal{F}$ consists of inductive non-empty subsets of $\omega$