On semitopological $\alpha$-bicyclic monoid

TL;DR

This paper studies the algebraic and topological properties of semitopological $eta$-bicyclic monoids, revealing their structure, isolated points, and the existence of non-discrete locally compact topologies, especially for certain ordinals.

Contribution

It characterizes the algebraic structure of $eta$-bicyclic monoids and constructs non-discrete locally compact topologies, challenging previous assumptions for specific ordinals.

Findings

$eta$-bicyclic monoids are isomorphic to semigroups of order isomorphisms

Certain elements are isolated points in the topology

Existence of non-discrete locally compact topologies for $eta=\omega+1$

Abstract

In this paper we consider a semitopological $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$ and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of the ordinal $\omega^\alpha$. We prove that for every ordinal $\alpha$ for every $(a,b)\in \mathcal{B_\alpha}$ if either $a$ or $b$ is a non-limit ordinal then $(a,b)$ is an isolated point in $\mathcal{B}_\alpha$. We show that for every ordinal $\alpha<\omega+1$ every locally compact semigroup topology on $\mathcal{B}_{\alpha}$ is discrete. However, we construct an example of a non-discrete locally compact topology $\tau_{lc}$ on $\mathcal{B}_{\omega+1}$ such that $(\mathcal{B}_{\omega+1},\tau_{lc})$ is a topological inverse semigroup. This example shows that there is a gap in \cite[Theorem~2.9]{Hogan-1984}, where is stated that for every ordinal $\alpha$ there is only discrete locally compact inverse semigroup topology on $\mathcal{B_\alpha}$.