On semitopological $\alpha$-bicyclic monoid

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 ${\mathcal{B}}_{\alpha }$.