On semitopological $α$-bicyclic monoid

Abstract

In this paper we consider a semitopological $α$-bicyclic monoid $\mathcal{B}_α$ and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of the ordinal $ω^α$. We prove that for every ordinal $α$ for every $(a,b)\in \mathcal{B_α}$ if either $a$ or $b$ is a non-limit ordinal then $(a,b)$ is an isolated point in $\mathcal{B}_α$. We show that for every ordinal $α<ω+1$ every locally compact semigroup topology on $\mathcal{B}_α$ is discrete. However, we construct an example of a non-discrete locally compact topology $τ_{lc}$ on $\mathcal{B}_{ω+1}$ such that $(\mathcal{B}_{ω+1},τ_{lc})$ 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 $α$ there is only discrete locally compact inverse semigroup topology on $\mathcal{B_α}$.