On a locally compact semitopological $α$-bicyclic monoid

Serhii Bardyla

Published 2017-07-22Version 1

In this paper for every ordinal $\alpha<\omega+1$ we describe all locally compact Hausdorff topologies which make $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$ a semitopological semigroup. In particular, we prove that there exist exactly $k$ distinct locally compact Hausdorff topologies which make $\mathcal{B}_{k}$ a semitopological semigroup and the set of all distinct locally compact Hausdorff topologies which make $\mathcal{B}_{\omega}$ a semitopological semigroup is countable. Moreover, for every ordinal $\alpha<\omega+1$ the set of all locally compact Hausdorff topologies on $\mathcal{B}_{\alpha}$ which make $\mathcal{B}_{\alpha}$ a semitopological semigroup is linearly ordered by the inclusion. Also we prove that for each ordinal $\alpha$ the $\alpha+1$-bicyclic semigroup $\mathcal{B}_{\alpha+1}$ is isomorphic to the Bruck extension of the $\alpha$-bicyclic semigroup $\mathcal{B}_{\alpha}$.