On locally compact shift-continuous topologies on the $\alpha$-bicyclic monoid

A topology $\tau$ on a monoid $S$ is called {\em shift-continuous} if for every $a,b\in S$ the two-sided shift $S\to S$, $x\mapsto axb$, is continuous. For every ordinal $\alpha \le \omega$, we describe all shift-continuous locally compact Hausdorff topologies on the $\alpha$-bicyclic monoid ${\mathcal{B}}_{\alpha }$. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on ${\mathcal{B}}_{\alpha }$ is anti-isomorphic to the segment of $[1,\alpha ]$ of ordinals, endowed with the natural well-order. Also we prove that for each ordinal $\alpha$ the $\alpha +1$-bicyclic monoid ${\mathcal{B}}_{\alpha +1}$ is isomorphic to the Bruck extension of the $\alpha$-bicyclic monoid ${\mathcal{B}}_{\alpha }$.