On semitopological α-bicyclic monoid

Анотація

We consider a semitopological $\alpha$-bicyclic monoid ${\mathcal{B}}_{\alpha }$ and prove that it is algebraically isomorphic to the semigroup of all order isomorphisms between the principal upper sets of ordinal ${\omega }^{\alpha }$. We prove that for every ordinal $\alpha$ and every $(a,b)\in {\mathcal{B}}_{\alpha }$ if $a$ 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. On the other hand, we construct an example of a non-discrete locally compact topology ${\tau }_{lc}$ on ${\mathcal{B}}_{\omega +1}$ such that $\left({\mathcal{B}}_{\omega +1},{\tau }_{lc}\right)$ is a topological inverse semigroup. This example shows that there is a gap in the proof of Theorem 2.9 [16]  stating that for every ordinal $\alpha$ the semigroup ${\mathcal{B}}_{\alpha }$ does not admit non-discrete locally compact inverse semigroup topologies.