On semitopological interassociates of the bicyclic monoid

Анотація

Semitopological interassociates ${\mathcal{C}}_{m,n}$ of the bicyclic semigroup $\mathcal{C}(p,q)$ are studied. In particular, we show that for arbitrary non-negative integers $m, n$ and every Hausdorff topology $\tau$ on ${\mathcal{C}}_{m,n}$ such that $\left({\mathcal{C}}_{m,n},\tau \right)$ is a semitopological semigroup, is discrete. Also, we prove that if an interassociate of the bicyclic monoid ${\mathcal{C}}_{m,n}$ is a dense subsemigroup of a Hausdorff semitopological semigroup $\left(S,·\right)$ and $I=S\backslash {\mathcal{C}}_{m,n}\ne \varnothing$ then $I$ is a two-sided ideal of the semigroup $S$ and show that for arbitrary non-negative integers $m, n$, any Hausdorff locally compact semitopological semigroup ${\mathcal{C}}_{m,n}^0={\mathcal{C}}_{m,n}\bigsqcup \left\{0\right\}$ is either discrete or compact.