ON THE DICHOTOMY OF A LOCALLY COMPACT SEMITOPOLOGICAL BICYCLIC MONOID WITH ADJOINED ZERO

Анотація

We prove that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero ${\mathcal{C}}^0$ is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological bicyclic semigroup with an adjoined compact ideal and construct an example which witnesses that a counterpart of the statements does not hold when ${\mathcal{C}}^0$ is a $\stackrel{\vee }{C}$ech-complete metrizable topological inverse semigroup.