On semigroups which admit only discrete left-continuous Hausdorff topology

Abstract

We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid $\mathscr{C}_{+}(a,b)$ (resp., $\mathscr{C}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_\alpha\colon \alpha\in\mathfrak{c}\}$ of continuum many subsemigroups with the following properties: $(i)$ every left-continuous (resp., right-continuous) Hausdorff topology on $S_\alpha$ is discrete; $(ii)$ every semigroup $S_\alpha$ admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); $(iii)$ every semigroup $S_\alpha$ isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid $\mathscr{C}_{\mathbb{Z}}^+$ (resp., $\mathscr{C}_{\mathbb{Z}}^-$) of the extended bicyclic semigroup which contains a family $\{S_\alpha\colon \alpha\in\mathfrak{c}\}$ of continuum many subsemigroups with the above described properties.