On semigroups which admit only discrete left-continuous Hausdorff topology

We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid ${\mathcal{C}}_{+}(a,b)$ (resp., ${\mathcal{C}}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_{\alpha }\text{\hspace{0.17em}⁣}:\alpha \in \mathfrak{c}\}$ 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 ${\mathcal{C}}_{\mathbb{Z}}^{+}$ (resp., ${\mathcal{C}}_{\mathbb{Z}}^{-}$) of the extended bicyclic semigroup which contains a family $\{S_{\alpha }\text{\hspace{0.17em}⁣}:\alpha \in \mathfrak{c}\}$ of continuum many subsemigroups with the above described properties.