On semigroups which admit only discrete left-continuous Hausdorff topology

We give a sufficient condition under which every left-continuous (right-continuous) Hausdorff topology on a semigroup S is discrete. We construct a submonoid C+(a, b) (resp., C−(a, b)) of the bicyclic monoid which contains a family {Sα : α ∈ c} of continuum many subsemigroups with the following properties: (i) every left-continuous (resp., right-continuous) Hausdorff topology on Sα is discrete; (ii) every semigroup Sα admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); (iii) every semigroup Sα isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid C+ℤ (resp., C−ℤ) of the extended bicyclic semigroup which contains a family {Sα : α ∈ c} of continuum many subsemigroups with the above described properties.