On semigroups which admit only discrete left-continuous Hausdorff topology

TL;DR

This paper characterizes conditions under which semigroups admit only discrete left- or right-continuous Hausdorff topologies, and constructs specific submonoids with diverse topological properties and embeddings into compact semigroups.

Contribution

It provides sufficient conditions for semigroups to have only discrete Hausdorff topologies and constructs submonoids with rich topological and embedding properties.

Findings

Constructed submonoids with continuum many subsemigroups.

Identified conditions for all left- or right-continuous Hausdorff topologies to be discrete.

Demonstrated embeddings into Hausdorff compact topological semigroups.

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.