On non-topologizable semigroups

TL;DR

This paper explores the topological properties of certain submonoids of the bicyclic monoid, demonstrating the existence of both discrete and non-discrete topologies, and constructing specific examples with unique continuity properties.

Contribution

It introduces anti-isomorphic submonoids of the bicyclic monoid with unique topological characteristics and constructs non-discrete topologies that are not fully continuous.

Findings

Every Hausdorff left-continuous topology on _{+}(a,b) is discrete.

Existence of a compact Hausdorff topological monoid containing _{+}(a,b) as a submonoid.

Construction of non-discrete right-continuous topology _p^+ on _{+}(a,b).

Abstract

We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$ ($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).