On topologization of subsemigroups of the bicyclic monoid

TL;DR

This paper investigates topological structures on subsemigroups of the bicyclic monoid, showing that certain infinite idempotent-containing subsemigroups and Baire topologies must be discrete, revealing limitations on their topologization.

Contribution

It establishes that subsemigroups with infinitely many idempotents and certain Baire topologies on bicyclic monoids are necessarily discrete, advancing understanding of their topological properties.

Findings

Subsemigroups with infinitely many idempotents admit only discrete Hausdorff topologies.

Right- and left-continuous Baire topologies on certain bicyclic monoids are necessarily discrete.

The results restrict possible topologizations of these algebraic structures.

Abstract

We show that if a subsemigroup $S$ of the bicyclic monoid ${\mathscr{C}}(p,q)$ contains infinitely many idempotents then $S$ admits only the discrete Hausdorff shift-continuous topology. Also we proof that every right-continuous (left-continuous\emph) Hausdorff Baire topology on the semigroup $\mathscr{C}_+(a,b)$ $(\mathscr{C}_-(a,b))$ is discrete and the same statement holds for the bicyclic monoid.