On some generalization of the bicyclic semigroup: the topological version
Matija Cencelj, Oleg Gutik, Dušan Repovš
TL;DR
This work analyzes topologizations of the non-idempotent-free simple semigroup $\mathcal{C}=\langle a,b\mid a^2b=a,ab^2=b\rangle$ within semitopological and topological semigroup frameworks. It shows that any Hausdorff Baire topology rendering $\mathcal{C}$ semitopological must be discrete, while also constructing a nondiscrete, Tychonoff topology that makes $\mathcal{C}$ a topological semigroup. The paper further investigates closure properties and embeddings, proving that $\mathcal{C}$ cannot be embedded into any semigroup with a countably compact square, and provides an explicit extension that highlights limitations on forming compact-like closures. Together with a canonical map to the bicyclic semigroup, these results position $\mathcal{C}$ within the broader landscape of semigroup topology and clarify intrinsic obstructions to closure and embedding.
Abstract
We show that every Hausdorff Baire topology $τ$ on $\mathcal{C}=\langle a,b\mid a^2b=a, ab^2=b\rangle$ such that $(\mathcal{C},τ)$ is a semitopological semigroup is discrete and we construct a nondiscrete Hausdorff semigroup topology on $\mathcal{C}$. We also discuss the closure of a semigroup $\mathcal{C}$ in a semitopological semigroup and prove that $\mathcal{C}$ does not embed into a topological semigroup with the countably compact square.