On non-topologizable semigroups
Oleg Gutik
TL;DR
The paper addresses non-topologizable behavior in substructures of the bicyclic monoid, focusing on the anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$. It shows that any Hausdorff left-continuous topology on $\mathscr{C}_{+}(a,b)$ (respectively right-continuous topology on $\mathscr{C}_{-}(a,b)$) is necessarily discrete, while also constructing a compact Hausdorff topological monoid $S$ in which one of these submonoids embeds densely by adjoining a zero. The authors also exhibit explicit non-discrete one-sided topologies $\tau_p^+$ and $\tau_p^-$ that fail the opposite continuity, demonstrating limits of topologization in this setting. These results advance understanding of non-discrete topologizations and embeddings of semigroups into compact-like topological semigroups and answer questions about universal one-sided continuity for these anti-isomorphic submonoids.
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 $τ_p^+$ ($τ_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).