On topologization of the extended bicyclic semigroup
Oleg Gutik, Marharyta Zolotar, Oleksandra Lysetska
TL;DR
The paper investigates topologizations of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$, addressing the construction of non-discrete $T_1$ topologies and providing comprehensive discreteness criteria for shift-continuous topologies. It presents explicit non-discrete topologies $\tau_1$ (non-discrete $T_1$ inverse semigroup), $\tau_2$ (locally compact $T_1$ inverse semigroup via $V_n(i,j)$-neighborhoods), and $\tau_{\mathrm{B}}$ (non-discrete left-topological $T_1$ topology generated by a base $\mathscr{P}_{\mathrm{B}}$ with components $\mathscr{P}_0,\mathscr{P}_1,\mathscr{P}_2,\mathscr{P}_3$); it also analyzes the dual topology $\tau_{\mathrm{B}}^{\mathrm{d}}$, quasi-regularity, and the (non-)existence of countably compact non-discrete left/right topologies. A central contribution is a complete characterization of when a left or right $T_1$ topology on $\mathscr{C}_{\mathbb{Z}}$ is discrete, namely that discreteness is equivalent to the existence of a sequence of isolated points with strictly monotone coordinates. These results illuminate how topological properties interact with the semigroup operations and translations in the extended bicyclic setting, with implications for topologization problems in inverse semigroups and related structures.
Abstract
Non-discrete semigroup $T_1$-topologies on the extended bicyclic semigroup $\mathscr{C}_\mathbb{Z}$ are constructed. Also, we present topological conditions, when a semigroup (shift-continuous) $T_1$-topology on $\mathscr{C}_\mathbb{Z}$ is discrete.