On topologization of the extended bicyclic semigroup

TL;DR

This paper explores the topological structures on the extended bicyclic semigroup, constructing non-discrete $T_1$-topologies and identifying conditions for discreteness in shift-continuous topologies.

Contribution

It introduces new non-discrete $T_1$-topologies on the extended bicyclic semigroup and characterizes when such topologies are necessarily discrete.

Findings

Construction of non-discrete $T_1$-topologies on $\, ext{extended bicyclic semigroup}$

Topological conditions for discreteness in shift-continuous $T_1$-topologies

Characterization of when a $T_1$-topology on $\, ext{extended bicyclic semigroup}$ is discrete

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.