On a semitopological extended bicyclic semigroup with adjoined zero

TL;DR

The paper investigates the topological structures of the extended bicyclic semigroup with zero, showing discreteness of certain topologies and constructing unique minimal topologies among many possible Hausdorff locally compact shift-continuous topologies.

Contribution

It establishes the discreteness of all Hausdorff locally compact semigroup topologies on the semigroup and constructs unique minimal topologies, revealing the rich topological diversity.

Findings

All Hausdorff locally compact semigroup topologies are discrete.

Existence of c5c5c5 many different Hausdorff locally compact shift-continuous topologies.

Construction of the unique minimal shift continuous and inverse semigroup topologies.

Abstract

In the paper it is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many different Hausdorff locally compact shift-continuous topologies. Also, it is constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift continuous topology and the unique minimal inverse semigroup topology.