Cone topologies of paratopological groups

TL;DR

This paper introduces cone topologies for paratopological groups, providing a versatile method to construct counterexamples with specific compact-like properties and discontinuous inversion.

Contribution

It establishes a connection between algebraic properties of cone subsemigroups and the topological properties of the resulting cone topologies.

Findings

Cone topologies can produce compact-like paratopological groups with discontinuous inversion.

A simple relationship exists between algebraic properties of subsemigroups and topological properties of cone topologies.

The framework aids in constructing counterexamples in the study of paratopological groups.

Abstract

We introduce so-called cone topologies of paratopological groups, which are a wide way to construct counterexamples, especially of examples of compact-like paratopological groups with discontinuous inversion. We found a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two basic semigroup topologies generated by S on the group G.