On a topological simple Warne extension of a semigroup

TL;DR

This paper introduces and studies topological $bZ$-Bruck-Reilly and $bZ$-Bruck extensions of (semi)topological monoids, analyzing their topologizations and characterizations of certain classes of semigroups.

Contribution

It generalizes existing topological extensions of monoids to include $bZ$-Bruck-Reilly and $bZ$-Bruck extensions and provides conditions for their topological properties.

Findings

Conditions for the extension to admit only the direct sum topology.

Criteria for coarsening the direct sum topology.

Topological characterizations of $I$-bisimple (semi)topological semigroups.

Abstract

In the paper we introduce topological $\mathbb{Z}$-Bruck-Reilly and topological $\mathbb{Z}$-Bruck extensions of (semi)topological monoids which are generalizations of topological Bruck-Reilly and topological Bruck extensions of (semi)topological monoids and study their topologizations. The sufficient conditions under which the topological $\mathbb{Z}$-Bruck-Reilly ($\mathbb{Z}$-Bruck) extension admits only the direct sum topology and conditions under which the direct sum topology can be coarsened are given. Also, topological characterizations of some classes of $I$-bisimple (semi)topological semigroups are given.