Openly factorizable spaces and compact extensions of topological semigroups

TL;DR

This paper demonstrates that the semigroup operation of a pseudocompact openly factorizable space extends continuously to its Stone- compactification, providing a spectral characterization and exploring properties of such spaces.

Contribution

It introduces the concept of openly factorizable spaces and proves the extension of semigroup operations to their Stone- compactifications, offering new insights into their structure.

Findings

Semigroup operation extends to  compactification for openly factorizable spaces.

Spectral characterization of openly factorizable spaces.

Properties of openly factorizable spaces are established.

Abstract

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-\v{C}ech compactification $\beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.