Quotient space of $\mathcal{LMC}$-compactification as a space of $z-$filters

TL;DR

This paper presents an internal construction of the semigroup compactification of a semitopological semigroup using a space of filters, complementing previous external construction methods.

Contribution

It introduces a novel internal approach to semigroup compactification via $z$-filters, expanding the theoretical framework for semitopological semigroup analysis.

Findings

Constructed a space of $z$-filters as a semigroup compactification

Provided an internal perspective complementing external methods

Enhanced understanding of $ ext{LMC}$-compactification structure

Abstract

The left multiplicative continuous compactification of a semitopological semigroup is the universal semigroup compactification. In this paper an internal construction of a semigroup compactification of a semitopological semigroup is constructed as a space of filters. In \cite{Akbari}, we described an external construction of a semigroup compactification of a semitopological semigroup.