Filters and Semigroup Compactification Properties

TL;DR

This paper explores the role of filters in the study of semigroup compactifications, extending known results from Stone-Cech compactifications to more general semigroup compactifications.

Contribution

It reviews filter characterizations of semigroup compactifications and extends results from Stone-Cech to Hausdorff semitopological semigroups.

Findings

Filters are crucial in semigroup compactification analysis.

Extended results from Stone-Cech to Hausdorff semitopological semigroups.

Provided new characterizations of semigroup compactifications.

Abstract

Stone-$\breve{C}$ech compactifications derived from a discrete semigroup $S$ can be considered as the spectrum of the algebra $\mathcal{B}(S)$ or as a collection of ultrafilters on $S$. What is certain and indisputable is the fact that filters play an important role in the study of Stone-$\breve{C}$ech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too. In the present paper, first we review the characterizations of semigroup compactifications in terms of filters and then extend some of the results in \cite{Talin1} concerning the Stone-$\breve{C}$ech compactification to a semigroup compactification associated with a Hausdorff semitopological semigroup.