Semigroup compactifications in terms of filters

TL;DR

This paper explores the structure of semigroup compactifications of semitopological semigroups using filters, providing characterizations of subsemigroups, ideals, and special points within these compactifications.

Contribution

It introduces a filter-based framework to characterize substructures and specific points in semigroup compactifications, extending previous algebraic and topological methods.

Findings

Characterization of closed subsemigroups and ideals via filters.

Identification of points belonging to the smallest ideal.

Connection between filters and ideals of $m$-admissible subalgebras.

Abstract

We present a study of semigroup compactifications of a semitopological semigroup $S$ using certain filters on $S$. We characterize closed subsemigroups and closed left, right, and two-sided ideals in any semigroup compactification of any semitopological semigroup $S$ in terms of these filters and in terms of ideals of the corresponding $m$-admissible subalgebra of $C(S)$. Furthermore, we characterize those points in any semigroup compactification of $S$ which belong either to the smallest ideal of the semigroup compactification or to the closure of this smallest ideal.