On pseudocompact topological Brandt $\lambda^0$-extensions of semitopological monoids

TL;DR

This paper studies the topological properties of Brandt $ ext{λ}^0$-extensions of semitopological monoids with zero, establishing existence, uniqueness, and compactification results for various classes of such monoids.

Contribution

It proves the existence and uniqueness of pseudocompact, countably compact, and compact extensions of semitopological monoids with zero, and describes their Stone-ech and Bohr compactifications.

Findings

Existence of unique semiregular pseudocompact extensions for Tychonoff pseudocompact monoids

Characterization of Stone-ech and Bohr compactifications of these extensions

Description of a category of ingredients for constructing various compact extensions

Abstract

In the paper we investigate topological properties of a topological Brandt $\lambda^0$-extension $B^0_{\lambda}(S)$ of a semitopological monoid $S$ with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid $S$ with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension $B^0_{\lambda}(S)$ of $S$ and establish theirs Stone-\v{C}ech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt $\lambda^0$-extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.