Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions of topological semigroups

TL;DR

This paper investigates the properties of topological semigroups related to matrix units and Brandt extensions, establishing conditions for $H$-closedness and describing the structure of countably compact extensions.

Contribution

It provides new criteria for $H$-closedness in topological semigroups of matrix units and characterizes countably compact Brandt $ ext{lambda}^0$-extensions of topological monoids.

Findings

A topological semigroup of finite partial bijections with a compact idempotent subsemigroup is absolutely $H$-closed.

Countably compact topological semigroups do not contain certain matrix unit semigroups as subsemigroups.

Conditions are given for $ ext{I}_ ext{lambda}^1$ to be non-$H$-closed.

Abstract

We show that a topological semigroup of finite partial bijections $\mathscr{I}_\lambda^n$ of an infinite set with a compact subsemigroup of idempotents is absolutely $H$-closed and any countably compact topological semigroup does not contain $\mathscr{I}_\lambda^n$ as a subsemigroup. We give sufficient conditions onto a topological semigroup $\mathscr{I}_\lambda^1$ to be non-$H$-closed. Also we describe the structure of countably compact Brandt $\lambda^0$-extensions of topological monoids and study the category of countably compact Brandt $\lambda^0$-extensions of topological monoids with zero.