On the Brandt $\lambda^0$-extensions of monoids with zero

TL;DR

This paper investigates the algebraic and topological properties of Brandt -extensions of monoids with zero, including their structure, homomorphisms, and topological variants like compact and countably compact extensions.

Contribution

It introduces and analyzes finite, compact, and countably compact topological Brandt -extensions, and describes the category of their construction ingredients.

Findings

Characterization of algebraic properties of Brandt -extensions.

Description of continuous homomorphisms between topological extensions.

Structure theorems for finite, compact, and countably compact topological extensions.

Abstract

We study algebraic properties of the Brandt $\lambda^0$-extensions of monoids with zero and non-trivial homomorphisms between the Brandt $\lambda^0$-extensions of monoids with zero. We introduce finite, compact topological Brandt $\lambda^0$-extensions of topological semigroups and countably compact topological Brandt $\lambda^0$-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt $\lambda^0$-extensions of topological monoids with zero. We also describe a category whose objects are ingredients in the constructions of finite (compact, countably compact) topological Brandt $\lambda^0$-extensions of topological monoids with zeros.