Quotients of monoid extensions and their interplay with Baer sums

TL;DR

This paper investigates cosetal extensions of monoids, exploring their structure, relationships with cohomology, and how they form inverse semigroups and categories, extending concepts from group theory.

Contribution

It introduces a detailed analysis of cosetal monoid extensions, relating them to second cohomology and constructing inverse semigroups and categories from their structures.

Findings

Characterization of the category of cosetal extensions

Relation to second cohomology groups

Construction of inverse semigroups and categories from extensions

Abstract

Cosetal extensions of monoids generalise extensions of groups, special Schreier extensions of monoids and Leech's normal extensions of groups by monoids. They share a number of properties with group extensions, including a notion of Baer sum when the kernel is abelian. However, unlike group extensions (with fixed kernel and cokernel) there may be nontrivial morphisms between them. We explore the structure of the category of cosetal extensions and relate it to an analogue of second cohomology groups. Finally, the order structure and additive structures are combined to give an indexed family of inverse semigroups of extensions. These in turn can be combined into an inverse category.