A Survey of Schreier-Type Extensions of Monoids

TL;DR

This survey reviews Schreier-type extensions of monoids, exploring their characterisations, relations, and relaxations, providing a unified framework for understanding various extension types in algebraic structures.

Contribution

It offers a comprehensive overview of Schreier extensions of monoids, including new characterisations and a unified approach using relaxed actions.

Findings

Characterisations of split extensions of groups

Extensions with abelian kernels

Unified account via relaxed actions

Abstract

We give an overview of a number of Schreier-type extensions of monoids and discuss the relation between them. We begin by discussing the characterisations of split extensions of groups, extensions of groups with abelian kernel and finally non-abelian group extensions. We see how these characterisations may be immediately lifted to Schreier split extensions, special Schreier extensions and Schreier extensions respectively. Finally, we look at weakenings of these Schreier extensions and provide a unified account of their characterisation in terms of relaxed actions.