Weakly Schreier extensions for general algebras

TL;DR

This paper extends the concept of weakly Schreier split extensions from monoids to general algebraic structures, introducing a parameterized framework that unifies and broadens existing extension classes.

Contribution

It provides a new, parameterized approach to defining and studying split extensions across various algebraic structures, generalizing known classes like weakly Schreier extensions.

Findings

Unified framework for split extensions in general algebras

Introduction of a parameter θ to characterize extensions

Discovery of a more general class of monoid extensions

Abstract

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term $\theta$). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the $\theta$ appearing in their syntactical characterisation). Restricting again to the case of monoids, a different choice of $\theta$ leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.