# Split extensions and semidirect products of unitary magmas

## TL;DR

This paper develops a theory of split extensions and semidirect products for unitary magmas, establishing categorical equivalences and exploring properties like pullback stability.

## Contribution

It introduces a formal framework for split extensions of unitary magmas and characterizes them via semidirect products, including subclasses with desirable stability properties.

## Key findings

- Split extensions are pullback stable.
- Not all split extensions are closed under composition.
- Two subclasses of split extensions are identified with better stability.

## Abstract

We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) actions of unitary magmas on unitary magmas. The class of split extensions is pullback stable but not closed under composition. We introduce two subclasses of it that have both of these properties.