Split epimorphims and Baer sums of left skew braces

TL;DR

This paper explores the structure of split epimorphisms in categories of algebraic objects called digroups and left skew braces, revealing that left skew braces form a strongly protomodular category and analyzing their Baer sums.

Contribution

It demonstrates that the category of left skew braces is strongly protomodular and characterizes Baer sums of exact sequences with abelian kernels.

Findings

Left skew braces form a strongly protomodular category.

Baer sums of exact sequences with abelian kernels are described.

Distinct properties of split epimorphisms in digroups and skew braces are analyzed.

Abstract

We investigate the split epimorphisms in the categories of digroups and left skew braces. We show that, unlike the category DiGp of digroups, the category SkB of left skew braces is strongly protomodular. From that, we describe the expected Baer sums of exact sequences of left skew braces with abelian kernel.