Aspects of the Category SKB of Skew Braces

TL;DR

This paper explores the algebraic structure of the category SKB of left skew braces, analyzing commutators, ideals, and centralizers to deepen understanding of their categorical properties.

Contribution

It introduces new results on the commutator of ideals, shows Huq=Smith equivalence, and provides generators for ideal commutators in left skew braces.

Findings

Huq=Smith holds for left skew braces

A set of generators for the commutator of two ideals is provided

Every ideal has a centralizer in the category SKB

Abstract

We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.