On quiver skew braces, their ideals and products

TL;DR

This paper develops the theory of quiver skew braces by defining ideals, quotients, and semidirect products, revealing their richer structure compared to traditional groupoids.

Contribution

It introduces ideals, quotients, and two types of semidirect products for quiver skew braces, expanding their algebraic framework.

Findings

No decomposition of connected quiver skew braces into group of loops and vertices.

Defined ideals and quotients for quiver skew braces.

Established classical and categorical semidirect products for these structures.

Abstract

Quiver skew braces or skew bracoids are equivalent to braided groupoids, that is, groupoids with a constraint of abelianity. They are the quiver-theoretic version of skew braces, an increasingly studied structure lying in the intersection of group and ring theory. In this paper, we define ideals and quotients for quiver skew braces, with respect to two notions of morphisms. Following the track of a previous work of ours (2025), we define a classical semidirect product \`a la Brown, and a categorical semidirect product \`a la Bourn and Janelidze, for the category of quiver skew braces. It is known that connected groupoids can be expressed as the datum of a group of loops and a set of vertices. We demonstrate how no such decomposition holds for quiver skew braces, which makes their theory richer than the theory of groupoids.