Skew braces, near-rings, skew rings, dirings

TL;DR

This paper introduces a novel algebraic perspective on set-theoretic solutions to the Yang-Baxter equation by defining structures like left skew braces, dirings, and skew rings, emphasizing dual operations.

Contribution

It presents a new approach to classical algebraic notions by replacing single multiplication with two operations, leading to the discovery of related structures and their categorical equivalences.

Findings

Categories of left skew rings and left weak rings are canonically isomorphic.

New algebraic structures provide alternative frameworks for Yang-Baxter solutions.

The approach unifies existing concepts under a dual-operation paradigm.

Abstract

We introduce a new point of view to present classical notions related to set-theoretic solutions of the Yang-Baxter equation: left skew braces, dirings, left skew rings. The idea is to replace the single multiplication on a left near-ring by two operations, one associative and the other left distributive. Two algebraic structures naturally appear: left skew rings and left weak rings, whose categories turn out to be canonically isomorphic.