Skew braces, near-rings, skew rings, dirings

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, in the definition of a left near-ring $G$, to replace the multiplication of $G$ with two operations on the additive group $G$, 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.