Set-theoretic solutions to the Yang-Baxter equation, skew-braces, and related near-rings

摘要

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang-Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang-Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field K are characterized as K-braces with a compatible near-ring structure. Under this correspondence, K-braces are radicals of local near-rings with radical square zero.