Skew lattices and set-theoretic solutions of the Yang-Baxter equation

TL;DR

This paper explores the connection between skew lattices and set-theoretic solutions of the Yang-Baxter equation, providing new characterizations and constructions for these solutions.

Contribution

It introduces the first known relation between skew lattices and the Yang-Baxter equation, offering a description of solutions from arbitrary skew lattices and a method to construct specific skew lattices.

Findings

Set-theoretic solutions derived from skew lattices are generally degenerate.

A main result describes solutions obtained from any skew lattice.

A construction method for cancellative and distributive skew lattices is provided.

Abstract

In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in general, and thus different from solutions obtained from braces and other algebraic structures. Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.