Retracts of degenerate solutions of the Yang-Baxter equation

TL;DR

This paper extends the concept of multipermutation solutions to degenerate solutions of the Yang-Baxter equation, providing a new axiomatic framework and proving results in a universal algebra setting.

Contribution

It introduces a generalized definition of multipermutation solutions for degenerate cases and offers an axiomatic description that broadens the understanding of Yang-Baxter solutions.

Findings

Defined multipermutation solutions for degenerate cases

Provided an axiomatic set characterizing these solutions

Proved results in a universal algebra framework

Abstract

Most of the set-theoretical solutions of the Yang-Baxter equation studied in the past years were non-degenerate multipermutation solutions. For degenerate solutions, a correct definition of multipermutation solutions has not been established so far. We fill here this gap providing a definition of multipermutation solutions that generalizes the one for non-degenerate solutions and we find an axiomatic description of this class by a set of equations that generalizes the equations describing non-degenerate multipermutation solutions. It turned out that the results do not need all the properties of solutions of the Yang-Baxter equation and therefore we prove them in a general universal-algebraic setting.