The retraction relation for biracks

TL;DR

This paper generalizes the concept of retraction from involutive solutions of the Yang-Baxter equation to all solutions using biracks, proving that the retraction relation is a congruence in this broader context.

Contribution

It introduces the generalized retraction relation on biracks and proves it is a congruence, extending the retraction concept beyond involutive solutions.

Findings

The generalized retraction relation is a congruence on biracks.

Retraction is well-defined for all non-degenerate solutions, not just involutive.

Provides a complete algebraic proof of the congruence property.

Abstract

In {\it Set-theoretical solutions to the quantum Yang-Baxter equation} (Duke Math. J. {\bf 100} (1999), 169--209), Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution $(X,\sigma,\tau)$ of the Yang-Baxter equation, the equivalence relation $\sim$ defined on the set $X$ and they considered a new non-degenerate involutive induced \emph{retraction} solution defined on the quotient set $X^{\sim}$. It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra called a \emph{birack}. In the paper we introduce the \emph{generalized retraction} relation $\approx$ on a birack, which is equal to $\sim$ in an involutive case. We present a complete algebraic proof that the relation $\approx$ is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions.