On the cabling of non-involutive set-theoretic solutions of the Yang--Baxter equation

TL;DR

This paper generalizes the cabling method for set-theoretic solutions of the Yang--Baxter equation from involutive to non-involutive cases by working within the monoid structure, leading to new criteria for decomposability.

Contribution

It extends the cabling approach to non-involutive solutions using the monoid, ensuring functoriality and preserving decomposability, which was not previously established.

Findings

Cabling is functorial on biquandles.

Diagonal map transforms as q↦q^k.

Square-free solutions with nilpotent derived monoid are decomposable.

Abstract

We extend the cabling method by Lebed, Ram\'irez and Vendramin from involutive to bijective non-degenerate set-theoretic solutions of the Yang--Baxter equation by working in the Yang--Baxter monoid $M(X,r)$ rather than the group $G(X,r)$. This shift in approach overcomes the obstruction that, for non-involutive solutions, the canonical map from $X$ to the Yang--Baxter group $G(X,r)$ need not be injective and yields a well-defined cabling. We prove that cabling is functorial on biquandles and that the diagonal map transforms as $q\mapsto q^k$. We also show that decomposability is preserved by injectivization and by passing to the associated biquandle, allowing us to work within that class without loss of generality. This leads to criteria for (in)decomposability. As an application, we obtain that square-free solutions with nilpotent derived monoid are decomposable.