Solutions of the Yang-Baxter equation and strong semilattices of skew braces

TL;DR

This paper demonstrates that solutions to the Yang-Baxter equation linked to dual weak braces can be decomposed into strong semilattices of non-degenerate solutions, connecting algebraic structures to solution properties.

Contribution

It introduces a novel description of dual weak braces as strong semilattices of skew braces and explores their ideals and nilpotency.

Findings

Solutions form strong semilattices of non-degenerate bijective solutions.

Ideals of the solutions are characterized in terms of the underlying skew braces.

Nilpotency of solutions is linked to the nilpotency of constituent skew braces.

Abstract

We prove that any set-theoretic solution of the Yang-Baxter equation associated to a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace $S$ we provide in terms of strong semilattice $Y$ of skew braces $B_\alpha$, with $\alpha \in Y$. Additionally, we describe the ideals of $S$ and study its nilpotency by correlating it to that of each skew brace $B_\alpha$.