Solutions of the Yang–Baxter Equation and Strong Semilattices of Skew Braces

Abstract

We prove that any set-theoretic solution of the Yang–Baxter equation associated with 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 $$B_\alpha $$ B α , with $$\alpha \in Y$$ α ∈ Y . Additionally, we describe the ideals of S and study its nilpotency by correlating it to that of each skew brace $$B_\alpha $$ B α .