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

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 \(B_\alpha \) $B_{\alpha }$ , with \(\alpha \in Y\) $\alpha \in Y$ . Additionally, we describe the ideals of S and study its nilpotency by correlating it to that of each skew brace \(B_\alpha \) $B_{\alpha }$ .