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 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 $B_{\alpha }$.

References in corpus (6)

• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces
• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation

Cited by in corpus (1)

• Deformed solutions of the Yang-Baxter equation associated to dual weak braces