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

Francesco Catino, Marzia Mazzotta, Paola Stefanelli

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 $B_α$, with $α\in Y$. Additionally, we describe the ideals of $S$ and study its nilpotency by correlating it to that of each skew brace $B_α$.

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 we provide in terms of strong semilattice of skew braces , with . Additionally, we describe the ideals of and study its nilpotency by correlating it to that of each skew brace .
Paper Structure (6 sections, 35 theorems, 48 equations)

This paper contains 6 sections, 35 theorems, 48 equations.

Table of Contents

  1. Basics on weak braces
  2. A description of dual weak braces and their solutions
  3. Ideals of dual weak braces
  4. Left ideals of dual weak braces
  5. Right nilpotent dual weak braces
  6. Annihilator nilpotency of dual weak braces

Key Result

Lemma 1.3

Let $(S,+, \circ)$ be a weak brace. Then, the following hold: for all $a,b \in S$.

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Proposition 1.7
  • proof
  • ...and 63 more