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On left braces in which every subbrace is an ideal II

A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, V. Pérez-Calabuig

TL;DR

This work extends the study of Dedekind left braces to the non-periodic additive setting, establishing practical abelianity criteria and a detailed structural description for multipermutational level 2. It shows that if every element is $2$-nilpotent under the star operation, or if a hypermultipermutational Dedekind brace has a torsion-free socle, then the brace is abelian, with implications for corresponding YBE twist solutions. The authors connect these results to set-theoretic Yang–Baxter solutions and provide a comprehensive decomposition framework for non-periodic, mp–level-2 Dedekind braces, highlighting how socle layers govern global structure and abelianity. Collectively, the results contribute to the classification of YBE solutions through tractable algebraic criteria and a clear structural picture of non-periodic Dedekind braces.

Abstract

The aim of this paper is to take the study of Dedekind braces, that is, left braces for which every subbrace is an ideal, started in a previous paper, further. Dedekind braces $A$ whose additive group is non-periodic are analysed. We prove sufficient conditions for $A$ to be abelian: it is enough that every element is $2$-nilpotent for the star operation; and, if $A$ is hypermultipermutational, it suffices that the additive group of the socle is torsion-free. Both conditions can be translated in terms of set-theoretical solutions of the Yang-Baxter equation. In addition, we prove a structural theorem for the case of $A$ to be a multipermutational brace of level $2$.

On left braces in which every subbrace is an ideal II

TL;DR

This work extends the study of Dedekind left braces to the non-periodic additive setting, establishing practical abelianity criteria and a detailed structural description for multipermutational level 2. It shows that if every element is -nilpotent under the star operation, or if a hypermultipermutational Dedekind brace has a torsion-free socle, then the brace is abelian, with implications for corresponding YBE twist solutions. The authors connect these results to set-theoretic Yang–Baxter solutions and provide a comprehensive decomposition framework for non-periodic, mp–level-2 Dedekind braces, highlighting how socle layers govern global structure and abelianity. Collectively, the results contribute to the classification of YBE solutions through tractable algebraic criteria and a clear structural picture of non-periodic Dedekind braces.

Abstract

The aim of this paper is to take the study of Dedekind braces, that is, left braces for which every subbrace is an ideal, started in a previous paper, further. Dedekind braces whose additive group is non-periodic are analysed. We prove sufficient conditions for to be abelian: it is enough that every element is -nilpotent for the star operation; and, if is hypermultipermutational, it suffices that the additive group of the socle is torsion-free. Both conditions can be translated in terms of set-theoretical solutions of the Yang-Baxter equation. In addition, we prove a structural theorem for the case of to be a multipermutational brace of level .
Paper Structure (6 sections, 23 theorems, 47 equations)

This paper contains 6 sections, 23 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proof of Theorem \ref{['ateo:2.11nou']}
  5. Proof of Theorem \ref{['ateo:Ahypersoc_torsionfreee->Aabelian']}
  6. Proof of Theorem \ref{['ateo:2.17nou']}

Key Result

Lemma 1

Let $a \in A$ such that $a\ast a = 0$. Then, $\langle a \rangle = \langle a \rangle_+ = \langle a \rangle_{\boldsymbol{\cdot}}$ is an abelian subbrace of $A$.

Theorems & Definitions (43)

  • Lemma 1: BallesterEstebanKurdachenkoPerezC-dedekind-leftbraces
  • Proposition 2: BallesterFerraraPerezCTrombetti23-arXiv-note
  • Lemma 3: BallesterEstebanKurdachenkoPerezC-dedekind-leftbraces
  • Proposition 4
  • proof
  • Theorem A
  • Corollary 5
  • Theorem B
  • Corollary 6
  • Theorem C
  • ...and 33 more