On left braces in which every subbrace is an ideal II

TL;DR

This paper advances the understanding of Dedekind braces, focusing on conditions that ensure abelian structure and analyzing the case of multipermutational braces of level 2, with implications for solutions to the Yang-Baxter equation.

Contribution

It provides new sufficient conditions for Dedekind braces to be abelian and offers a structural theorem for multipermutational braces of level 2.

Findings

Dedekind braces with non-periodic additive groups can be abelian under certain nilpotency conditions.

Torsion-free socle additive groups imply abelian structure in hypermultipermutational braces.

Structural characterization of multipermutational braces of level 2.

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 $2$.