On left braces in which every subbrace is an ideal

TL;DR

This paper introduces Dedekind and extraspecial left braces, exploring their structure, properties, and nilpotency, with implications for the classification of certain algebraic objects related to braces.

Contribution

It defines Dedekind and extraspecial left braces, proves finite Dedekind braces are centrally nilpotent, and characterizes those with elementary abelian additive groups.

Findings

Finite Dedekind left braces are centrally nilpotent.

Complete description of Dedekind braces with elementary abelian additive groups.

Hypermultipermutational Dedekind braces with elementary abelian groups are of level 2.

Abstract

The aim of this paper is to introduce and study the class of all left braces in which every subbrace is an ideal. We call them Dedekind left braces. It is proved that every finite Dedekind left brace is centrally nilpotent. Structural results about Dedekind left braces and a complete description of those ones whose additive group is elementary abelian are also shown. As a consequence, every hypermultipermutational Dedekind left brace whose additive group is elementary abelian is multipermutational of level $2$. A new class of left braces, the extraspecial left braces, is introduced and plays a prominent role in our approach.