Structural theorem for Dedekind left braces

Establish a structural theorem that characterizes Dedekind left braces, namely left braces in which every subbrace is an ideal.

Background

The paper introduces Dedekind left braces, defined as left braces for which every subbrace is an ideal, motivated by the analogy with Dedekind groups. The authors emphasize the relevance of understanding their structure for the classification of set-theoretic solutions of the Yang–Baxter equation.

They show that every finite Dedekind left brace is centrally nilpotent and obtain complete structural results for the subclass whose additive group is an elementary abelian p-group, introducing extraspecial left braces as a key tool. However, a general structural description beyond these cases is not provided, which they indicate as an open problem.