  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.
