… of left braces by ideals with trivial structure. This … left braces. To explicitly construct them, we
define the matched product of two left braces, which is a useful method to recover a left brace …

… Given a skew left brace B , we introduce the notion of an “opposite” skew left brace B ′ ,
which is closely related to the concept of the opposite of a group, and provide several …

… A new class of left braces, the extraspecial left braces, is introduced and play a … a left brace.
Then, it is abundantly clear that the more we know about the algebraic structure of left braces …

… For (a), we prove some general results about the possible additive group of a left brace of …
So we settle (b) for some p without using a computer. However, observe that the argument …

… We refer to the compatibility condition (1.1) as the brace distributive law. A skew left brace
is called simply a left brace provided that (A, ) is an Abelian group. Since their introduction and …

… some technical troubles when we define the actions of H on I. Note that the socle of B is a
trivial brace, and that there are left braces … to construct new left braces from two given braces, …

… In this paper we study left braces B with some special conditions … a left brace B satisfies lri
if the associated solution ( B , r B ) satisfies lri, that is ( b a ) a = b , for all a , b ∈ B . A left brace …

… factorisations of skew left braces and their impact on the skew left brace structure. As a …
ideal of a left brace. Our approach depends strongly on a description of a skew left brace in terms …

… of finite simple left braces. To explicitly construct them, we define the matched product of
two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups. …

… in the associated solution of a particular left brace. Hence, left braces are an appropriate
tool to study the YBE. In Chapter 1, we study the structure of left brace in order to enlarge our …