Asymmetric product of left braces and simplicity; new solutions of the Yang-Baxter equation

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation (YBE) recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the YBE. In this paper, we continue the study of finite simple left braces with the emphasis on the application of the asymmetric product of left braces in order to construct new classes of simple left braces. We do not only construct new classes but also we interpret all previously known constructions as asymmetric products. Moreover, a construction is given of finite simple left braces with a multiplicative group that is solvable of arbitrary derived length.