Free Skew Braces and Free Solutions of the Yang--Baxter Equation
We offer a workable construction of the free right nilpotent skew braces of arbitrary class which allows us to prove (among many other things) that this free object has free additive/multiplicative groups, and that it must also be residually finite and Hopfian. We introduce the class of right nilpotent solutions, which correspond to right nilpotent skew braces. As a consequence of our construction, the free solutions in this class have a solvable Word Problem, and every law holding for finite solutions of the previous type also holds for every solution of the same type. In the remainder of the paper, we present further explicit realizations of free objects and explore their consequences. Among these are free two-sided skew braces of abelian type (with an abelian multiplicative group) and free centrally nilpotent skew braces of class 2.