Near braces and p-deformed braided groups

Motivated by recent findings on the derivation of parametric noninvolutive solutions of the Yang-Baxter equation, we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi-parametric, nondegenerate, noninvolutive solutions of the set-theoretic Yang-Baxter equation. These solutions are generalizations of the known ones coming from braces and skew braces. Bijective maps associated to the inverse solutions are also constructed. Furthermore, we introduce the generalized notion of p-deformed braided groups and p-braidings and we show that every p-braiding is a solution of the braid equation. We also show that certain multi-parametric maps within the near braces provide special cases of p-braidings.