  Motivated by recent findings on the derivation of parametric non-involutive
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, non-degenerate, non-involutive solutions of the
set-theoretic Yang-Baxter equation. These solutions are generalisations 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.
