# Near braces and p-deformed braided groups

## TL;DR

This paper introduces near braces and p-deformed braided groups to generate new solutions to the Yang-Baxter equation, expanding the algebraic structures used in quantum algebra and integrable systems.

## Contribution

It reconstructs near braces from recent findings, introduces p-deformed braided groups, and produces new multi-parametric solutions to the set-theoretic Yang-Baxter equation.

## Key findings

- New multi-parametric, non-involutive solutions to the Yang-Baxter equation.
- Introduction of near braces as a generalization of braces and skew braces.
- Definition of p-deformed braided groups and their relation to braid solutions.

## Abstract

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.