Set-theoretic type solutions of the braid equation

TL;DR

This paper introduces a new framework for set-theoretic solutions to the braid equation, connecting various algebraic structures and extending existing theories in the field.

Contribution

It develops a unified theory that encompasses set-theoretical solutions, q-cycle sets, q-braces, skew-braces, and related algebraic structures.

Findings

Relationships between key algebraic structures are preserved in the new setting.

Set-theoretical solutions are included as fundamental examples.

The theory broadens the understanding of solutions to the braid equation.

Abstract

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible $1$-cocycles remain valid in our setting.