Inverse semi-braces and the Yang-Baxter equation

TL;DR

This paper introduces inverse semi-braces, a new algebraic structure, to find set-theoretical solutions to the Yang-Baxter equation, including non-bijective and idempotent solutions, expanding the scope of existing methods.

Contribution

It proposes inverse semi-braces as a novel framework for constructing solutions to the Yang-Baxter equation, extending the classical brace theory and exploring non-bijective solutions.

Findings

Constructed new inverse semi-braces for solution generation

Provided solutions that differ from previously known ones

Expanded the algebraic toolkit for Yang-Baxter solutions

Abstract

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.