Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation

TL;DR

This paper develops the theory of Rota-Baxter operators on Clifford semigroups to construct solutions to the Yang-Baxter equation, revealing structural properties and methods for dual weak braces.

Contribution

It introduces Rota-Baxter operators on Clifford semigroups and connects them to solutions of the Yang-Baxter equation, advancing the structural understanding of dual weak braces.

Findings

Methods for constructing dual weak braces

Structural insights into Clifford semigroups

Set-theoretic solutions close to bijectivity

Abstract

In this paper, we introduce the theory of Rota-Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples $\left(S,+,\circ\right)$ where $\left(S,+\right)$ and $\left(S,\circ\right)$ are Clifford semigroups such that $a\circ\left(b+c\right) = a\circ b - a +a\circ c$ and $a\circ a^- = -a+a$, for all $a,b,c\in S$. To each algebraic structure is associated a set-theoretic solution of the Yang-Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal.