Set-theoretical solutions to the Hom-Yang-Baxter equation and Hom-cycle sets

TL;DR

This paper explores set-theoretic solutions to the Hom-Yang-Baxter equation, introducing Hom-cycle sets and establishing their properties and relations to existing algebraic structures, filling a gap in the current research.

Contribution

It introduces Hom-cycle sets and characterizes solutions to the HYBE, expanding the algebraic framework for set-theoretic solutions to this equation.

Findings

Characterization of left non-degenerate involutive solutions to HYBE

Establishment of relations between Hom-cycle sets and cycle sets

Connections drawn among Hom-cycle sets, cycle sets, and the Yang-Baxter equation

Abstract

Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic solutions to the Hom-Yang-Baxter equation (HYBE for short). This paper investigates set-theoretic solutions to HYBE and associated algebraic system, called Hom-cycle sets. We characterize left non-degenerate involutive set-theoretic solutions to HYBE and Hom-cycle sets, and establish their relations. We discuss connections among Hom-cycle sets, cycle sets, left non-degenerate involutive set-theoretic solutions to HYBE and the Yang-Baxter equation.