  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.
