Reflections to set-theoretic solutions of the Yang-Baxter equation
Andrea Albano, Marzia Mazzotta, Paola Stefanelli
TL;DR
The paper investigates how to determine reflections for non-degenerate set-theoretic solutions (X,r) of the Yang–Baxter equation, emphasizing their interplay with derived solutions and shelf structures. It develops precise criteria for left- and right-non-degenerate reflections, connects reflections to endomorphisms of associated shelves, and then extends these ideas to bijective non-degenerate solutions via retracts and anti-isomorphisms between left and right shelves. It also provides constructive principles for producing reflections under standard solution-building operations—matched products, strong semilattices, and twisted unions—showing how reflections on components induce reflections on the composite objects. Finally, it presents computational data for skew-brace–related bijective non-degenerate solutions, illustrating the breadth of reflections in small-order cases and motivating further study of their structure and classification.
Abstract
The main aim of this paper is to determine reflections to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left non-degenerate solutions in terms of a family of automorphisms of their associated left rack. In some cases, we show that the study of reflections for bijective and non-degenerate solutions can be reduced to those of derived type. Moreover, we extend some results obtained in the literature for reflections of involutive non-degenerate solutions to more arbitrary solutions. Besides, we provide ways for defining reflections for solutions obtained by employing some classical construction techniques of solutions. Finally, we gather some numerical data on reflections for bijective non-degenerate solutions associated with skew braces of small order.