Quasi racks, quasi bijective and quasi non-degenerate set-theoretic solutions of the Yang-Baxter equation

TL;DR

This paper systematically studies quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation, extending classical notions and characterizing their structure, especially those derived from dual weak braces.

Contribution

It introduces the concepts of quasi rack and derived solution, extending classical definitions, and characterizes quasi racks as Plonka sums of racks.

Findings

Solutions from dual weak braces are quasi bijective and quasi non-degenerate.

Introduces and examines quasi rack and derived solutions.

Characterizes quasi racks as Plonka sums of racks.

Abstract

This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces belong to these classes. The notions of quasi rack and derived solution are introduced and examined, extending the classical definitions. Additionally, a family of quasi left non-degenerate solutions is described in terms of quasi racks and g-twists, analogous to the left non-degenerate case. Furthermore, we completely characterize a class of quasi racks that are Plonka sum of racks.