Graph quandles: Generalized Cayley graphs of racks and right quasigroups

TL;DR

This paper develops a graph-theoretic framework for studying right quasigroups, racks, and quandles, introducing invariants and characterizations of Cayley graphs that generalize geometric group theory concepts.

Contribution

It introduces a novel approach to analyze right quasigroups and related algebraic structures via graph invariants and Cayley graph characterizations.

Findings

All right quasigroups are realizable by edgeless and complete graphs.

Cayley (di)graphs of right quasigroups can be characterized using Schreier graphs.

Solved two open problems of Valeriy Bardakov regarding realizability.

Abstract

This article lays the foundations for an analogue of geometric group theory that studies actions on graphs by right quasigroups, including racks and quandles. We study markings of graphs that realize racks, and we introduce (di)graph invariants based on such markings. We show that all right quasigroups are realizable by edgeless graphs and complete (di)graphs. Using Schreier (di)graphs, we also characterize Cayley (di)graphs of right quasigroups Q that realize Q. In particular, all racks are realizable by their full Cayley (di)graphs. This solves two problems of Valeriy Bardakov. Finally, we give graph-theoretic characterizations of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.