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Graph quandles: Generalized Cayley graphs of racks and right quasigroups

Luc Ta

Abstract

We solve two open problems of Valeriy Bardakov about Cayley graphs of racks and graph-theoretic realizations of right quasigroups. We also extend Didier Caucal's classification of labeled Cayley digraphs to right quasigroups and related algebraic structures like quandles. First, we characterize markings of graphs that realize racks. As an application, we construct rack-theoretic (di)graph invariants from permutation representations of graph automorphism groups. We describe how to compute these invariants with general results for path graphs and cycle graphs. Second, 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. Finally, we give a graph-theoretic characterization of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.

Graph quandles: Generalized Cayley graphs of racks and right quasigroups

Abstract

We solve two open problems of Valeriy Bardakov about Cayley graphs of racks and graph-theoretic realizations of right quasigroups. We also extend Didier Caucal's classification of labeled Cayley digraphs to right quasigroups and related algebraic structures like quandles. First, we characterize markings of graphs that realize racks. As an application, we construct rack-theoretic (di)graph invariants from permutation representations of graph automorphism groups. We describe how to compute these invariants with general results for path graphs and cycle graphs. Second, 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. Finally, we give a graph-theoretic characterization of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.

Paper Structure

This paper contains 39 sections, 24 theorems, 27 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivating discussion
  3. Quandle theory and generalizations
  4. Racks as symmetries
  5. Characterization of labeled Cayley digraphs
  6. Structure of the paper
  7. Notation
  8. Algebraic preliminaries
  9. Magmas
  10. Racks
  11. Examples
  12. Preliminary results
  13. Graph-theoretic preliminaries
  14. Graphs and digraphs
  15. Marked graphs
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a (di)graph with vertex set $V$, and let $R:V\to \mathop{\mathrm{Aut}}\nolimits\Gamma$ be a marking (resp. q-marking) of $V$. Then the right quasigroup $V^\Gamma_R$ realized by $\Gamma$ is a rack (resp. quandle) if and only if $R$ is a magma homomorphism from $V^\Gamma_R$ to $\operat

Figures (5)

  • Figure 1: Full Cayley digraph and full Cayley graph of the right quasigroup from Example \ref{['ex:not']}.
  • Figure 2: Partial and full Cayley digraphs and full Cayley graph of the quandle from Example \ref{['ex:3quandle']}.
  • Figure 3: Full Cayley digraphs of the two right quasigroups from Example \ref{['ex:different']} and their shared full Cayley graph.
  • Figure 4: Full Cayley digraph and full Cayley graph of the right quasigroup from Example \ref{['ex:conj']}.
  • Figure 5: Partial Cayley digraph and underlying Cayley graph of the right quasigroup from Example \ref{['ex:5quandle']}.

Theorems & Definitions (79)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 69 more