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Complementation of Subquandles

K. J. Amsberry, J. A. Bergquist, T. A. Horstkamp, M. H. Lee, D. N. Yetter

TL;DR

The paper addresses when subquandle lattices $\mathcal{L}(Q)$ are complemented, extending finite-quandle results to infinite, ind-finite, and profinite contexts. It introduces strongly complemented subquandles and leverages two natural actions of the inner automorphism group $\mathrm{Inn}(Q)$ to obtain multiple equivalent characterizations, including a mesh-based semidisjoint decomposition and a subquotient relation $\mathrm{Inn}(Q')\cong S_{Q'}/K_{Q'}$. It proves transitivity-type results for chains of strongly complemented subquandles, provides constructive complements, and presents ind-finite examples where complementability fails, while proposing a conjecture that profinite quandles have complemented subquandle lattices. Together, these results provide a structural framework for decomposing quandles via orbit- and mesh-based decompositions, with implications for understanding subquandle lattices in knot-theoretic and algebraic contexts.

Abstract

Saki and Kiani proved that the subrack lattice of a rack $R$ is necessarily complemented if $R$ is finite but not necessarily complemented if $R$ is infinite. In this paper, we investigate further avenues related to the complementation of subquandles. Saki and Kiani's example of an infinite rack without complements is a quandle, which is neither ind-finite nor profinite. We provide an example of an ind-finite quandle whose subobject lattice is not complemented, and conjecture that profinite quandles have complemented subobject lattices. Additionally, we provide a complete classification of subquandles whose set-theoretic complement is also a subquandle, which we call \textit{strongly complemented}, and provide a partial transitivity criterion for the complementation in chains of strongly complemented subquandles. One technical lemma used in establishing this is of independent interest: the inner automorphism group of a subquandle is always a subquotient of the inner automorphism group of the ambient quandle.

Complementation of Subquandles

TL;DR

The paper addresses when subquandle lattices are complemented, extending finite-quandle results to infinite, ind-finite, and profinite contexts. It introduces strongly complemented subquandles and leverages two natural actions of the inner automorphism group to obtain multiple equivalent characterizations, including a mesh-based semidisjoint decomposition and a subquotient relation . It proves transitivity-type results for chains of strongly complemented subquandles, provides constructive complements, and presents ind-finite examples where complementability fails, while proposing a conjecture that profinite quandles have complemented subquandle lattices. Together, these results provide a structural framework for decomposing quandles via orbit- and mesh-based decompositions, with implications for understanding subquandle lattices in knot-theoretic and algebraic contexts.

Abstract

Saki and Kiani proved that the subrack lattice of a rack is necessarily complemented if is finite but not necessarily complemented if is infinite. In this paper, we investigate further avenues related to the complementation of subquandles. Saki and Kiani's example of an infinite rack without complements is a quandle, which is neither ind-finite nor profinite. We provide an example of an ind-finite quandle whose subobject lattice is not complemented, and conjecture that profinite quandles have complemented subobject lattices. Additionally, we provide a complete classification of subquandles whose set-theoretic complement is also a subquandle, which we call \textit{strongly complemented}, and provide a partial transitivity criterion for the complementation in chains of strongly complemented subquandles. One technical lemma used in establishing this is of independent interest: the inner automorphism group of a subquandle is always a subquotient of the inner automorphism group of the ambient quandle.
Paper Structure (10 sections, 25 theorems, 14 equations, 2 figures)

This paper contains 10 sections, 25 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. Subquandle Lattices
  4. Strongly Complemented Subquandles
  5. Chains of Strongly Complemented Subquandles
  6. Infinite Quandles
  7. Ind-finite Quandles
  8. Profinite Quandles
  9. Acknowledgments
  10. Bibliography

Key Result

Theorem 2.14

EGTY2008 A matrix of group homomorphisms $M = (g)_{ij}:\mathrm{Adconj}(Q_i)\to \mathrm{Aut}(Q_j)$ is a mesh if and only if the following conditions are satisfied are met for all $i,j,k$ distinct with $1\le i,j,k\le n$,

Figures (2)

  • Figure 1: The multiplication table for $\mathbf{T}_3$
  • Figure 2: Canonical group homomorphism $\varphi_i$

Theorems & Definitions (74)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 64 more