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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.09747 |
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Table of Contents:
- 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.