Complementation of Subquandles
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.
Cite
@article{arxiv.2304.09747,
title = {Complementation of Subquandles},
author = {K. J. Amsberry and J. A. Bergquist and T. A. Horstkamp and M. H. Lee and D. N. Yetter},
journal= {arXiv preprint arXiv:2304.09747},
year = {2025}
}