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Complementation
of subquandles
Kieran Amsberry, August Bergquist, Thomas Horstkamp, Meghan Lee and David Yetter |
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Vol. 18 (2025), No. 3, 417–435
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Abstract |
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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 subquandle lattice is not complemented, and conjecture that profinite quandles have complemented subquandle lattices. Additionally, we provide a complete classification of subquandles whose set-theoretic complement is also a subquandle, which we call 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. |
Keywords
quandle, subobject, lattice
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Mathematical Subject Classification
Primary: 08B99, 57K12
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Milestones
Received: 17 April 2023
Revised: 3 November 2023
Accepted: 11 January 2024
Published: 28 April 2025
Communicated by Joel Foisy
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Authors |
| Kieran Amsberry | |
| Department of Mathematics Kansas State University Manhattan, KS United States |
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| August Bergquist | |
| Department of Mathematics and
Statistics University of New Hampshire Durham, NH United States |
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| Thomas Horstkamp | |
| Department of Mathematical
Sciences Carnegie Mellon University Pittsburgh, PA United States |
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| Meghan Lee | |
| Department of Mathematics University of California, Santa Barbara Santa Barbara, CA United States |
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| David Yetter | |
| Department of Mathematics Kansas State University Manhattan, KS United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
