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Invariant universality for quandles and fields

Volume 251 / 2020

Andrew D. Brooke-Taylor, Filippo Calderoni, Sheila K. Miller Fundamenta Mathematicae 251 (2020), 1-16 MSC: Primary 03E15; Secondary 12F05, 12F20, 20N99. DOI: 10.4064/fm862-2-2020 Published online: 8 April 2020


We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.


  • Andrew D. Brooke-TaylorSchool of Mathematics
    University of Leeds
    Leeds, LS2 9JT, United Kingdom
  • Filippo CalderoniDepartment of Mathematics, Statistics,
    and Computer Science
    University of Illinois at Chicago
    Chicago, IL 60613, U.S.A.
  • Sheila K. Miller28 Archuleta Road
    Ranchos de Taos, NM 87557, U.S.A.

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