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<p>We investigate the independence of axioms defining a quandle – an algebraic structure important in knot theory and combinatorial algebra. It is proved that all four axioms included in the standard definition of a quandle are independent. For finite quandles, one of the solvability axioms turns out to be a consequence of the others, owing to which it is possible to construct a minimal system of three axioms. An example of a right-distributive left quasigroup is constructed. As an application, for the constructed right-distributive systems, set-theoretic solutions to the Yang–Baxter equations and the corresponding associated groups are indicated.</p>

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Independence of Quandle Axioms

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • A. N. Borodin,
  • M. V. Neshchadim,
  • A. A. Simonov

摘要

We investigate the independence of axioms defining a quandle – an algebraic structure important in knot theory and combinatorial algebra. It is proved that all four axioms included in the standard definition of a quandle are independent. For finite quandles, one of the solvability axioms turns out to be a consequence of the others, owing to which it is possible to construct a minimal system of three axioms. An example of a right-distributive left quasigroup is constructed. As an application, for the constructed right-distributive systems, set-theoretic solutions to the Yang–Baxter equations and the corresponding associated groups are indicated.