Abstract: We define and study properties of triangular and near-trivial quandles. A quandle is a nonempty set equipped with a binary operation satisfying three algebraic axioms [1, 2], which formalize the three Reidemeister moves [3] of planar knot diagrams in three-dimensional space. If only the second and third Reidemeister moves are considered, one obtains an algebraic structure known as a rack. Historically, the concept of a quandle or a self-distributive groupoid is associated with the names of S. V. Matveev and D. Joyce. In 1982, Matveev [1] introduced an algebraic system that he called a distributive groupoid. He proved that every classical knot can be naturally associated with a right-distributive right quasigroup, which serves as an algebraic invariant of the knot and determines it up to an isotopy and a mirror reflection. This result provided a universal algebraic invariant of classical knots. In the same year, Joyce [2] independently obtained the same result, introducing the term quandle for this structure. While investigating the differentiability of solutions to functional equations, Ryll–Nardzewski [4] in 1949 introduced the symmetric mean and obtained a distributive quasigroup. Later, in 1953, Gossu [5] considered nonsymmetric means. Subsequently, in [6, 7], he not only axiomatized this object, but also constructed its representation over an arbitrary group.

Cite: Borodin A.N. , Neshchadim M.V. , Simonov A.A.
Triangular and Near-Trivial Quandles
Algebra and Logic. 2025. V.64. N2. P.67-83. DOI: 10.1007/s10469-026-09813-9 WOS Scopus OpenAlex

Original: Бородин А.Н. , Нещадим М.В. , Симонов А.А.
Треугольные и почти - тривиальные квандлы
Алгебра и логика. 2025. Т.64. №2.

Dates:

Submitted:	Mar 24, 2025
Accepted:	Oct 8, 2025
Published print:	Feb 11, 2026
Published online:	Feb 11, 2026

Identifiers:

≡ Web of science:	WOS:001686877100001
≡ Scopus:	2-s2.0-105030063693
≡ OpenAlex:	W7128520068

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