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Quandles up to order 8 were determined first by Sam Nelson and co-authors (see references). Nelson's results were confirmed independently by the submitter, and extended to order 9.

Both names "abelian" and "medial" refer to the identity (xy)(uv)=(xu)(yv). A (left) quandle is involutory (aka symmetric, kei) if all (left) translations have order at most 2, i.e., x(xy)=y is satisfied.

Enumerates intersection of the classes enumerated in A165200, A178432.

Both names "abelian" and "medial" refer to the identity (xy)(uv)=(xu)(yv). A quandle is called 2-reductive if all orbits are projection quandles.

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

a(n) is also the number of medial Legendrian racks of order n up to isomorphism; see Ta, "Equivalences of...," Theorem 1.1.

a(n) is also the number of medial generalized Legendrian quandles (also called GL-quandles or bi-Legendrian quandles) of order n up to isomorphism; see Ta, "Generalized Legendrian...," Theorem 5.5.

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; cf. Cattabriga and Nasybullov, Section 3.2.

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; see Cattabriga and Nasybullov, Section 3.2.

A rack or quandle is medial if it satisfies the identity (xy)(uv) = (xu)(yv).

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of medial racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.

a(n) is also the number of Legendrian kei (i.e., kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.

a(n) is also the number of symmetric kei (i.e., kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

A virtual quandle is a quandle equipped with a distinguished quandle automorphism. Two virtual quandles (Q1,f1), (Q2,f2) are isomorphic if there exists a quandle isomorphism g: Q1 -> Q2 such that g*f1 = f2*g.

A good involution f of a rack R is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (R,f) a symmetric rack. A symmetric rack isomorphism is a rack isomorphism that intertwines good involutions.