Quandles

Abbreviation: Qnd

Definition

A quandle is a structure $\mathbf{Q}=\langle Q,\triangleright,\triangleleft\rangle$ of type $\langle 2,2\rangle$ such that

$\triangleright$ is left-selfdistributive: $x\triangleright(y\triangleright z)=(x\triangleright y)\triangleright(x\triangleright z)$

$\triangleleft$ is right-selfdistributive: $(x\triangleleft y)\triangleleft z=(x\triangleleft z)\triangleleft(y\triangleleft z)$

$(x\triangleright y)\triangleleft x=y$

$x\triangleright (y\triangleleft x)=y$

$\triangleright$ is idempotent: $x\triangleright x=x$

Remark: The last identity can equivalently be replaced by $\triangleleft$ is idempotent: $x\triangleleft x=x$

Morphisms

Let $\mathbf{Q}$ and $\mathbf{R}$ be quandles. A morphism from $\mathbf{Q}$ to $\mathbf{R}$ is a function $h:Q\rightarrow R$ that is a homomorphism: $h(x \triangleright y)=h(x) \triangleright h(y)$ and $h(x \triangleleft y)=h(x) \triangleleft h(y)$.

Examples

Example 1: If $\langle G,\cdot,^{-1},1\rangle$ is a group and $x\triangleright y=xyx^{-1}$, $x\triangleleft y=x^{-1}yx$ (conjugation) then $\langle G,\triangleright,\triangleleft\rangle$ is a quandle.

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &7\\ f(5)= &22\\ f(6)= &73\\ f(7)= &298\\ f(8)= &1581\\ f(9)= &11079\\ f(10)= &\\ \end{array}$

Subclasses

Superclasses

References