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## 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.

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety No No Yes, $n=2$

### Finite members

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