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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.

### Basic results

### 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.

### Finite members

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

### Subclasses

### Superclasses

### References

Trace: » quandles