Dihedral quandle

Example

Let $n \ge 1$ and let $R_n = \Z/n\Z$ be the integers modulo $n$, with $$a \triangleright b = 2a - b.$$ Then $(R_n, \triangleright)$ is a quandle, the dihedral quandle of order $n$.

Check of the three axioms:

1. Self-distributivity: $a \triangleright (b \triangleright c) = 2a - 2b + c$ and $(a \triangleright b) \triangleright (a \triangleright c) = 2(2a - b) - (2a - c) = 2a - 2b + c$.
2. For fixed $a$ and $b$, the equation $a \triangleright c = b$ has the unique solution $c = 2a - b$.
3. Idempotence: $a \triangleright a = 2a - a = a$.

The name comes from the dihedral group $D_n$. Let $s_a$ be the reflection $x \mapsto 2a - x$ of $\Z/n\Z$. Then $s_a s_b s_a^{-1} = s_{2a - b}$, so the operation of $R_n$ is conjugation of reflections. For odd $n$ the map $a \mapsto s_a$ is injective, and $R_n$ is the conjugacy class of reflections in $D_n$ with conjugation.