Rack and quandle

Remarks

Clause 2 says that for each $a$ the map $x \mapsto a \triangleleft x$ is a bijection of $X$, and clause 1 says that this bijection is an automorphism of $(X, \triangleleft)$.

Example: the conjugation quandle of a group $G$, with $a \triangleleft b = a b a^{-1}$, the conjugate of $b$ by $a$. Clause 1: both sides equal $a b c b^{-1} a^{-1}$, since $(aba^{-1})(aca^{-1})(aba^{-1})^{-1} = abcb^{-1}a^{-1}$. Clause 2: $c = a^{-1} b a$ is the only solution. Clause 3: $a a a^{-1} = a$.

Example: the dihedral quandle on $\Z/n\Z$, with $a \triangleleft b = 2a - b$. Both sides of clause 1 equal $2a - 2b + c$, clause 2 has the solution $c = 2a - b$, and $2a - a = a$.

Non-example: a rack that is not a quandle. Let $\sigma$ be a permutation of $X$ other than the identity, and let $a \triangleleft b = \sigma(b)$. Both sides of clause 1 equal $\sigma(\sigma(c))$, and clause 2 holds since $\sigma$ is a bijection. But $a \triangleleft a = \sigma(a) \ne a$ for each $a$ moved by $\sigma$, so clause 3 fails.

Some authors write the operation on the other side, with $(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright (b \triangleright a)$.