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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]] with a [[binary operation|0M418FE7BB]] $\\triangleleft$.\nThe pair $(X, \\triangleleft)$ is a *rack* if:\n\n1. $a \\triangleleft (b \\triangleleft c) = (a \\triangleleft b) \\triangleleft (a \\triangleleft c)$ for all $a, b, c \\in X$.\n2. For all $a, b \\in X$ there is exactly one $c \\in X$ with $a \\triangleleft c = b$.\n\nA rack is a *quandle* if moreover:\n\n3. $a \\triangleleft a = a$ for all $a \\in X$.\n\n## Remarks\n\nClause 2 says that for each $a$ the map $x \\mapsto a \\triangleleft x$ is a [[bijection|0GG9TVW90M]] of $X$, and clause 1 says that this bijection is an automorphism of $(X, \\triangleleft)$.\n\nExample: the conjugation quandle of a [[group|G5S1M9]] $G$, with $a \\triangleleft b = a b a^{-1}$, the [[conjugate|0DXD4QSZ8A]] of $b$ by $a$.\nClause 1: both sides equal $a b c b^{-1} a^{-1}$, since $(aba^{-1})(aca^{-1})(aba^{-1})^{-1} = abcb^{-1}a^{-1}$.\nClause 2: $c = a^{-1} b a$ is the only solution.\nClause 3: $a a a^{-1} = a$.\n\nExample: the dihedral quandle on $\\Z/n\\Z$, with $a \\triangleleft b = 2a - b$.\nBoth sides of clause 1 equal $2a - 2b + c$, clause 2 has the solution $c = 2a - b$, and $2a - a = a$.\n\nNon-example: a rack that is not a quandle.\nLet $\\sigma$ be a permutation of $X$ other than the identity, and let $a \\triangleleft b = \\sigma(b)$.\nBoth sides of clause 1 equal $\\sigma(\\sigma(c))$, and clause 2 holds since $\\sigma$ is a bijection.\nBut $a \\triangleleft a = \\sigma(a) \\ne a$ for each $a$ moved by $\\sigma$, so clause 3 fails.\n\nSome authors write the operation on the other side, with $(c \\triangleright b) \\triangleright a = (c \\triangleright a) \\triangleright (b \\triangleright a)$.\n",
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