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  "body": "\n## Definition\n\nA *heap* is a [[set|0SJ3RX9TXA]] $H$ with a [[ternary operation|WC47MFR6GW]] $[\\cdot, \\cdot, \\cdot] \\colon H \\times H \\times H \\to H$ such that, for all $a, b, c, d, e \\in H$:\n\n1. $[\\,[a, b, c], d, e] = [a, b, [c, d, e]\\,] = [a, [d, c, b], e]$ (para-associativity);\n2. $[a, a, b] = b$ and $[b, a, a] = b$ (identity law).\n\n## Remarks\n\nExample: a [[group|G5S1M9]] $G$ with $[x, y, z] = x y^{-1} z$.\nFor clause 1, all three expressions equal $a b^{-1} c d^{-1} e$; the third one because $(d c^{-1} b)^{-1} = b^{-1} c d^{-1}$.\nFor clause 2, $a a^{-1} b = b$ and $b a^{-1} a = b$.\n\nNon-example: $\\Z$ with $[x, y, z] = x + y + z$.\nClause 1 holds, since all three expressions equal $a + b + c + d + e$.\nClause 2 fails: $[0, 1, 1] = 2 \\ne 0$.\n\nConversely, let $H$ be a nonempty heap and fix $e \\in H$.\nThen $x \\cdot y = [x, e, y]$ makes $H$ a group with identity $e$ and inverse $x^{-1} = [e, x, e]$, and the heap operation is recovered as $[x, y, z] = x y^{-1} z$.\nDifferent choices of $e$ give isomorphic groups.\nSome authors allow the empty heap, which corresponds to no group.\n",
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