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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\le$ be a [[partial order|0QV2JS9RJ3]] on $X$.\nA *linear extension* of $\\le$ is a [[total order|0FRXBVVR03]] $\\preceq$ on $X$ such that $x \\le y$ implies $x \\preceq y$ for all $x, y \\in X$.\n\n## Remarks\n\nExample: on $\\set{1, 2, 3, 6}$ ordered by divisibility, the usual order $1 \u003c 2 \u003c 3 \u003c 6$ is a linear extension, because each divisor of a positive integer is at most that integer.\nNon-example: the order $1 \\prec 3 \\prec 6 \\prec 2$ is total but not a linear extension, because $2 \\mid 6$ and yet $6 \\prec 2$.\nIf $X$ is finite with $n$ elements, a linear extension is the same as a bijection $f \\colon X \\to \\set{1, \\ldots, n}$ with $f(x) \u003c f(y)$ whenever $x \u003c y$.\nThe number of linear extensions of a finite poset $P$ is often written $e(P)$.\nA total order has exactly one linear extension, itself, and an antichain with $n$ elements has $n!$.\n",
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