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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nA *realizer* of $P$ is a set $\\mathcal{R}$ of [[linear extensions|W8DZ5KGGHN]] of $\\le$ with the following property.\nFor all $x, y \\in P$, $x \\le y$ holds if and only if $x \\preceq y$ for every $\\preceq$ in $\\mathcal{R}$.\nThe *order dimension* $\\dim P$ is the least cardinality of a realizer of $P$.\n\n## Remarks\n\nThe set of all linear extensions is a realizer, by the [[Szpilrajn extension theorem|T9CJHV6JW3]], so the dimension is defined.\nExample: an antichain $\\set{a, b, c}$ has dimension $2$.\nThe orders $a \\prec b \\prec c$ and $c \\prec b \\prec a$ form a realizer, since each pair of distinct elements appears in both directions, and one linear order alone would make two elements comparable.\nNon-example of a realizer: for the same antichain, the set $\\set{a \\prec b \\prec c,\\ b \\prec a \\prec c}$ is not a realizer, since $a$ precedes $c$ in both orders but $a \\not\\le c$.\nA poset has dimension $1$ exactly when it is a total order.\nEquivalently, $\\dim P$ is the least $d$ such that $P$ embeds in a product of $d$ chains.\n",
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