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  "body": "\n## Definition\n\nLet $(P, \\le_P)$ and $(Q, \\le_Q)$ be [[partially ordered sets|011C29Q1S1]].\nA map $f \\colon P \\to Q$ is an *order embedding* when, for all $x, y \\in P$,\n$$x \\le_P y \\iff f(x) \\le_Q f(y).$$\n\n## Remarks\n\nThe inclusion $\\Q \\to \\R$ is an order embedding, since the order of $\\Q$ is the restriction of the order of $\\R$.\nLet $P = \\set{a, b}$ with $a$ and $b$ incomparable, and let $Q = \\set{0, 1}$ with $0 \u003c 1$.\nThe map $f(a) = 0$, $f(b) = 1$ is [[monotone|0WFF6AZPBP]], but it is not an order embedding: $f(a) \\le f(b)$ while $a \\not\\le b$.\nAn order embedding is injective: $f(x) = f(y)$ gives $x \\le y$ and $y \\le x$, so $x = y$ by antisymmetry.\nSo $f$ is an [[order isomorphism|0Y74KN99A5]] from $P$ onto its image $f(P)$ with the induced order.\nFor total orders, a strictly increasing map is the same as an order embedding.\n\n## Explanations\n\n### L1 human\n\nAn order embedding keeps every comparison and creates no new ones.\nIt shows one ordered set as an ordered part of another.\n",
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