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  "body": "\n## Summary\n\nAn order embedding is a map between [[partially ordered sets|011C29Q1S1]] that both preserves and reflects the order.\nIt places a copy of one poset inside another, with exactly the same comparisons.\nAn order embedding is automatically injective, and a surjective order embedding is an [[order isomorphism|0Y74KN99A5]].\n\n## Explanations\n\n### L1 human\n\nAn order embedding copies one ordered set into another without changing any comparison.\nThe map $n \\mapsto 2n$ from the natural numbers to the natural numbers is an order embedding: $m \\le n$ exactly when $2m \\le 2n$.\nThe inclusion of the rationals into the reals is another one.\nA map that only keeps \"smaller\" as \"smaller\", but can make incomparable elements comparable, is not an order embedding.\n",
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