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  "body": "\n## Definition\n\nLet $(P, \\le_P)$ and $(Q, \\le_Q)$ be [[partially ordered sets|011C29Q1S1]].\nAn order isomorphism from $P$ to $Q$ is a [[bijective function|0SNQEF7Z53]] $f \\colon P \\to Q$ such that, for all $a, b \\in P$,\n$$a \\le_P b \\iff f(a) \\le_Q f(b).$$\n\n## Remarks\n\nThe inverse of an order isomorphism is an order isomorphism.\nThe map $x \\mapsto 2x$ is an order isomorphism from $\\mathbb{Z}$ to the even integers.\nThe ordered sets $\\mathbb{N}$ and $\\mathbb{Z}$ are not order isomorphic, because $\\mathbb{N}$ has a least element and $\\mathbb{Z}$ has none.\nA [[monotone|0WFF6AZPBP]] bijection need not be an order isomorphism, when the orders are not total.\n\n## Explanations\n\n### L1 human\n\nOrder isomorphic sets have the same shape as ordered sets.\nOnly the names of the elements differ.\n",
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