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  "body": "\n## Definition\n\nLet $L$ be a signature and let $\\mathcal{M}$ and $\\mathcal{N}$ be $L$-[[structures|983GZBQYT4]] with domains $M$ and $N$.\nA [[map|0EPAT51DXQ]] $j \\colon M \\to N$ is an elementary embedding of $\\mathcal{M}$ into $\\mathcal{N}$ when, for every $L$-[[formula|HYJ10AF6RT]] $\\varphi(x_1, \\dots, x_n)$ and all $a_1, \\dots, a_n \\in M$,\n$$\\mathcal{M} \\models \\varphi(a_1, \\dots, a_n) \\iff \\mathcal{N} \\models \\varphi(j(a_1), \\dots, j(a_n)).$$\n\n## Remarks\n\nAn elementary embedding is injective: apply the condition to the formula $x_1 = x_2$.\nIt is an embedding of structures: apply the condition to the atomic formulas.\nIts image is the domain of an [[elementary substructure|8WZF9SBP39]] of $\\mathcal{N}$.\n\nExample: the inclusion $\\Q \\to \\R$ is an elementary embedding of $(\\Q, \u003c)$ into $(\\R, \u003c)$, because $(\\Q, \u003c)$ is an elementary substructure of $(\\R, \u003c)$.\n\nNon-example: the map $j(n) = n + 1$ from $(\\N, \u003c)$ to $(\\N, \u003c)$ preserves $\u003c$ and is injective.\nIt is not elementary: for $\\varphi(x) = \\exists y \\, (y \u003c x)$, $(\\N, \u003c) \\models \\neg\\varphi(0)$ but $(\\N, \u003c) \\models \\varphi(j(0))$.\n\n## Explanations\n\n### L1 human\n\nAn elementary embedding places a copy of one structure inside another so that no first-order statement changes its truth value.\nEach elementary embedding is an isomorphism onto an elementary substructure.\n",
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