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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]] and let $\\mathcal{M}$ and $\\mathcal{N}$ be $L$-[[structures|983GZBQYT4]] with domains $M$ and $N$.\nThe structure $\\mathcal{N}$ is a substructure of $\\mathcal{M}$, written $\\mathcal{N} \\subseteq \\mathcal{M}$, when the following hold.\n\n1. $N$ is a [[subset|0BJR9QR08E]] of $M$.\n2. For each $n$-ary function symbol $f$, the map $f^{\\mathcal{N}}$ is the restriction of $f^{\\mathcal{M}}$ to $N^n$.\n3. For each $n$-ary relation symbol $R$, $R^{\\mathcal{N}} = R^{\\mathcal{M}} \\cap N^n$.\n4. For each constant symbol $c$, $c^{\\mathcal{N}} = c^{\\mathcal{M}}$.\n\n## Remarks\n\nClause 2 forces $N$ to be closed under each $f^{\\mathcal{M}}$, and clause 4 forces $N$ to contain each $c^{\\mathcal{M}}$.\nConversely, a nonempty subset of $M$ with these two closure properties is the domain of exactly one substructure of $\\mathcal{M}$.\n\nExample: in the signature $\\set{+, -, 0}$, the set $2\\Z$ of even integers is closed under $+$ and $-$ and contains $0$.\nSo it carries a substructure of $(\\Z, +, -, 0)$.\n\nNon-example: the set $\\N$ is not the domain of a substructure of $(\\Z, +, -, 0)$, because $-1 \\notin \\N$ although $1 \\in \\N$.\nIn the smaller signature $\\set{+, 0}$, the set $\\N$ is the domain of a substructure of $(\\Z, +, 0)$.\n\n## Explanations\n\n### L1 human\n\nA substructure is a part of a structure that is a structure on its own, with the same operations and relations.\nIn the signature of groups, the substructures of a group are its subgroups.\n",
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