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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]]: a set of function symbols, relation symbols and constant symbols.\nEach function symbol and each relation symbol has an arity $n \\ge 1$.\nAn $L$-structure $\\mathcal{M}$ consists of the following data.\n\n1. A nonempty [[set|0BSJ7RQ9BW]] $M$, the domain or universe of $\\mathcal{M}$.\n2. For each $n$-ary function symbol $f$ of $L$, a [[function|0EPAT51DXQ]] $f^{\\mathcal{M}} \\colon M^n \\to M$, where $M^n$ is the [[Cartesian product|09CEDJYBKP]] of $n$ copies of $M$.\n3. For each $n$-ary relation symbol $R$ of $L$, a subset $R^{\\mathcal{M}} \\subseteq M^n$.\n4. For each constant symbol $c$ of $L$, an element $c^{\\mathcal{M}} \\in M$.\n\nThe objects $f^{\\mathcal{M}}$, $R^{\\mathcal{M}}$ and $c^{\\mathcal{M}}$ are the interpretations of the symbols in $\\mathcal{M}$.\n\n## Remarks\n\nExample: let $L = \\set{+, -, 0}$ with $+$ binary, $-$ unary and $0$ a constant.\nThe integers $\\Z$ with addition, negation and the number $0$ form an $L$-structure, because each symbol receives a function of the right arity on $\\Z$ or an element of $\\Z$.\n\nNon-example: for the same signature, the natural numbers $\\N$ with $x \\mapsto -x$ do not form an $L$-structure, because $x \\mapsto -x$ is not a function from $\\N$ to $\\N$.\n\nA structure is often written as a tuple, such as $(\\Z, +, -, 0)$.\nSome authors, and the Mathlib library, allow the empty set as domain.\n\n## Explanations\n\n### L1 human\n\nA structure gives a meaning to each symbol of a formal language.\nThe symbol $\u003c$ alone says nothing; in the structure $(\\Q, \u003c)$ it means the usual order of the rational numbers.\nOnce each symbol has a meaning, each sentence of the language becomes true or false in the structure.\n",
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