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  "body": "\n## Definition\n\nA *signature* $\\sigma$ consists of:\n\n1. a set $F$ of function symbols;\n2. a set $R$ of relation symbols, disjoint from $F$;\n3. an arity [[function|0EPAT51DXQ]] $\\operatorname{ar} \\colon F \\cup R \\to \\N$ that gives the number of arguments of each symbol.\n\nA function symbol of arity $0$ is a *constant symbol*.\nThe symbols of $\\sigma$ are the non-logical symbols of a first-order language; the variables, connectives, quantifiers and equality are not part of $\\sigma$.\n\n## Remarks\n\nThe signature of rings is $\\set{+, \\cdot, -, 0, 1}$ with arities $2, 2, 1, 0, 0$; it has no relation symbols.\nThe signature of ordered sets has one relation symbol $\u003c$ of arity $2$ and no function symbols.\nThe list \"a binary operation $\\cdot$ that is associative\" is not a signature: associativity is an axiom, and a signature contains only symbols and their arities.\nSome books list the constant symbols as a third set instead of treating them as functions of arity $0$.\nIn Mathlib a signature is `FirstOrder.Language`.\n",
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