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  "body": "\n## Definition\n\nLet $\\sigma$ be a [[signature|WDVVEYD6AT]] whose symbols are all function symbols, each with an arity $n \\ge 0$.\nAn *algebraic structure* of signature $\\sigma$, or $\\sigma$-algebra, is a pair $\\mathbf{A} = (A, (f^{\\mathbf{A}})_{f \\in \\sigma})$ such that:\n\n1. $A$ is a nonempty set, the universe of $\\mathbf{A}$;\n2. for each symbol $f \\in \\sigma$ of arity $n$, $f^{\\mathbf{A}} \\colon A^n \\to A$ is an $n$-ary [[finitary operation|EJ8YJVM9BW]] on $A$.\n\nAn operation of arity $0$ is identified with an element of $A$, a constant.\nSo a $\\sigma$-algebra is the same as a [[structure|983GZBQYT4]] for the purely functional signature $\\sigma$.\n\n## Remarks\n\nExample: $(\\Z, +, -, 0)$ is an algebraic structure of signature $(2, 1, 0)$.\nAddition is a map $\\Z^2 \\to \\Z$, negation is a map $\\Z \\to \\Z$, and $0 \\in \\Z$ is a constant.\n\nNon-example: $(\\N, +, -, 0)$ with $-$ the map $x \\mapsto -x$ is not an algebraic structure, because $-1 \\notin \\N$, so clause 2 fails for the unary symbol.\n\nA *class* of algebras, such as the class of groups, is given by a signature together with axioms that the operations must satisfy.\nThe axioms are not part of the definition of an algebraic structure.\nSome authors allow the empty set as universe when $\\sigma$ has no constants.\n",
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