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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nFor $X \\subseteq A$, let $\\operatorname{Sg}(X)$ denote the [[subuniverse generated by|2TQD8NMG9P]] $X$.\nIt is the smallest subset of $A$ that contains $X$ and is closed under all fundamental operations of $\\mathbf{A}$.\nThe algebra $\\mathbf{A}$ is *finitely generated* when there is a [[finite set|070R982G4B]] $X \\subseteq A$ with $\\operatorname{Sg}(X) = A$.\n\n## Remarks\n\nExample: the group $(\\Z, +, -, 0)$ is finitely generated.\nIndeed $\\operatorname{Sg}(\\set{1})$ contains $0$, all sums $1 + \\cdots + 1$, and their negatives, so it is all of $\\Z$.\n\nNon-example: the group $(\\Q, +, -, 0)$ is not finitely generated.\nIf $X = \\set{p_1/q, \\dots, p_k/q}$ with a common denominator $q \\ge 1$, then every element of $\\operatorname{Sg}(X)$ is an integer multiple of $1/q$, so $1/(2q) \\notin \\operatorname{Sg}(X)$.\n\nEvery finite algebra is finitely generated, with $X = A$.\nIn model theory the same notion is used for structures in a signature with relation symbols, where the relations play no role in generation.\n",
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