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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be a [[structure|983GZBQYT4]] with domain $A$ for a signature $\\sigma$, and let $X \\subseteq A$.\nA subset $B \\subseteq A$ is *closed* if it contains $c^{\\mathbf{A}}$ for each constant symbol $c$ and $f^{\\mathbf{A}}(b_1, \\dots, b_n) \\in B$ for each $n$-ary function symbol $f$ and all $b_1, \\dots, b_n \\in B$.\nThe set generated by $X$ is\n$$\\operatorname{Sg}(X) = \\bigcap \\set{B \\subseteq A : X \\subseteq B \\text{ and } B \\text{ is closed}},$$\nthe [[intersection|0ZNMMA9TF5]] of all closed subsets that contain $X$.\nWhen $\\operatorname{Sg}(X)$ is nonempty, the *subalgebra generated by $X$* is the [[substructure|676H7EGTKJ]] of $\\mathbf{A}$ with domain $\\operatorname{Sg}(X)$.\n\n## Remarks\n\nThe set $A$ itself is closed and an intersection of closed sets is closed, so $\\operatorname{Sg}(X)$ is the smallest closed subset containing $X$.\nFor finitary operations, $\\operatorname{Sg}(X) = \\bigcup_{k \\ge 0} X_k$, where $X_0 = X$ and $X_{k+1}$ is $X_k$ together with the constants and all values $f^{\\mathbf{A}}(b_1, \\dots, b_n)$ with $b_1, \\dots, b_n \\in X_k$.\n\nExample: in $(\\Z, +, -, 0)$, $\\operatorname{Sg}(\\set{4, 6}) = 2\\Z$.\nThe set $2\\Z$ is closed and contains $4$ and $6$; and any closed set containing $4$ and $6$ contains $6 - 4 = 2$, hence all multiples of $2$.\n\nNon-example: the set $4\\Z \\cup 6\\Z$ contains $\\set{4, 6}$ and $0$ but is not $\\operatorname{Sg}(\\set{4, 6})$, because it is not closed: $4 + 6 = 10$ is not in it.\n\nWhen $\\sigma$ has no constants, $\\operatorname{Sg}(\\emptyset) = \\emptyset$.\n",
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