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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]], let $I$ be a nonempty set, and let $\\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on $I$.\nFor each $i \\in I$ let $M_i$ be an [[$L$-structure|983GZBQYT4]] with nonempty domain.\nLet $P = \\prod_{i \\in I} M_i$ be the [[Cartesian product|09CEDJYBKP]] of the domains.\nFor $a, b \\in P$ write $a \\sim_{\\mathcal{U}} b$ when $\\set{i \\in I : a(i) = b(i)} \\in \\mathcal{U}$; this is an [[equivalence relation|0G8VEZ3WTK]] on $P$, and $[a]$ denotes the class of $a$.\nThe *ultraproduct* $\\prod_{i \\in I} M_i / \\mathcal{U}$ is the $L$-structure with domain $P / {\\sim_{\\mathcal{U}}}$ and the following interpretations:\n\n1. for an $n$-ary function symbol $f$, $f([a_1], \\dots, [a_n]) = [b]$, where $b(i) = f^{M_i}(a_1(i), \\dots, a_n(i))$;\n2. for an $n$-ary relation symbol $R$, $R([a_1], \\dots, [a_n])$ holds if and only if $\\set{i \\in I : (a_1(i), \\dots, a_n(i)) \\in R^{M_i}} \\in \\mathcal{U}$;\n3. for a constant symbol $c$, its value is $[b]$, where $b(i) = c^{M_i}$.\n\nIf all $M_i$ equal one structure $M$, the ultraproduct is called the *ultrapower* $M^I / \\mathcal{U}$.\n\n## Remarks\n\nThe interpretations do not depend on the chosen representatives, because $\\mathcal{U}$ is closed under finite intersections and supersets.\n\nExample: let $\\mathcal{U}$ be a non-principal ultrafilter on $\\N$ and let $M_i = \\R$ for all $i$.\nThe ultrapower $\\R^{\\N} / \\mathcal{U}$ is the ordered field of [[hyperreal numbers|421QMTFG9T]].\nThe class of $a = (0, 1, 2, 3, \\dots)$ is larger than the class of each constant sequence $(r, r, r, \\dots)$, since $\\set{i : i \u003e r}$ is cofinite and so lies in $\\mathcal{U}$.\n\nNon-example: replace $\\mathcal{U}$ by the Fréchet filter $\\mathcal{F}$ of cofinite subsets of $\\N$, which is a filter but not an ultrafilter.\nThe same construction gives a ring $\\R^{\\N} / \\mathcal{F}$, but it is not a field.\nThe classes of $(1, 0, 1, 0, \\dots)$ and $(0, 1, 0, 1, \\dots)$ are both nonzero, because neither sequence is zero on a cofinite set, but their product is the class of $0$.\n\nIf $\\mathcal{U}$ is the principal ultrafilter at $i_0 \\in I$, the ultraproduct is isomorphic to $M_{i_0}$.\n",
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