Ultraproduct

Definition

Let $L$ be a signature, let $I$ be a nonempty set, and let $\mathcal{U}$ be an ultrafilter on $I$. For each $i \in I$ let $M_i$ be an $L$-structure with nonempty domain. Let $P = \prod_{i \in I} M_i$ be the Cartesian product of the domains. For $a, b \in P$ write $a \sim_{\mathcal{U}} b$ when $\set{i \in I : a(i) = b(i)} \in \mathcal{U}$; this is an equivalence relation on $P$, and $[a]$ denotes the class of $a$. The ultraproduct $\prod_{i \in I} M_i / \mathcal{U}$ is the $L$-structure with domain $P / {\sim_{\mathcal{U}}}$ and the following interpretations:

1. for an $n$-ary function symbol $f$, $f([a_1], \dots, [a_n]) = [b]$, where $b(i) = f^{M_i}(a_1(i), \dots, a_n(i))$;
2. for an $n$-ary relation symbol $R$, $R([a_1], \dots, [a_n])$ holds if and only if $\set{i \in I : (a_1(i), \dots, a_n(i)) \in R^{M_i}} \in \mathcal{U}$;
3. for a constant symbol $c$, its value is $[b]$, where $b(i) = c^{M_i}$.

If all $M_i$ equal one structure $M$, the ultraproduct is called the ultrapower $M^I / \mathcal{U}$.