Definition:Ultraproduct
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Definition
Let $\LL$ be a first-order language and let $I$ be an infinite set.
Let $\UU$ be an ultrafilter on $I$.
Let $\MM_i$ be an $\LL$-structure for each $i \in I$.
The ultraproduct:
- $\ds \MM := \paren {\prod_{i \mathop \in I} \MM_i } / \UU$
is an $\LL$-structure defined as follows:
- $(1): \quad$ The universe of $\MM$:
Let $X$ be the Cartesian product:
- $\ds \prod_{i \mathop \in I} \MM_i$
Define an equivalence relation $\sim$ on $X$ by:
- $\family {a_i}_{i \mathop \in I} \sim \family {b_i}_{i \mathop \in I}$ if and only if $\set {i \in I: a_i = b_i} \in \UU$
The universe of $\MM$ is the set of equivalence classes of $X$ modulo $\sim$.
These are essentially sequences taken modulo the equivalence relation above, and are sometimes denoted $\eqclass {m_i} \UU$.
- $(2): \quad$ Interpretation of non-logical symbols of $\LL$ in $\MM$:
For each constant symbol $c$, we define $c^\MM$ to be $\eqclass {c^{\MM_i} } \UU$.
For each $n$-ary function symbol $f$, we define $f^\MM$ by setting:
- $\map {f^\MM} {\eqclass {m_{1, i} } \UU, \dotsc, \eqclass {m_{n, i} } \UU}$
to be:
- $\eqclass {\map {f^{\MM_i} } {m_{1, i}, \dotsc, m_{n, i} } } \UU$
For each $n$-ary relation symbol $R$, we define $R^\MM$ to be the set of $n$-tuples:
- $\tuple {\eqclass {m_{1, i} } \UU, \dots, \eqclass {m_{n, i} } \UU}$ from $\MM$
such that:
- $\set {i \in I: \tuple {m_{1, i}, \dotsc, m_{n, i} } \in R^{\MM_i} } \in \UU$