= Reduced product
{title2=$\prod_{i\in I}\mathcal M_i/F$}
Given nonempty <first-order structures> $\mathcal M_i$ and a proper <filter on a set> $F$, a reduced product identifies product functions $f,g$ when $\{i:f(i)=g(i)\}\in F$. Operations are interpreted coordinatewise, and relations hold when their coordinate truth sets belong to $F$. For an <ultrafilter> this is an <ultraproduct>. <Primitive positive formulas> transfer through an arbitrary proper filter, whereas <logical disjunction> and <logical negation> can fail to transfer.
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