Variations on the Feferman-Vaught theorem, with applications to \(\prod _p \mathbb {F}_p\)

Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove that for families of structures with certain properties, including families of integral domains, the pure Boolean generalized product is definable in the direct product structure. We use these results to obtain characterizations of the definable subsets of \(\prod _p \mathbb {F}_p\) ${\prod }_p{\mathbb{F}}_p$ —in particular, every formula is equivalent to a Boolean combination of \(\exists \forall \exists \) $\exists \forall \exists$ formulae.