# Variations on the Feferman-Vaught Theorem, with applications to $\prod_p \mathbb{F}_p$

## TL;DR

This paper explores the Feferman-Vaught Theorem's implications for product structures, showing definable sets are Boolean combinations of open sets and characterizing definable subsets of products of finite fields.

## Contribution

It extends the Feferman-Vaught Theorem with a converse for certain structure families and characterizes definable sets in products of finite fields.

## Key findings

- Definable subsets are Boolean combinations of open sets in the product topology.
- Every formula in the structure is equivalent to a Boolean combination of x orall x formulas.
- Provides a characterization of definable subsets of igp _p structures.

## Abstract

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 a converse of the Feferman-Vaught theorem for families of structures with certain properties, including families of integral domains. We use these results to obtain characterizations of the definable subsets of $\prod_p \mathbb{F}_p$ -- in particular, every formula is equivalent to a Boolean combination of $\exists \forall \exists$ formulae.