What does Feferman-Vaught say $\mathbf{exactly}$ about definable subsets of a direct product of two structures? Below I reproduce a consequence of the Feferman-Vaught theorem, taken from Wilfrid Hodges' book Model Theory:

Corollary 9.6.4: Let $L$ be a first-order language, let $A$ and $B$ be $L$-structures and let $\phi(\overline{x})$ be a formula of $L$. Then there is a finite set $\bigl\{\bigl(\theta_i(\overline{x}),\chi_i(\overline{x})\bigr): i<n\bigr\}$ of pairs of formulas of $L$, such that for all tuples $\overline{a}=(a_0,a_1,\ldots),\overline{b}=(b_0,b_1,\ldots)$ from $A,B$ respectively,
$$A\times B\models\phi\bigl((a_0,b_0),(a_1,b_1),\ldots\bigr)\iff\ \style{font-family:inherit;}{\text{for some}}\ i<n, A\models\theta_i(\overline{a})\ \style{font-family:inherit;}{\text{and}}\ B\models\chi_i(\overline{b}).$$

I have two questions about this result:


*

*Does this result say that a definable subset in a product of two structures is the finite union of "definable rectangles" (that is, Cartesian produts of definable subsets in the respective components)?


*It is true, conversely, that every definable rectangle is a definable subset of the product structure?

Note that Question 2 is not trivial, because the statement of the corollary assumes that a definable subset of the product is given.
Bonus question: Does the results above hold for definable subsets with parameters in a uniform way? that is, if $\phi$ has parameters, can formulas $\theta_i$ and $\chi_i$ be chosen depending on the components of such parameters? I ask this because the proof of the result is left to the reader ("Proof: Unpick what the theorem says."), which amounts to reexamine the previous material.
 A: Re: $(1)$, yes, that's exactly what the result is saying.
And this holds uniformly since we can always look at expansions of our starting structures:

Suppose $d_1=(a_1,b_1),..., d_k=(a_k,b_k)\in A\times B$ and $X\subseteq (A\times B)^n$ is definable over $d_1,..., d_k$ in the structure $A\times B$. Let $u_1,...,u_k$ be new constant symbols and let $A'$ and $B'$ be the expansions of $A$ and $B$ gotten by interpreting $u_i$ as $a_i$ and $b_i$, respectively. Then $X$ is parameter-freely definable in the structure $A'\times B'$.


Re: $(2)$, perhaps surprisingly, the answer is no - in general, products of definable sets are not definable. This is because we "lose the coordinatization." For example, suppose both $A$ and $B$ are the group $(\mathbb{Z};+)$, and consider the set $$X=\{(a,b)\in\mathbb{Z}^2: a=0\}.$$ Since both $\mathbb{Z}$ and $\{0\}$ are definable in $(\mathbb{Z};+)$, the set $X$ is a product of definable sets, but it is clearly not definable since it is not fixed by all automorphisms (consider the map $(u,v)\mapsto (v,u)$, for example).
