Baur-Monk quantifier elimination (BG-invariants in 1-free variable)

Question

$\DeclareMathOperator\Inv{Inv}$Baur-Monk quantifier elimination implies that a sentence in the language of modules is a combination of BG invariant statements.

A BG invariant sentence is a boolean combination of $ \Inv(A,\phi, \psi) * k$ where $*$ is one of $=,>,<$ where $\phi$ and $\psi$ are positive primitive formulae.

We define $\Inv(A,\phi, \psi)$ as the cardinality of the quotient group $\phi(A) / \psi(A)$ and $\Inv(A,\phi, \psi) > k$ is equivalent modulo the theory of modules $T$ to the following first-order statement:

\begin{equation*} \forall \overline{v_1},...,\overline{v_k} \exists \overline{v} (\phi(\overline{v}) \wedge {\bigwedge_{i=1}^{k}} \neg \psi (\overline{v}-\overline{v_i})). \end{equation*}

It is then concluded from quantifier elimination that $\Inv(A,\phi, \psi)=\Inv(B,\phi, \psi)$ iff $A \equiv B$ (elementarily) where $\phi$ and $\psi$ are positive primitive in one free variable. My question is why it follows that we can restrict to invariants with formulas in one free variable? (Instead of just invariants). I assume that it is because the positive primitive formulas are conjunction closed. Is this correct?

Answer 1

This doesn't follow directly from the statement of QE (and I don't think it has anything to do with the fact that pp formulas are conjunction-closed). To understand it, you really have to look at the proof of QE. For example, see the proof of Theorem 1.1 on p. 155 of Ziegler's paper Model Theory of Modules.

The proof of QE works one quantifier at a time. For a fixed module $M$, you show that if $\varphi(x,\overline{y})$ is equivalent (in $M$) to a Boolean combination of pp formulas, then so is $\forall x\, \varphi(x,\overline{y})$. Here $x$ is a single variable. By induction, it follows that every formula is equivalent (in $M$) to a Boolean combination of pp formulas.

But now, examining the proof, you observe that the proof almost works uniformly in $M$: when you show that $\forall x\, \varphi(x,\overline{y})$ is equivalent to the boolean combination of pp formulas $\theta(\overline{y})$, the formula $\theta$ only depends on the values of finitely many $\mathrm{Inv}(M,\psi,\chi)$ where $\psi$ and $\chi$ are pp formulas in the single variable $x$. So if you know all the values of the BG invariants $\mathrm{Inv}(M,\psi,\chi)$ when $\psi$ and $\chi$ are pp formulas in a single variable, then you know which quantifier-free formula every formula is equivalent to. In particular, these values determine whether every sentence is true or false in $M$.