Baur-Monk quantifier elimination theorem asserts that any formula in the language of modules is modulo the theory a boolean combination of BG-Invariants and positive primitive formulas. However, in p.54 Model theory of Modules by Prest, just right after proving the theorem, it was immediately concluded as Cor 2.15 that every sentence is a boolean combination of BG-Invariant conditions. Why does it follow? Thanks.
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1$\begingroup$ Isn't this just because a positive primitive sentence (i.e. a positive primitive formula with no free variables) is true in every module? So the Boolean combination of BG-Invariants and positive primitive sentences can be simplified to remove the positive primitive sentences. $\endgroup$– Alex KruckmanCommented Jun 6, 2023 at 4:48
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$\begingroup$ Thanks for this. However, I could not see how a positive primitive sentence would be true in every module. Why would that be the case? $\endgroup$– Eladio VuenteCommented Jun 6, 2023 at 14:18
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$\begingroup$ A pp sentence has the form $\exists x_1\dots\exists x_n\, \varphi(x_1,\dots,x_n)$, where $\varphi$ is a finite conjunction of equations between linear combinations of the variables $x_1,\dots,x_n$. Since $\varphi(0,\dots,0)$ is true in every module, the pp sentence is true in every module. $\endgroup$– Alex KruckmanCommented Jun 6, 2023 at 16:01
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$\begingroup$ Thank you. I thought of it as quantifying the variables $v_i$ in the definition: $\begin{equation*} \exists x_1,...,x_n {\bigwedge_{j=1}^{m}} (\sum_{i=1}^{p} v_i r_{ij} + \sum_{k=1} ^{n} x_k s_{kj} = 0).\end{equation*}$ It turns out we remove them altogether, and the formula is always satisfied by (0,...,0) $\endgroup$– Eladio VuenteCommented Jun 6, 2023 at 16:50
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By Baur-Monk QE, every sentence is equivalent to a boolean combination of BG-invariants and positive primitive sentences (which have no free variables).
A positive primitive sentence $\psi$ has the form $\exists x_1\dots\exists x_n\,\varphi(x_1,\dots,x_n)$, where $\varphi$ is a finite conjunction of equations between linear combinations of the variables $x_1,\dots,x_n$. Since $\varphi(0,\dots,0)$ is true in every module, $\psi$ is true in every module.
It follows that the Boolean combination of BG-invariants and positive primitive sentences can be simplified to remove the positive primitive sentences.
answered Jun 6, 2023 at 17:10