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A note on the pp conjecture for sheaves of spaces of orderings. (English) Zbl 1396.11060
A space of orderings \((X,G)\) in the sense of M. Marshall [Spaces of orderings and abstract real spectra. Berlin: Springer (1996; Zbl 0866.12001)] consists of a nonempty set \(X\), a subgroup \(G\) of \(\{ 1,-1\}^X\) that contains the constant function \(-1\) and separates points in \(X\) and such that \(X\) and \(G\) satisfy some further axioms. Recall that the pp conjecture asks whether a positive primitive formula holds in a space of orderings provided the formula holds in every finite subspace of that space of orderings. It is known that the answer is negative in general as shown by M. Marshall and the author in [J. Algebra Appl. 6, No. 2, 245–257 (2007; Zbl 1113.11024)]. Let now \(I\) be a Boolean space and \((X_i,G_i)\), \(i\in I\), be spaces of orderings. M. Marshall and the author [Fundam. Math. 229, No. 3, 255–275 (2015; Zbl 1379.11033)] defined a space of orderings \((X,G)\) called the sheaf of the spaces \((X_i,G_i)\), where \(X\) is the disjoint union of the \(X_i\) equipped with a suitable topology and where \(G\) consists of those continuous maps \(\phi:X\to\{\pm 1\}\) with \(\phi|_{X_i}\in G_i\). The main goal of the present paper is to show that if the pp conjecture holds in every \((X_i,G_i)\), then it also holds in their sheaf \((X,G)\). This theorem was originally proved by V. Astier [Arch. Math. Logic 46, No. 5–6, 481–488 (2007; Zbl 1176.03017)] using methods from model theory. In the present note, the author gives a new proof within the framework of the theory of spaces of orderings by invoking only basic topology and set theory.
MSC:
11E10 Forms over real fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
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