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Open questions in the theory of spaces of orderings. (English) Zbl 1018.11017
In the theory of the spaces of orderings it is a proven strategy to reduce questions to the finite case via the isotropy theorem and its extended version. Nevertheless there are problems that do not appear to be solved in this way. Four of those are the topic of the present paper:
Let \((X,G)\) be a space of orderings.
1) Does every positive primitive formula that holds in every finite subspace of \((X,G)\) necessarily hold in \((X,G)\)?
2) Is there a certain \(\operatorname {mod}2^n\)-version of the representation theorem?
3) Does the formula \(\text{Cont} (X,2^n\mathbb{Z})\cap \text{Witt} (X,G)= I^n(X,G)\) hold for \(n\geq 1\)?
4) Is the separating depth of a constructible set \(C\) in \(X\) necessarily bounded by the stability index of \((X,G)\)?
Although all these questions are interesting in there own right, the first one is the crucial one, since the author proves in this paper that the other ones hold, if question 1) is answered positively. He also shows that 1) holds if \((X,G)\) has stability index 1 or is a direct sum or group extension of such spaces of orderings. This also implies that 1) holds for all spaces of orderings of finite chain length.

MSC:
11E10 Forms over real fields
12J15 Ordered fields
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