# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Quadratic and Hermitian forms pp 231– (1984) [2] Constructible sets in real geometry (1996) · Zbl 0873.14044 [3] Spaces of orderings and abstract real spectra (1996) · Zbl 0866.12001 [4] DOI: 10.1007/BF01832992 · Zbl 0829.14026 · doi:10.1007/BF01832992 [5] DOI: 10.1080/00927878408823025 · Zbl 0533.10018 · doi:10.1080/00927878408823025 [6] Journal of Algebra pp 68– (1978) [7] DOI: 10.4153/CJM-1979-035-4 · Zbl 0412.10012 · doi:10.4153/CJM-1979-035-4 [8] DOI: 10.4153/CJM-1980-047-0 · Zbl 0433.10009 · doi:10.4153/CJM-1980-047-0 [9] Memoirs of the American Mathematical Society (2000) [10] DOI: 10.4153/CJM-1979-061-4 · Zbl 0419.10024 · doi:10.4153/CJM-1979-061-4 [11] Transactions of the American Mathematical Society pp 505– (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.