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Local Stability and Saturation in Spaces of Orderings

Published online by Cambridge University Press:  20 November 2018

Niels Schwartz
Niels Schwartz*
Affiliation:
Universität München, München, W. Germany
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If k is a f.r. (= formally real) field which is partially ordered with positive cone, P, XP denotes the space of total orders T of k with P ⊂ T. Suppose you have a subset A ⊂ XP and an element T ∈ XP, T ∉ A. Then the main question investigated in this paper is the following: How can T be separated from A by using elements of k? To be more specific, this is split up into two different questions.

Question 1. Suppose A is closed. Then there is an n ∈ N and elements a1, …, an ∈ k such that the basic open set H = H(a1, …, an) is a neighborhood of T and has empty intersection with A. Now, if T is given, what is the least n ∊ N (if it exists) such that T has a neighborhood basis consisting of basic open sets of the form H(a1, …, an)?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 454 - 477
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Becker, E. and Bröcker, L., On the description of the reduced Witt ring, J. Alg. 52 (1978), 328–346.Google Scholar
2. Becker, E. and Köing, E., Reduzierte quadratische Formen, Abh. Math. Sem. Univ. Hamb. 46 (1977), 143–177.Google Scholar
3. Bröcker, L., Zur Theorie der quadratischen Formen über formal reellen Körpern, Math. Ann. 270 (1974), 233–256.Google Scholar
4. Bröcker, L., Characterization of fans and hereditarily pythagorean fields, Math. Z. 151 (1976), 149–163.Google Scholar
5. Bröcker, L., Über die Anzahl der Anordnungen eines kommutativen Körpers, Arch. Math. 29 (1977), 458–464.Google Scholar
6. Coste, M. and Coste-Roy, M.-F., La topologie du spectre reel, Preprint.Google Scholar
7. Craven, T., Stability in Witt rings, Transactions AMS 225 (1977), 227–242.Google Scholar
8. Dubois, D. W., Infinite primes and ordered fields, Dissertationes Math. 69 (1970).Google Scholar
9. Elman, R. and Lam, T. Y., Quadratic forms over formally real fields and pythagorean fields, Amer. J. M. 94 (1972), 1155–1194.Google Scholar
10. Elman, R., Lam, T. Y. and Wadsworth, A., Pfister ideals in Witt-rings, Math. Ann. 245 (1979), 219–245.Google Scholar
11. Kleinstein, J., Abstract Witt rings, Thesis.Google Scholar
12. Kleinstein, J., Abstract Witt rings, In: Conference on quadratic forms – 1976, Kingston, Ontario (1977).Google Scholar
13. Kleinstein, J. and Rosenberg, A., Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings, Can. J. Math. 30 (1978), 872–895.Google Scholar
14. Knebusch, M., Rosenberg, A. and Ware, R., Signatures on semilocal rings, J. Alg. 26 (1973), 208–250.Google Scholar
15. Lam, T. Y., The algebraic theory of quadratic forms (Benjamin, Reading, Mass., 1973).Google Scholar
16. Marshall, M., Classification of finite spaces of orderings, Can. J. Math. 31 (1979), 320–330.Google Scholar
17. Marshall, M., The Witt ring of a space of orderings, Transactions AMS 258 (1980), 505–521.Google Scholar
18. Marshall, M., Spaces of orderings IV, Preprint.CrossRefGoogle Scholar
19. Prestel, A., Lectures on formally real fields, IMPA, Rio de Janeiro (1975).Google Scholar
20. Schülting, H.-W., Über reelle Stellen eines Körpers und ihren Holomorphiering, Dissertation. Dortmund (1979).Google Scholar