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Some model theory for almost real closed fields

Published online by Cambridge University Press:  12 March 2014

Françoise Delon  and
Rafel Farré
Françoise Delon
Affiliation:
U. F. R. de Mathématiques, Université Paris VII, 2, Place Jussieu, 75251 Paris Cedex 05, France, E-mail: delon@logique.jussieu.fr
Rafel Farré
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo, 5, 08028 Barcelona, Spain, E-mail: farre@ma2.upc.es
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Abstract

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

Keywords

Henselian fieldsreal closed fieldsordered abelian groupsdecidability
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 4 , December 1996 , pp. 1121 - 1152
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Becker, E., Hereditarily-Pythagorean fields and orderings of higher level, IMPA Lecture Notes, no. 29, Rio de Janeiro, 1978.Google Scholar
[2]Becker, E., Summen n-ter Potenzen in Körpern, Journal für die reine und angewandte Mathematik, vol. 307–308 (1979), pp. 8–30.Google Scholar
[3]Becker, E., Berr, R., and Gondard, D., Henselian fields with real-closed residue field, preprint.Google Scholar
[4]Brown, R., Craven, T., and Pelling, M.J., Ordered fields satisfying Rolle's theorem, Illinois Journal of Mathematics (1986), pp. 66–79.Google Scholar
[5]Chang, C. C. and Keisler, H. J., Model theory, Studies in Logic and the Foundations of Mathematics, no. 73, North-Holland, Amsterdam-London, 1973.Google Scholar
[6]Delon, F., Quelques propriétés des corps valués en théorie des modèles, Thèse d'état, Université Paris VII, 1982.Google Scholar
[7]Delon, F., Corps et anneaux de Rolle, Proceedings of the American Mathematical Society (1986), pp. 315–320.Google Scholar
[8]Delon, F. and Lucas, F., Inclusions et produits de groupes abéliens ordonnés étudiés au premier ordre, this Journal, vol. 54 (1989), pp. 499–511.Google Scholar
[9]Farré, R., Some model theory for valued and ordered fields, and applications, 1993.CrossRefGoogle Scholar
[10]Farré, R., A transfer theorem for henselian valued and ordered fields, this Journal, vol. 58 (1993), no. 3, pp. 915–930.Google Scholar
[11]Feferman, S. and Vaught, R. L., The first-order properties of products of algebraic systems, Fundament a Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[12]Gurevich, Y., Elementary properties of ordered Abelian groups, Translations of the American Mathematical Society, vol. 46 (1965), pp. 165–192.Google Scholar
[13]Harman, J., Chains of higher level orderings, Contemporary Mathematics, vol. 8 (1982), pp. 141–174.CrossRefGoogle Scholar
[14]Jacob, B., The model theory of Pythagorean fields, Ph.D. thesis, Princeton, 1979.Google Scholar
[15]Jacob, B., The model theory of generalized real-closed fields, Journal für die reine und angewandte Mathematik, vol. 323 (1981), pp. 213–220.Google Scholar
[16]Knebusch, M. and Wright, M., Bewertungen mit reeller Henselisierung, Journal für die reine und angewandte Mathematik, vol. 286–287 (1976), pp. 314–321.Google Scholar
[17]Kuhlmann, F. V. and Prestel, A., On places of algebraic function fields, Journal für die reine und angewandte Mathematik, vol. 353 (1984), pp. 181–195.Google Scholar
[18]Prestel, A., Lectures on formally real fields, Lecture Notes in Mathematics, no. 1093, Springer-Verlag, Berlin-Heidelberg, 1984.CrossRefGoogle Scholar
[19]Ribenboim, P., Théorie des Valuations, Les Presses de l'Université de Montréal, Montréal, 1964.Google Scholar
[20]Schmitt, P. H., Elementary properties of ordered abelian groups, preprint.Google Scholar
[21]Schmitt, P. H., Model and substructure-complete theories of ordered Abelian groups, Models and sets (Müller, G. and Richter, M., editors), Lecture Notes in Mathematics, no. 1103, Springer-Verlag, Berlin, 1984, pp. 389–418.CrossRefGoogle Scholar
[22]Schmitt, P.H., Model theory of ordered Abelian groups, Habilitationsschrift, Heidelberg, 1982.Google Scholar
[23]Schwartz, N., Signatures and real closures of a field, vol. 1, pp. 65–78, Publications de l'Université de Paris 7, vol. 33, pp. 65–78.Google Scholar
[24]Schwartz, N., Chain signatures and real closures, Journal für die reine und angewandte Mathematik, vol. 347 (1984), pp. 1–20.Google Scholar
[25]Szmielew, W., Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.CrossRefGoogle Scholar
[26]van den Dries, L., Model theory of fields, Ph.D. thesis, Utrecht, 1978.Google Scholar
[27]Weispfenning, V., Quantifier elimination and decision procedures for valued fields, Models and sets (Müller, G. and Richter, M., editors), Lecture Notes in Mathematics, no. 1103, Springer-Verlag, Berlin, 1984, pp. 419–472.CrossRefGoogle Scholar
[28]Zakon, E., Generalized archimedean groups, Transactions of the American Mathematical Society, vol. 99 (1961), pp. 21–48.CrossRefGoogle Scholar
[29]Ziegler, M., Einige unentscheidbare Körpertheorien, L'enseignement mathématique, vol. XXVIII (1982), pp. 269–280.Google Scholar