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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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