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T-convexity and tame extensions

Published online by Cambridge University Press:  12 March 2014

Lou van den Dries  and
Adam H. Lewenberg
Adam H. Lewenberg
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61820, E-mail: adam@math.uiuc.edu
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Abstract

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (ℛ, V) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.

Keywords

Primary 03C1012J1012J15Secondary 03C35Quantifier eliminationT-convexityo-minimal theoriestame pairsvalued fieldspolynomially bounded theories
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 74 - 102
Copyright
Copyright © Association for Symbolic Logic 1995

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