T-Convexity and Tame Extensions

Journal of Symbolic Logic 60 (1):74 - 102 (1995)

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 $\mathscr{R} \models T$ 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 (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete
Keywords Quantifier elimination   $T$-convexity   $o$-minimal theories   tame pairs   valued fields   polynomially bounded theories
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DOI 10.2307/2275510
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References found in this work BETA

Real Closed Rings II. Model Theory.Gregory Cherlin & Max A. Dickmann - 1983 - Annals of Pure and Applied Logic 25 (3):213-231.
Expansions of the Real Field with Power Functions.Chris Miller - 1994 - Annals of Pure and Applied Logic 68 (1):79-94.
Model Completeness Results for Elliptic and Abelian Functions.Ricardo Bianconi - 1991 - Annals of Pure and Applied Logic 54 (2):121-136.

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Citations of this work BETA

Pseudo Completions and Completions in Stages of o-Minimal Structures.Marcus Tressl - 2006 - Archive for Mathematical Logic 45 (8):983-1009.
O-Minimal Residue Fields of o-Minimal Fields.Jana Maříková - 2011 - Annals of Pure and Applied Logic 162 (6):457-464.
The Elementary Theory of Dedekind Cuts in Polynomially Bounded Structures.Marcus Tressl - 2005 - Annals of Pure and Applied Logic 135 (1-3):113-134.
Non-Archimedean Stratifications of Tangent Cones.Erick García Ramírez - 2017 - Mathematical Logic Quarterly 63 (3-4):299-312.

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