Journal of Symbolic Logic

$T$-Convexity and Tame Extensions

Lou Van Den Dries and Adam H. Lewenberg

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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 $(\mathscr{R}, V)$ where $\mathscr{R} \models T$ and $V \neq \mathscr{R}$ is the convex hull of an elementary substructure of $\mathscr{R}$. 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 $(\mathscr{R}, \mathscr{N})$ with $\mathscr{R}$ a model of $T$ and $\mathscr{N}$ a proper elementary substructure that is Dedekind complete in $\mathscr{R}$. We deduce that the theory of such "tame" pairs is complete.

Article information

Source
J. Symbolic Logic, Volume 60, Issue 1 (1995), 74-102.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744679

Mathematical Reviews number (MathSciNet)
MR1324502

Zentralblatt MATH identifier
0856.03028

JSTOR
links.jstor.org

Subjects
Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 12J10: Valued fields 12J15: Ordered fields 03C35: Categoricity and completeness of theories

Keywords
Quantifier elimination $T$-convexity $o$-minimal theories tame pairs valued fields polynomially bounded theories

Citation

Dries, Lou Van Den; Lewenberg, Adam H. $T$-Convexity and Tame Extensions. J. Symbolic Logic 60 (1995), no. 1, 74--102. https://projecteuclid.org/euclid.jsl/1183744679


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