## Journal of Symbolic Logic

### $T$-Convexity and Tame Extensions

#### 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

https://projecteuclid.org/euclid.jsl/1183744679

Mathematical Reviews number (MathSciNet)
MR1324502

Zentralblatt MATH identifier
0856.03028

JSTOR
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