The Journal of Symbolic Logic

Research Article

T-convexity and tame extensions

Lou van den Dries and Adam H. Lewenberga1

a1 Department of Mathematics, University of Illinois, Urbana, Illinois 61820, E-mail: adam@math.uiuc.edu

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.

(Received February 22 1993)

(Revised March 21 1994)

1991 Mathematics Subject Classification

  • Primary 03C10;
  • 12J10;
  • 12J15;
  • Secondary 03C35

Key words and phrases:

  • Quantifier elimination;
  • T-convexity;
  • o-minimal theories;
  • tame pairs;
  • valued fields;
  • polynomially bounded theories