The Journal of Symbolic Logic

Research Article

T-convexity and tame extensions II

Lou van den Dries

Department of Mathematics, University of Illinois, Urbana, IL 61820, USA, E-mail:

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular, T will denote a complete o-minimal theory extending RCF, the theory of real closed fields. Let ( , V) ⊨ T convex, let = V/m(V) be the residue field, with residue class map x : V , and let υ: → Γ be the associated valuation. “Definable” will mean “definable with parameters”. The main goal of this article is to determine the structure induced by ( , V) on its residue field and on its value group Γ. In [9] we expanded the ordered field to a model of T as follows. Take a tame elementary substructure ′ of such that R′ ⊆ V and R′ maps bijectively onto under the residue class map, and make this bijection into an isomorphism ′ ≌ of T-models. (We showed such ′ exists, and that this gives an expansion of to a T-model that is independent of the choice of ′.).

(Received March 21 1995)

(Revised August 30 1995)