Model completeness of o-minimal structures expanded by Dedekind cuts

Contents 1. Introduction. 2. Heirs. 3. The invariance group of a cut. 4. Review of T-convex valuation rings. 5. The invariance valuation ring of a cut. 6. A method for producing cuts with prescribed signature. 7. Existentially closed extensions. 1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (p L , p R) of subsets p L , p R of M such that p L ∪ p R = M and p L < p R , i.e. a < b for all a ∈ p L , b ∈ p R. In this article we are looking for model completeness results of o-minimal structures M expanded by a set p L for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L (D)-structure (M, p L) which is model complete. The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z + for the cut p with p R = {a ∈ M | a > Z} and Z − for the cut q with q L = {a ∈ M | a < Z}. We call Z + the upper edge of Z and Z − the lower edge of Z. Then the Cherlin-Dickmann theorem says that (M, p L) is model complete if p is the upper edge of a convex valuation ring of a real closed field M. This theorem has been generalized by van den Dries and Lewenberg in [vdD-Lew], for o-minimal expansions M of real closed fields and so called T-convex subrings of M (where T is the theory of M ; a T-convex valuation ring is the convex hull of an elementary restriction of M , cf. (4.3)).