Model completeness of o-minimal structures expanded by Dedekind cuts

Abstract

§1. Introduction. LetMbe a totally ordered set. A (Dedekind) cutpofMis a couple (pL,pR) of subsetspL,pRofMsuch thatpL⋃pR=MandpL<pR, i.e.,a<bfor alla∈pL,b∈pR. In this article we are looking for model completeness results of o-minimal structuresMexpanded by a setpLfor a cutpofM. This means the following. LetMbe an o-minimal structure in the languageLand supposeMis model complete. LetDbe a new unary predicate and letpbe a cut of (the underlying ordered set of)M. Then we are looking for a natural, definable expansion of theL(D)-structure (M,pL) 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. IfZis a subset of an ordered setMwe writeZ+for the cutpwithpR= {a∈M∣a>Z} andZ−for the cutqwithqL= {a∈M∣a<Z}.