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Model completeness of o-minimal structures expanded by Dedekind cuts

Published online by Cambridge University Press:  12 March 2014

Marcus Tressl
Marcus Tressl*
Affiliation:
Universität Regensburg, NWF I - Mathematik, D-93040 Regensburg, Germany, E-mail: [email protected]
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§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL 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, 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. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 29 - 60
Copyright
Copyright © Association for Symbolic Logic 2005

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