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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

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    Tressl, Marcus 2006. Pseudo completions and completions in stages of o-minimal structures. Archive for Mathematical Logic, Vol. 45, Issue. 8, p. 983.


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T-convexity and tame extensions

  • Lou van den Dries and Adam H. Lewenberg (a1)
  • DOI: http://dx.doi.org/10.2307/2275510
  • Published online: 01 March 2014
Abstract
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.

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[Bi]Ricardo Bianconi , Model completeness results for elliptic and abelian functions, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 121136.

[B1]Ludwig Bröcker , On the reductions of semialgebraic sets by real valuations, Recent advances in real algebraic geometry and quadratic forms (W. B. Jacob , T.-Y. Lam , and R. O. Robson , editors), Contemporary Mathematics, vol. 155, 1994, pp. 7595.

[B2]Ludwig Bröcker , Families of semialgebraic sets and limits, Real algebraic geometry, proceedings (Rennes, 1991) (M. Coste , et al, editors), Lecture Notes in Mathematics no. 1524, Springer-Verlag, Berlin, 1992, pp. 145162.

[C-D]Gregory Cherlin and Max A. Dickmann , Real closed rings. II: Model theory, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 213231.

[D-vdD]J. Denef and L. van den Dries , p-adic and real subanalytic sets, Annals of Mathematics, ser. 2, vol. 128 (1988), pp. 79138.

[vdD2]Lou van den Dries , A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bulletin (New Series) of the American Mathematical Society, vol. 15 (1986), pp. 189193.

[vdD-Mi]Lou van den Dries and Chris Miller , On the real exponential field with restricted analytic functions, Israel Journal of Mathematics, vol. 85 (1994), pp. 1956.

[K-P-S]Julia F. Knight , Anand Pillay , and Charles Steinhorn , Definable sets in ordered structures, II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.

[Mi2]Chris Miller , Expansions of the real field with power functions, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 7994.

[P-S]Anand Pillay and Charles Steinhorn , Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.

[Pr]Alexander Prestel , Einführung in die Mathematische Logik und Modelltheorie, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1986.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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