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Archive for Mathematical Logic

, Volume 45, Issue 8, pp 983–1009 | Cite as

Pseudo completions and completions in stages of o-minimal structures

  • Marcus Tressl
Article

Abstract

For an o-minimal expansion R of a real closed field and a set \(\fancyscript{V}\) of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to \(\fancyscript{V}\). This is an elementary extension S of R generated by all completions of all the residue fields of the \(V \in \fancyscript{V}\), when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to an R-isomorphism. For polynomially bounded R we can iterate the construction of the pseudo completion in order to get a “completion in stages” S of R with respect to \(\fancyscript{V} \). S is the “smallest” extension of R such that all residue fields of the unique extensions of all \(V \in \fancyscript{V}\) to S are complete.

Mathematics Subject Classification (2000)

Primary 03C64 Primary 12J10 Primary 12J15 Secondary 13B35 

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References

  1. 1.
    Bochnak J., Coste M., Roy M.F.(1998). Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 36. Springer, Berlin Heidelberg New YorkGoogle Scholar
  2. 2.
    van den Dries L., Lewenberg A.H.(1995). T-convexity andtame extension. J. Symb. Logic 60(1): 74–101zbMATHCrossRefGoogle Scholar
  3. 3.
    van den Dries L., Speissegger P.(2000). The field of realswith multisummable series and the exponential function. Proc. Lond. Math. Soc. 81(3): 513–565zbMATHCrossRefGoogle Scholar
  4. 4.
    Hodges W.(1993). Model Theory. Encyclopedia of mathematics and its applications, vol. 42. Cambridge university Press, CambridgeGoogle Scholar
  5. 5.
    Marker D., Steinhorn C.(1994). Definable types in o-minimal theories. J. Symb. Logic 59, 185–198zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Pillay A., Steinhorn C.(1986). Definable sets in ordered structures I. Trans. Am. Math. Soc. 295, 565–592zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Prieß-Crampe S.(1983). Angeordnete Strukturen: Gruppen,Körper, projektive Ebenen. Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 98. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. 8.
    Ribenboim P.(1964). Théorie des valuations. Les Presses de l’Université de Montréal, MontrealzbMATHGoogle Scholar
  9. 9.
    Tressl M.(2005). Model Completeness of o-minimal Structures expanded by Dedekind Cuts. J. Symb. Logic 70(1): 29–60zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    van der Waerden B.L.(1966). Algebra I. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.NWF-I MathematikUniversität RegensburgRegensburgGermany

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