Archive for Mathematical Logic

, Volume 52, Issue 5–6, pp 667–688 | Cite as

Cell decomposition for semibounded p-adic sets

Article
  • 72 Downloads

Abstract

We study a reduct \({\mathcal{L}_*}\) of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the \({\mathcal{L}_*}\) -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, \({\mathcal{L}_*}\)) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.

Keywords

Cell decomposition Quantifier elimination p-adic numbers p-minimality o-minimality Definability Restricted multiplication 

Mathematics Subject Classification (2000)

03C07 03C10 03C64 11U09 12J12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cluckers R., Leenknegt E.: A version of p-adic minimality. J. Symb. Log. 77(2), 621–630 (2012)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Denef J.: The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77, 1–23 (1984)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Denef J.: p-adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369, 154–166 (1986)MathSciNetMATHGoogle Scholar
  4. 4.
    Haskell D., Macpherson D.: A version of o-minimality for the p-adics. J. Symb. Log. 62(4), 1075–1092 (1997)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Leenknegt E.: Cell decomposition and definable functions for weak p-adic structures. MLQ Math. Log. Q. 58(6), 482–497 (2012a)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Leenknegt E.: Cell decomposition for semi-affine structures on p-adic fields. J. Log. Anal. 4(14), 1–25 (2012b)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Leenknegt, E.: Reducts of p-adically closed fields. Preprint, February (2012c)Google Scholar
  8. 8.
    Liu N.: Semilinear cell decomposition. J. Symb. Log. 59(1), 199–208 (1994)MATHCrossRefGoogle Scholar
  9. 9.
    Macintyre A.: On definable subsets of p-adic fields. J. Symb. Log. 41, 605–610 (1976)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Mourgues M.-H.: Cell decomposition for P-minimal fields. MLQ Math. Log. Q. 55(5), 487–492 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Peterzil Y.: A structure theorem for semibounded sets in the reals. J. Symb. Log. 57(3), 779–794 (1992)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Peterzil Y.: Reducts of some structures over the reals. J. Symb. Log. 58(3), 955–966 (1993)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Pillay, A., Scowcroft, P., Steinhorn, C.: Between groups and rings. Rocky Mt. J. Math. 19(3), Summer (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest-LafayetteUSA

Personalised recommendations