Skip to main content
SpringerLink
Log in

On VC-minimal fields and dp-smallness

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, H.: Theories controlled by formulas of Vapnik-Chervonenkis codimension 1. Preprint (2008)

  2. Aschenbrenner, A., Dolich, A., Haskell, D., MacPherson, H.D., Starchenko, S.: Vapnik-Chervonenkis density in some theories without the independence property, II. Notre Dame J. Form. Log. 54(3–4), 311–363 (2013)

    Google Scholar 

  3. Cotter S., Starchenko S.: Forking in VC-minimal theories. J. Symb. Log. 77(4), 1257–1271 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dolich A., Goodrick J., Lippel D.: Dp-minimality: Basic facts and examples. Notre Dame J. Form. Log. 52(3), 267–288 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Flenner, J., Guingona, V.: Canonical forests in directed families. Proc. Am. Math. Soc. (2013) (to appear)

  6. Flenner, J., Guingona, V.: Convexly orderable groups and valued fields. J. Symbolic Logic (2012) (to appear)

  7. Goodrick J.: A monotonicity theorem for dp-minimal densely ordered groups. J. Symb. Log. 75(1), 221–238 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Guingona V., Laskowski M.C.: On VC-minimal theories and variants. Arch. Math. Log. 52(7), 743–758 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Krupinski K., Pillay A.: On stable fields and weight. J. Inst. Math. Jussieu 10(2), 349–358 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Macintyre A., McKenna K., Van den Dries L.: Elimination of quantifiers in algebraic structures. Adv. Math. 47, 74–87 (1983)

    Article  MATH  Google Scholar 

  11. Marker D., MacPherson D., Steinhorn C.: Weakly o-minimal structures and real closed fields. Trans. Am. Math. Soc. 352(12), 5435–5483 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Onshuus A., Usvyatsov A.: On dp-minimality, strong dependence, and weight. J. Symb. Log. 76(3), 737–758 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ribenboim, P.: Theorie des Valuations, vol. 40, pp. 2670. Les Presses de l’Universite de Montreal, Montreal (1964)

  14. Simon P.: On dp-minimal ordered structures. J. Symb. Log. 76(2), 448–460 (2011)

    Article  MATH  Google Scholar 

  15. Wood C.: Notes on the stability of separably closed fields. J. Symb. Log. 44, 412–416 (1979)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN, 46556, USA

    Vincent Guingona

Authors
  1. Vincent Guingona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Guingona.

Additional information

The author was supported by NSF Grant DMS-0838506.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guingona, V. On VC-minimal fields and dp-smallness. Arch. Math. Logic 53, 503–517 (2014). https://doi.org/10.1007/s00153-014-0376-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-014-0376-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation