Skip to main content
SpringerLink
Log in

Elementary equivalence of some rings of definable functions

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

Abstract

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R n to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.

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. Bergman G.M.: Hereditary commutative rings and centres of hereditary rings. Proc. Lond. Math. Soc. 23(3), 214–236 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  2. Comer S.D.: Elementary properties of structures of sections. Bol. Soc. Mat. Mexicana (2) 19(2), 78–85 (1974)

    MATH  MathSciNet  Google Scholar 

  3. Feferman S., Vaught R.L.: The first order properties of products of algebraic systems. Fund. Math. 47, 57–103 (1959)

    MATH  MathSciNet  Google Scholar 

  4. Hodges W.: Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  5. Koppelberg, S.: Handbook of Boolean algebras, vol. 1. In: Donald Monk J., Bonnet R. (eds.) North-Holland Publishing Co., Amsterdam (1989)

  6. Lipshitz L., Saracino D.: The model companion of the theory of commutative rings without nilpotent elements. Proc. Am. Math. Soc. 38, 381–387 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  7. Macintyre, A.: Model-completeness for sheaves of structures. Fund. Math. 81(1), 73–89 (Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I) (1973)

  8. Pierce, R.S.: Modules over commutative regular rings. Memoirs of the American Mathematical Society, No. 70. American Mathematical Society, Providence (1967)

  9. Prestel, A., Schwartz, N.: Model theory of real closed rings. In: Valuation theory and its applications, vol. I (Saskatoon, SK, 1999), Fields Inst. Commun., vol. 32, pp. 261–290. American Mathematical Society, Providence (2002)

  10. Schwartz, N., Madden, J.J.: Semi-algebraic function rings and reflectors of partially ordered rings. Lecture Notes in Mathematics, vol. 1712. Springer, Berlin (1999)

  11. Tressl, M.: Algebraization of rings of continuous functions (preprint)

  12. Weispfenning V.: Model-completeness and elimination of quantifiers for subdirect products of structures. J. Algebra 36(2), 252–277 (1975)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, D203, Universität Konstanz, 78457, Konstanz, Germany

    Vincent Astier

Authors
  1. Vincent Astier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Astier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Astier, V. Elementary equivalence of some rings of definable functions. Arch. Math. Logic 47, 327–340 (2008). https://doi.org/10.1007/s00153-008-0085-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-008-0085-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation