Israel Journal of Mathematics

, Volume 81, Issue 1–2, pp 1–30 | Cite as

Linear O-minimal structures

  • James Loveys
  • Ya’acov Peterzil


A linearly ordered structure\(\mathcal{M} = (M,< , \cdot \cdot \cdot )\) is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an\(\mathcal{M}\) is aCF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if\(\mathcal{M}\) is aCF structure which expands an (interval in an) ordered group, then it is elementary equivalent to a reduct of an (interval in an) ordered vector space. Along the way we prove several quantifier-elimination results for expansions and reducts of ordered vector spaces.


Division Ring Finite Union Elementary Extension Quantifier Elimination Order Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [vdD] L. van den Dries,Tame topology and σ-minimal structures, preliminary version, 1991.Google Scholar
  2. [HL] E. Hrushovski and J. Loveys,Weakly minimal groups of bounded exponent, in preparation.Google Scholar
  3. [KPS] J. Knight, A. Pillay and C. Steinhorn,Definable sets in ordered structures II, Trans. Amer. Math. Soc.295 (1986), 593–605.MATHCrossRefMathSciNetGoogle Scholar
  4. [P1] Y. Peterzil,Some definability questions in structures over the reals and in general o-minimal structures, Ph.D. thesis, University of California, Berkeley, 1991.Google Scholar
  5. [P2] Y. Peterzil,Zilber’s Conjecture for some o-minimal structures over the reals, to appear in Ann. Pure Appl. Logic.Google Scholar
  6. [P3] Y. Peterzil,Constructing a group-interval in o-minimal structures, preprint.Google Scholar
  7. [Pi] A. Pillay,On groups and fields definable in o-minimal structures, J. Pure Appl. Algebra53 (1988), 239–255.MATHCrossRefMathSciNetGoogle Scholar
  8. [PSS] A. Pillay, P. Scowcroft and C. Steinhorn,Between groups and rings, Rocky Mountain J.9 (1989), 871–885.MathSciNetCrossRefGoogle Scholar
  9. [PS] A. Pillay and C. Steinhorn,Definable sets in ordered structures I, Trans. Amer. Math. Soc.295, (1986), 565–592.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1993

Authors and Affiliations

  • James Loveys
    • 1
  • Ya’acov Peterzil
    • 1
  1. 1.Department of MathematicsMcGill UniversityMontrealCanada

Personalised recommendations