Skip to main content
SpringerLink
Log in

Structure theorems in tame expansions of o-minimal structures by a dense set

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study sets and groups definable in tame expansions of o-minimal structures. Let \(\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle \) be an expansion of an o-minimal \({\cal L}\)-structure ℳ by a dense set P. We impose three tameness conditions on \(\widetilde{\cal M}\) and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such \(\widetilde{\cal M}\), as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an \({\cal L}\)-definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an \({\cal L}\)-definable map.

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. O. Belegradek and B. Zilber, The model theory of the field of reals with a subgroup of the unit circle, Journal of the London Mathematical Society 78 (2008), 563–579.

    MathSciNet  MATH  Google Scholar 

  2. A. Berenstein, A. Dolich and A. Onshuus, The independence property in generalized dense pairs of structures, Journal of Symbolic Logic 76 (2011), 391–404

    MathSciNet  MATH  Google Scholar 

  3. A. Berenstein, C. Ealy and A. Günaydin, Thorn independence in the field of real numbers with a small multiplicative group, Annals of Pure and Applied Logic 150 (2007), 1–18.

    MathSciNet  MATH  Google Scholar 

  4. T. Blossier and A. Martin-Pizarro, Des beaux groupes, Confluentes Mathematici 6 (2014), 3–13.

    MathSciNet  MATH  Google Scholar 

  5. A. V. Borovik and A. Nesin, Groups of Finite Morley Rank, Oxford Logic Guides, Vol. 26, The Clarendon Press, Oxford University Press, New York, 1994.

    MATH  Google Scholar 

  6. G. Boxall and P. Hieronymi, Expansions which introduce no new open sets, Journal of Symbolic Logic 77 (2012), 111–121.

    MathSciNet  MATH  Google Scholar 

  7. A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Advances in Mathematics 231 (2012), 605–623.

    MathSciNet  MATH  Google Scholar 

  8. A. Dolich, C. Miller and C. Steinhorn, Structures having o-minimal open core, Transactions of the American Mathematical Society 362 (2010), 1371–1411.

    MathSciNet  MATH  Google Scholar 

  9. A. Dolich, C. Miller and C. Steinhorn, Expansions of o-minimal structures by dense independent sets, Annals of Pure and Applied Logic 167 (2016), 684–706.

    MathSciNet  MATH  Google Scholar 

  10. L. van den Dries, Remarks on Tarski’s problem concerning (R, +, −, exp), in Logic Colloquium’ 82 (Florence, 1982), Studies in logic and the Foundations of Mathematics, Vol. 112, North-Holland, Amsterdam, 1984, pp. 97–121.

    Google Scholar 

  11. L. van den Dries, Dimension of definable sets, algebraic boundedness and henselian fields, Annals of Pure and Applied Logic 45 (1989), 189–209.

    MathSciNet  MATH  Google Scholar 

  12. L. van den Dries, Dense pairs of o-minimal structures, Fundamenta Mathematicae 157 (1988), 61–78.

    MathSciNet  MATH  Google Scholar 

  13. L. van den Dries, Tame Topology and o-Minimal Structures, London Mathematical Society Lecture Note Series, Vol. 248, Cambridge University Press, Cambridge, 1998.

    MATH  Google Scholar 

  14. L. van den Dries and A. Günaydin, The fields of real and complex numbers with a small multiplicative group, Proceedings of the London Mathematical Society 93 (2006), 43–81.

    MathSciNet  MATH  Google Scholar 

  15. M. Edmundo, Structure theorems for o-minimal expansions of groups, Annals of Pure and Applied Logic 102 (2000), 159–181.

    MathSciNet  MATH  Google Scholar 

  16. M. Edmundo and M. Otero, Definably compact abelian groups, Journal of Mathematical Logic 4 (2004), 163–180.

    MathSciNet  MATH  Google Scholar 

  17. P. Eleftheriou, Local analysis for semi-bounded groups, Fundamenta Mathematicae 216 (2012), 223–258.

    MathSciNet  MATH  Google Scholar 

  18. P. Elefheriou, Non-standard lattices and o-minimal groups, Bulletin of Symbolic Logic 19 (2013), 56–76.

    MathSciNet  Google Scholar 

  19. P. Eleftheriou, A. Güunaydin and P. Hieronymi, The choice property in tame expansions of o-minimal structures, Mathematical Logic Quarterly 66 (2020), 239–246.

    MathSciNet  MATH  Google Scholar 

  20. P. Eleftheriou, Characterizing o-minimal groups in tame expansions of o-minimal structures, Journal of the Institute of Mathematics of Jussieu, to appear, https://doi.org/10.1017/S1474748019000392.

  21. P. Eleftheriou, Product cones in dense pairs, https://arxiv.org/abs/1708.03894.

  22. P. Eleftheriou, Counting algebraic points in expansions of o-minimal structures by a dense set, https://arxiv.org/abs/1708.03936.

  23. P. Eleftheriou and Y. Peterzil, Definable groups as homomorphic images of semilinear and field-definable groups, Selecta Mathematica 18 (2012), 905–940.

    MathSciNet  MATH  Google Scholar 

  24. P. Eleftheriou and S. Starchenko, Groups definable in ordered vector spaces over ordered division rings, Journal of Symbolic Logic 72 (2007), 1108–1140.

    MathSciNet  MATH  Google Scholar 

  25. J.-H. Everste, H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative subgroup, Annals of Mathematics 155 (2002), 807–836.

    MathSciNet  MATH  Google Scholar 

  26. J. Gagelman, Stability in geometric theories, Annals of Pure and Applied Logic 132 (2005), 313–326.

    MathSciNet  MATH  Google Scholar 

  27. A. Günaydin and P. Hieronymi, The real field with the rational points of an elliptic curve, Fundamenat Mathematicae 215 (2011). 167–175.

    MathSciNet  MATH  Google Scholar 

  28. A. Günaydin and P. Hieronymi, Dependent pairs, Journal of Symbolic Logic 76 (2011), 377–390.

    MathSciNet  MATH  Google Scholar 

  29. P. Hieronymi, The real field with an irrational power function and a dense multiplicative subgroup, Journal of the London Mathematical Societry 83 (2011), 153–167.

    MathSciNet  MATH  Google Scholar 

  30. E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal of the American Mathematical Society 9 (1996), 667–690.

    MathSciNet  MATH  Google Scholar 

  31. E. Hrushovski, Y. Peterzil and A. Pillay, Groups, measures, and the NIP, Journal of the American Mathematical Society 21 (2008), 563–596.

    MathSciNet  MATH  Google Scholar 

  32. E. Hrushovski and A. Pillay, On NIP and invariant measures, Journal of the European Mathematical Society 13 (2011), 1005–1061.

    MathSciNet  MATH  Google Scholar 

  33. J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Transactions of the American Mathematical Society 295 (1986), 593–605.

    MathSciNet  MATH  Google Scholar 

  34. C. Miller and P. Speissegger, Expansions of the real line by open sets: o-minimality and open cores, Fundamenta Mathematicae 162 (1999), 193–208.

    MathSciNet  MATH  Google Scholar 

  35. M. Otero, A survey on groups definable in o-minimal structures, in Model Theory with Applications to Algebra and Analysis, Vol. 2, London Mathematical Society Lecture Note Series, Vol. 350, Cambridge University Press, Cambridge, 2008, pp. 177–206.

    Google Scholar 

  36. Y. Peterzil, A structure theorem for semibounded sets in the reals, The Journal of Symbolic Logic 57 (1992), 779–794.

    MathSciNet  MATH  Google Scholar 

  37. Y. Peterzil, A. Pillay and S. Starchenko, Definably simple groups in o-minimal structures, Transactions of the American Mathematical Society 352 (2000), 4397–4419.

    MathSciNet  MATH  Google Scholar 

  38. Y. Peterzil and C. Steinhorn, Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society 69 (1999), 769–786.

    MathSciNet  MATH  Google Scholar 

  39. J. Pila, O-minimality and the André-Oort conjecture for ℂn, Annals of Mathematics 173 (2011), 1779–1840.

    MathSciNet  MATH  Google Scholar 

  40. A. Pillay, Some remarks on definable equivalence relations in o-minimal structures, Journal of Symbolic Logic 51 (1986), 709–714.

    MathSciNet  MATH  Google Scholar 

  41. A. Pillay, On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra 53 (1988), 239–255.

    MathSciNet  MATH  Google Scholar 

  42. A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Transactions of the American Mathematical Society 295 (1986), 565–592.

    MathSciNet  MATH  Google Scholar 

  43. B. Poizat, Groupes Stables, Nur al-Mantiq wal-Ma’riah, Vol. 2, Launay, Paris, 1987; English translation: Stable Groups, Mathematical Surveys and Monographs, Vol. 87, American Mathematical Society, Providence, RI, 2001.

    Google Scholar 

  44. L. Pontrjagin, Topological Groups, Princeton Mathematical Series, Vol. 2, Princeton University Press, Princeton, NJ, 1939.

    MATH  Google Scholar 

  45. A. Strzebonski, Euler charateristic in semialgebraic and other o-minimal groups, Journal of Pure and Applied Algebra 96 (1994), 173–201.

    MathSciNet  MATH  Google Scholar 

  46. M. Tychonievich, Tameness results for expansions of the real field by groups, Ph.D. Thesis, Ohio State University, 2013.

  47. J. Tyne, Functions definable in dense pairs, unpublished note.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, Box 216, 78457, Konstanz, Germany

    Pantelis E. Eleftheriou

  2. Department of Mathematics, Boğaziçi University, Bebek, Istanbul, Turkey

    Ayhan Günaydin

  3. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Philipp Hieronymi

Authors
  1. Pantelis E. Eleftheriou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ayhan Günaydin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philipp Hieronymi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pantelis E. Eleftheriou.

Additional information

The first author was supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.

The second author was partially supported by TUBITAK Career Grant 113F119.

The third author was partially supported by NSF grant DMS-1300402.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eleftheriou, P.E., Günaydin, A. & Hieronymi, P. Structure theorems in tame expansions of o-minimal structures by a dense set. Isr. J. Math. 239, 435–500 (2020). https://doi.org/10.1007/s11856-020-2058-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-020-2058-0

Search

Navigation