Strong cell decomposition property in o-minimal traces


Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an o-minimal trace.

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


  1. 1.

    Baizhanov, B.S.: Expansion of a model of weakly o-minimal theory by family of unary predicates. J. Symb. Logic 63, 570–578 (1998)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Berenstein, A., Vassiliev, E.: On lovely pairs of geometric structures. Ann. Pure Appl. Logic 161, 866–878 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dickmann, M.A.: Elimination of quantifiers for ordered valuation rings. J. Symb. Logic 52, 116–128 (1987)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Eivazloo, J.S., Tari, S.: Tame properties of sets and functions definable in weakly o-minimal structures. Arch. Math. logic 53, 433–447 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Eleftheriou, P.E., Hasson, A., Keren, G.: On definable Skolem functions in weakly o-minimal nonvalutional structures. J. Symb. Logic 82, 1482–1495 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hasson, A., Onshuus, A.: Embeded o-minimal structures. Bull. London Math. Soc 42, 64–74 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Knight, J., Pillay, A., Steinhorn, C.: Definable sets in ordered structures. II. Trans. Am. Math. Soc 295, 593–605 (1986)

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

    Pillay, A., Steinhorn, C.: Definable sets in ordered structures. I. Trans. Am. Math. Soc 295, 565–592 (1986)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Pillay, A., Steinhorn, C.: Definable sets in ordered structures. III. Trans. Am. Math. Soc 309, 469–476 (1988)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Tari, S.: A note on prime models in weakly o-minimal structures. MLQ 63, 109–113 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    van den Dries, L.: Dense pairs of o-minimal structures. Fund. Math. 157, 61–78 (1998)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    van den Dries, L.: Tame Topology and o-Minimal Structures. London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  14. 14.

    Wencel, R.: On the strong cell decomposition property for weakly o-minimal structures. MLQ 59, 379–493 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Wencel, R.: Topological properties of sets definable in weakly O-minimal structures. J. Symb. Logic 75, 841–867 (2010)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wencel, R.: Weakly o-minimal non-valuational structures. Ann. Pure Appl. Logic 154, 139–162 (2008)

    MathSciNet  Article  Google Scholar 

Download references


This research was in part supported by a grant from IPM (No. 96030031). The author thank the anonymous reviewers whose useful comments helped to improve the paper.

Author information



Corresponding author

Correspondence to Somayyeh Tari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tari, S. Strong cell decomposition property in o-minimal traces. Arch. Math. Logic (2020).

Download citation


  • O-minimal trace
  • Irrational nonvaluational cut
  • Dense pair
  • Strong cell decomposition

Mathematics Subject Classification

  • 03C64