Abstract
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.
Similar content being viewed by others
References
Baizhanov, B.S.: Expansion of a model of weakly o-minimal theory by family of unary predicates. J. Symb. Logic 63, 570–578 (1998)
Berenstein, A., Vassiliev, E.: On lovely pairs of geometric structures. Ann. Pure Appl. Logic 161, 866–878 (2010)
Dickmann, M.A.: Elimination of quantifiers for ordered valuation rings. J. Symb. Logic 52, 116–128 (1987)
Eivazloo, J.S., Tari, S.: Tame properties of sets and functions definable in weakly o-minimal structures. Arch. Math. logic 53, 433–447 (2014)
Eleftheriou, P.E., Hasson, A., Keren, G.: On definable Skolem functions in weakly o-minimal nonvalutional structures. J. Symb. Logic 82, 1482–1495 (2017)
Hasson, A., Onshuus, A.: Embeded o-minimal structures. Bull. London Math. Soc 42, 64–74 (2010)
Knight, J., Pillay, A., Steinhorn, C.: Definable sets in ordered structures. II. Trans. Am. Math. Soc 295, 593–605 (1986)
Macpherson, D., Marker, D., Steinhorn, C.: Weakly o-minimal structures and real closed fields. Trans. Am. Math. Soc 352, 5435–5483 (2000)
Pillay, A., Steinhorn, C.: Definable sets in ordered structures. I. Trans. Am. Math. Soc 295, 565–592 (1986)
Pillay, A., Steinhorn, C.: Definable sets in ordered structures. III. Trans. Am. Math. Soc 309, 469–476 (1988)
Tari, S.: A note on prime models in weakly o-minimal structures. MLQ 63, 109–113 (2017)
van den Dries, L.: Dense pairs of o-minimal structures. Fund. Math. 157, 61–78 (1998)
van den Dries, L.: Tame Topology and o-Minimal Structures. London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge (1998)
Wencel, R.: On the strong cell decomposition property for weakly o-minimal structures. MLQ 59, 379–493 (2013)
Wencel, R.: Topological properties of sets definable in weakly O-minimal structures. J. Symb. Logic 75, 841–867 (2010)
Wencel, R.: Weakly o-minimal non-valuational structures. Ann. Pure Appl. Logic 154, 139–162 (2008)
Acknowledgements
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
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tari, S. Strong cell decomposition property in o-minimal traces. Arch. Math. Logic 60, 135–144 (2021). https://doi.org/10.1007/s00153-020-00739-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-020-00739-2