Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-18T22:34:14.299Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly one-based geometric theories

Published online by Cambridge University Press:  12 March 2014

Alexander Berenstein  and
Evgueni Vassiliev
Alexander Berenstein
Affiliation:
Universidad De Los Andes, CRA 1 NO 18A-10, Bogota, Colombia, URL: http://www.matematicas.uniandes.edu.co/˜aberenst
Evgueni Vassiliev
Affiliation:
Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL A2H 6P9, Canada, E-mail: yvasilyev@grenfell.mun.ca, URL: http://sites.google.com/site/yevgvas/
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

Keywords

geometric structureslinear structuresrosy theoriesgeometrieslovely pairs

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 392 - 422
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1] Baisalov, Y. and Poizat, B., Paires de structures o-minimales, this Journal, vol. 63 (1998), pp. 570–578.Google Scholar
[2] Baldwin, J., First order theories of abstract dependence relations, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 215–243.Google Scholar
[3] Berenstein, A., Ealy, C., and Günaydin, A., Thorn independence in the field of real numbers with a small multiplicative group. Annals of Pure and Applied Logic, vol. 150 (2007), no. 1–3, pp. 1–18.Google Scholar
[4] Berenstein, A. and Vassiliev, E., On lovely pairs ofgeometric structures, Annals of Pure and Applied Logic, vol. 161 (2010), no. 7, pp. 866–878.Google Scholar
[5] Boxall, G., Lovely pairs of models of a þ-rank 1 theory, preprint.Google Scholar
[6] Boxall, Gareth, Bradley-Williams, David, Kestner, Charlotte, Aziz, Alexandra Omar, and Penazzi, Davide, Some proofs on weakly one-based geometric theories, notes.Google Scholar
[7] Buechler, S., Essential stability theory, Perspectives in Mathematical Logic, vol. 4, Springer-Verlag, Berlin, 1996.Google Scholar
[8] Chatzidakis, Z. and Pillay, A., Generic structures and simple theories, Annals of Pure and Applied Logic, (1998), pp. 71–92.Google Scholar
[9] de Piro, T. and Kim, B., The geometry of l-based minimal types, Transactions of the American Mathematical Society, vol. 355 (2003), pp. 4241–4263.CrossRefGoogle Scholar
[10] Ealy, C. and Onshuus, A., Characterizing rosy theories, this Journal, vol. 72 (2007), no. 3, pp. 919–940.Google Scholar
[11] Gagelman, J., Stability in geometric theories, Annals of Pure and Applied Logic, vol. 132 (2005), pp. 313–326.Google Scholar
[12] Hall, M., The theory of groups, Macmillan, New York, 1959.Google Scholar
[13] Hasson, A., Onshuus, A., and Peterzil, Y., Definable structures in o-minimal theories, Israel Journal of Mathematics, vol. 179 (2010), pp. 363–379.Google Scholar
[14] Lam, T. Y., A first course in non-commmutative rings, second ed., Springer Verlag, 2001.Google Scholar
[15] Loveys, J. and Peterzil, Y, Linear o-minimal structures, Israel Journal of Mathematics, vol. 81 (1993), pp. 1–30.Google Scholar
[16] Maalouf, F., Construction d'un groupe dans les structures C-minimales, this Journal, vol. 73 (2008), no. 3, pp. 957–968.Google Scholar
[17] Nübling, H., Expansion and reducts of simple and stable theories, Ph.D. thesis, University of East Anglia, Norwich, 2004.Google Scholar
[18] Peterzil, Y., Constructing a group-interval in o-minimal structures, Journal of Pure and Applied Algebra, vol. 94 (1994), pp. 85–100.Google Scholar
[19] Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society, (2000), pp. 481–523.Google Scholar
[20] Pillay, A., Geometric stability theory, Oxford Logic Guides 32, Clarendon Press, Oxford, 1996.Google Scholar
[21] Pillay, A., Canonical bases in o-minimal and related structures, this Journal, to be published.Google Scholar
[22] Pillay, A. and Vassiliev, E., On lovely pairs and the Ǝy ∈ quantifier, Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 491–501.CrossRefGoogle Scholar
[23] van den Dries, L., Dense pairs of o-minimal structures, Fundamenta Mathematicae, vol. 157 (1998), pp. 61–78.Google Scholar
[24] van den Dries, L. and Günaydin, A., The fields of real and complex numbers with a small multiplicative group, Proceedings of the London Mathematical Society. Series 3, vol. 93 (2006), no. 1, pp. 43–81.Google Scholar
[25] Vassiliev, E., Generic pairs of SU-rank 1 structures. Annals of Pure and Applied Logic, vol. 120 (2003), pp. 103–149.Google Scholar
[26] Wagner, F. O., Simple theories, Mathematics and its Applications, Kluwer Academic Publishers, 2000.Google Scholar
[27] Yaacov, I. Ben, Pillay, A., and Vassiliev, E., Lovely pairs of models, Annals of Pure and Applied Logic, vol. 122 (2003), pp. 235–261.Google Scholar