Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-18T02:15:56.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Oleg Belegradek ,
Ya'acov Peterzil  and
Frank Wagner
Oleg Belegradek
Affiliation:
Department of Mathematics and Computer Science, Istanbul Bilgi University, Inönü Caddesi No.28, Kuştepe 80310. Şişli, Istanbul, Turkey E-mail: olegb@bilgi.edu.tr
Ya'acov Peterzil
Affiliation:
Department of Mathematics, University of Haifa, Haifa, Israel E-mail: kobi@mathcs2.haifa.ac.il
Frank Wagner
Affiliation:
Institut Girard Desargues, Universite Claude Bernard, Mathematiques, Batiment 101, 43, Boulevard du 11 November 1918, 69622 Villeurbanne-Cedex, France E-mail: wagner@desargues.univ-lyonl.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A structure (M, <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1.

Keywords

Quasi-o-minimal theoryo-minimal theoryordered grouptheory of U-rank 1
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1115 - 1132
Copyright
Copyright © Association for Symbolic Logic 2000

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.)

References

REFERENCES

[1]Belegradek, O. V., Point, F., and Wagner, F. O., A quasi-o-minimal group without the exchange property, MSRI preprint series, 1998–051.Google Scholar
[2]Belegradek, O. V., Stolboushkin, A. P., and Taitslin, M. A., Generic queries over quasi-o-minimal domains, Logical Foundations of Computer Science (Proceedings 4th International Symposium LFCS'97, Yaroslavl, Russia, July 1997) (Adian, S. and Nerode, A., editors), Lecture Notes in-Computer Science 1234, Springer-Verlag, 1997, pp. 21–32.Google Scholar
[3]Dickmann, M. A., Elimination of quantifiers for ordered valuation rings, Proceedings of the 3rd Easter Conference on Model Theory (Groß Köris, April 8–13, 1985), Sektion Mathematik der Humboldt-Universität zu Berlin (DDR), Seminarbericht 70, 1985, pp. 64–88.Google Scholar
[4]Karpilovski, G., The Schur multipliers, LMS monographs, New Series, vol. 2, Clarendon Press, Oxford, 1987.Google Scholar
[5]Knight, J. F., Pillay, A., and Steinhorn, C., Definable sets in ordered structures II, Transactions of the American Mathematical Society, (1986), no. 295, pp. 565–592.Google Scholar
[6]Laskowski, M. C. and Steinhorn, C., On o-minimal expansions of archimedian ordered groups, this Journal, vol. 60 (1995), pp. 817–831.Google Scholar
[7]Michaux, C. and Villemaire, R., Presburger Arithmetic and recognizability of sets of natural numbers by automata, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 251–271.CrossRefGoogle Scholar
[8]Pillay, A., An introduction to stability theory, Oxford Logic Guides 8, Clarendon Press, Oxford, 1983.Google Scholar
[9]Pillay, A. and Steinhorn, C., Definable sets in ordered structures I, Transactions of the American Mathematical Society, (1986), no. 295, pp. 593–605.Google Scholar
[10]Pillay, A. and Steinhorn, C., Definable sets in ordered structures III, Transactions of the American Mathematical Society, (1988), no. 309, pp. 469–476.Google Scholar
[11]Shelah, S., Classification theory and the number of non-isomorphic models, 2nd ed., North-Holland, Amsterdam, 1990.Google Scholar
[12]van den Dries, L., O-minimal structures, Logic: from Foundations to Applications (European Logic Colloquium'93) (Hodges, W.et al., editors), Oxford University Press, 1996, pp. 133–185.Google Scholar
[13]Weispfenning, V., Elimination of quantifiers for certain ordered and lattice-ordered groups, Bulletin de la Société Mathématique de Belgique, (1981), no. 33, pp. 131–155, Ser. B.Google Scholar
[14]Zakon, E., Generalized archimedian groups, Transactions of the Amererican Mathematical Society, (1961), no. 99, pp. 21–40.Google Scholar