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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
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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

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References

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