Journal of Symbolic Logic

Quasi-O-Minimal Structures

Oleg Belegradek, Ya'Acov Peterzil, and Frank Wagner

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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.

Article information

J. Symbolic Logic, Volume 65, Issue 3 (2000), 1115-1132.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C67
Secondary: 06F15: Ordered groups [See also 20F60] 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] 03C45: Classification theory, stability and related concepts [See also 03C48]

Quasi-O-Minimal Theory O-Minimal Theory Ordered Group Theory of U -Rank 1


Belegradek, Oleg; Peterzil, Ya'Acov; Wagner, Frank. Quasi-O-Minimal Structures. J. Symbolic Logic 65 (2000), no. 3, 1115--1132.

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