Journal of Symbolic Logic 65 (3):1115-1132 (2000)

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 Theory   O-Minimal Theory   Ordered Group   Theory of U -Rank 1
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DOI 10.2307/2586690
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References found in this work BETA

Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.
An Introduction to Stability Theory.Anand Pillay - 1986 - Journal of Symbolic Logic 51 (2):465-467.

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Citations of this work BETA

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Presburger Sets and P-Minimal Fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.

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