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Definable sets in ordered structures. II. (English) Zbl 0662.03024
[This article is reviewed together with the preceding one (see Zbl 0662.03023).]
Let \({\mathcal L}\) be a first-order language including \(<\) and let \({\mathcal M}\) be an \({\mathcal L}\) structure in which \(<\) is a linear ordering. If every parametrically definable subset of \({\mathcal M}\) is a union of finitely many intervals, then \({\mathcal M}\) is said to be O-minimal. For example, this condition is easily seen to be the same as “abelian divisible” for ordered groups and “real closed” for ordered rings. An O-minimal structure can be conceived as a well-behaved case of an unstable structure. This case exhibits most of the properties of stable theories; e.g., the exchange principle, and the existence and uniqueness of prime models [cf. the second author: An introduction to stability theory (1983; Zbl 0526.03014)]. These are established in part I (the proofs are straightforward). More surprisingly, in part II an analysis of definable subsets of \({\mathcal M}^ n\) reveals that O-minimality is preserved under elementary equivalence. The key to this argument is that if (a,b) is an interval in \({\mathcal M}\) and f: (a,b)\(\to {\mathcal M}\) is a definable function, then there are \(a_ 0=a<...<a_ n=b\) in \({\mathcal M}\) such that \(f| (a_ i,a_{i+1})\) is constant or a (monotone) isomorphism for \(0\leq i\leq n-1.\) As well as this technical result, part I also includes the following results: (1) types over O-minimal theories have at most two coheirs (whence no O-minimal theory has the independence property); (2) if \({\mathcal L}\) is finite, any \(\aleph_ 0\) categorical O-minimal theory is finitely axiomatisable.

03C45 Classification theory, stability and related concepts in model theory
03C40 Interpolation, preservation, definability
06F99 Ordered structures
03C50 Models with special properties (saturated, rigid, etc.)
Full Text: DOI
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