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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 be a first-order language including < and let be an structure in which < is a linear ordering. If every parametrically definable subset of is a union of finitely many intervals, then 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 n reveals that O-minimality is preserved under elementary equivalence. The key to this argument is that if (a,b) is an interval in and f: (a,b) is a definable function, then there are a 0 =a<···<a n =b in such that f|(a i ,a i+1 ) is constant or a (monotone) isomorphism for 0in-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 is finite, any 0 categorical O-minimal theory is finitely axiomatisable.


MSC:
03C45Classification theory, stability etc. (model theory)
03C40Interpolation, preservation, definability
06F99Ordered structures (connections with other sections)
03C50Models with special properties