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

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