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Definable sets in ordered structures. II. (English) Zbl 0662.03024
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.