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

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