*(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 ${a}_{0}=a<\xb7\xb7\xb7<{a}_{n}=b$ in $\mathcal{M}$ such that $f\left|\right({a}_{i},{a}_{i+1})$ is constant or a (monotone) isomorphism for $0\le i\le n-1\xb7$ 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.

##### MSC:

03C45 | Classification theory, stability etc. (model theory) |

03C40 | Interpolation, preservation, definability |

06F99 | Ordered structures (connections with other sections) |

03C50 | Models with special properties |