Definable sets in ordered structures. II.

*(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 \(a_ 0=a<...<a_ n=b\) 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.

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.

##### MSC:

03C45 | Classification theory, stability and related concepts in model theory |

03C40 | Interpolation, preservation, definability |

06F99 | Ordered structures |

03C50 | Models with special properties (saturated, rigid, etc.) |

##### Keywords:

O-minimal structure; unstable structure; elementary equivalence; definable function; O-minimal theories; coheirs
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\textit{J. F. Knight} et al., Trans. Am. Math. Soc. 295, 593--605 (1986; Zbl 0662.03024)

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##### References:

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