Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0662.03024
Knight, Julia F.; Pillay, Anand; Steinhorn, Charles
Definable sets in ordered structures. II. (English)
[J] Trans. Am. Math. Soc. 295, 593-605 (1986). ISSN 0002-9947; ISSN 1088-6850/e

[This article is reviewed together with the preceding one (see Zbl 0662.03023).] \par 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.

MSC 2000:: *03C45 Stability (model theory)
03C40 Interpolation, etc. (model theory)
06F99 Ordered structures (connections with other sections)
03C50 Models with special properties

Keywords: O-minimal structure; unstable structure; elementary equivalence; definable function; O-minimal theories; coheirs

Citations: Zbl 0662.03023; Zbl 0526.03014

Cited in: Zbl 1051.91016 Zbl 0707.03024 Zbl 0662.03023

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]