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o-minimality (Definition)

Let $ M$ be an ordered structure. An interval in $ M$ is any subset of $ M$ that can be expressed in one of the following forms:

  • $ \{x:a<x<b\}$ for some $ a,b$ from $ M$
  • $ \{x:x>a\}$ for some $ a$ from $ M$
  • $ \{x:x<a\}$ for some $ a$ from $ M$

Then we define $ M$ to be o-minimal iff every definable subset of $ M$ is a finite union of intervals and points. This is a property of the theory of $ M$ i.e. if $ M \equiv N$ and $ M$ is o-minimal, then $ N$ is o-minimal. Note that $ M$ being o-minimal is equivalent to every definable subset of $ M$ being quantifier free definable in the language with just the ordering. Compare this with strong minimality.


The model theory of o-minimal structures is well understood, for an excellent account see Lou van den Dries, Tame topology and o-minimal structures, CUP 1998. In particular, although this condition is merely on definable subsets of $ M$ it gives very good information about definable subsets of $ M^{n}$ for $ n \in \omega$.



"o-minimality" is owned by Timmy.
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See Also: strongly minimal

Also defines:  o-minimal
Keywords:  definable set, semialgebraic, real algebraic, tame topology

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Cross-references: information, topology, model theory, ordering, language, quantifier free, theory, property, points, union, finite, definable, iff, subset, interval, structure
There are 4 references to this entry.

This is version 4 of o-minimality, born on 2003-01-23, modified 2003-02-11.
Object id is 3919, canonical name is OMinimality.
Accessed 3765 times total.

Classification:
AMS MSC: 03C64 (Mathematical logic and foundations :: Model theory :: Model theory of ordered structures; o-minimality)
 14P10 (Algebraic geometry :: Real algebraic and real analytic geometry :: Semialgebraic sets and related spaces)
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