ominimality
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:

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$$ for some $a,b$ from $M$

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$\{x:x>a\}$ for some $a$ from $M$

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$$ for some $a$ from $M$
Then we define $M$ to be ominimal 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 ominimal, then $N$ is ominimal. Note that $M$ being ominimal 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 ominimal structures is well understood, for an excellent account see Lou van den Dries, Tame topology and ominimal 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 $.
Title  ominimality 

Canonical name  Ominimality 
Date of creation  20130322 13:23:01 
Last modified on  20130322 13:23:01 
Owner  Timmy (1414) 
Last modified by  Timmy (1414) 
Numerical id  7 
Author  Timmy (1414) 
Entry type  Definition 
Classification  msc 03C64 
Classification  msc 14P10 
Related topic  StronglyMinimal 
Defines  ominimal 