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 $n \in \omega$ .