O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure ${\displaystyle (M,<,\dots )}$ that is totally ordered by ${\displaystyle <}$ is called an o-minimal structure if and only if every definable subset ${\displaystyle X\subseteq M}$ (with parameters taken from ${\displaystyle M}$) is a finite union of intervals and points.

O-minimality can be regarded as a weak form of quantifier elimination. A structure ${\displaystyle M}$ is o-minimal if and only if every formula with one free variable and parameters in ${\displaystyle M}$ is equivalent to a quantifier-free formula involving only the ordering, also with parameters in ${\displaystyle M}$. This is analogous to the minimal structures, which are exactly the analogous property down to equality.

A theory ${\displaystyle T}$ is an o-minimal theory if every model of ${\displaystyle T}$ is o-minimal. It is known that the complete theory ${\displaystyle T}$ of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure that is not minimal.