o-minimal theory

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In mathematical logic, and more specifically in model theory, a totally ordered structure $(M,<,\ldots)$ is o-minimal iff for every definable set $X \subset M$ (with parameters), X can be realized as a finite union of intervals and points.

A theory T is o-minimal if every model is o-minimal.

One can show that a complete theory T is o-minimal if any of its models is o-minimal. Some kind of ordering concept is implicit in the notion of an "interval." In other words, any set definable in M by an arbitrary formula is also definable via a quantifier free formula using only the ordering. This compares to strong minimality, where the definable sets in M are precisely those definable via quantifier free formulae using only equality.

Examples of o-minimal theories are:

RCOF, the axioms for the real closed fields;
The complete theory of the real field with a symbol for the exponential function;
The complete theory of the real numbers with restricted analytic functions added. (i.e. analytic functions on a neighborhood of $[0,1] n$ , restricted to $[0,1] n$ ; note that the unrestricted sine function has infinitely many roots, and so cannot be definable in a o-minimal structure.)
The complete theory of dense linear orders in the language with just the ordering.

In the first example, the definable sets are the semialgebraic sets. Thus the study of o-minimal structures and theories generalises Real algebraic geometry. A major line of current research is based on discovering expansions of the real ordered field that are o-minimal. Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures. There is a cell decomposition theorem, and a good notion of dimension and Euler characteristic.