O-minimal Theory - Examples

Examples

Examples of o-minimal theories are:

• The complete theory of dense linear orders in the language with just the ordering.
• RCF, the theory of real closed fields.
• The complete theory of the real field with restricted analytic functions added (i.e. analytic functions on a neighborhood of n, restricted to n; note that the unrestricted sine function has infinitely many roots, and so cannot be definable in an o-minimal structure.)
• The complete theory of the real field with a symbol for the exponential function by Wilkie's theorem. More generally, the complete theory of the real numbers with Pfaffian functions added.
• The last two examples can be combined: given any o-minimal expansion of the real field (such as the real field with restricted analytic functions), one can define its Pfaffian closure, which is again an o-minimal structure. (The Pfaffian closure of a structure is, in particular, closed under Pfaffian chains where arbitrary definable functions are used in place of polynomials.)

In the case of RCF, 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, Whitney and Verdier stratification theorems and a good notion of dimension and Euler characteristic.