O-minimal theory


In mathematical logic, and more specifically in model theory, an infinite structure which is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊂ M is a finite union of intervals and points.
O-minimality can be regarded as a weak form of quantifier elimination. A structure M is o-minimal if and only if every formula with one free variable and parameters in M is equivalent to a quantifier-free formula involving only the ordering, also with parameters in M. This is analogous to the minimal structures, which are exactly the analogous property down to equality.
A theory T is an o-minimal theory if every model of T is o-minimal. It is known that the complete theory 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 which is not minimal.

Set-theoretic definition

O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set M in a set-theoretic manner, as a sequence S = , n = 0,1,2,... such that
  1. Sn is a boolean algebra of subsets of Mn
  2. if A ∈ Sn then M × A and A ×M are in Sn+1
  3. the set is in Sn
  4. if A ∈ Sn+1 and π : Mn+1 → Mn is the projection map on the first n coordinates, then π ∈ Sn.
If M has a dense linear order without endpoints on it, say <, then a structure S on M is called o-minimal if it satisfies the extra axioms

  1. the set is in S2
  2. the sets in S1 are precisely the finite unions of intervals and points.
The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.

Model theoretic definition

O-minimal structures originated in model theory and so have a simpler — but equivalent — definition using the language of model theory. Specifically if L is a language including a binary relation <, and is an L-structure where < is interpreted to satisfy the axioms of a dense linear order, then is called an o-minimal structure if for any definable set X ⊆ M there are finitely many open intervals I1,..., Ir without endpoints in M ∪ and a finite set X0 such that

Examples

Examples of o-minimal theories are:
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.