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O minimal theory

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

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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 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 = (Sn), 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 {(x1,...,xn) ∈ Mn : x1 = xn} is in Sn
  4. if A ∈ Sn+1 and π : Mn+1 → Mn is the projection map on the first n coordinates, then π(A) ∈ 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 {(x,y) ∈ M2 : x < y} 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 (M,<,...) is an L-structure where < is interpreted to satisfy the axioms of a dense linear order, then (M,<,...) is called an o-minimal structure if for any definable set X ⊆ M there are finitely many open intervals I1,..., Ir with endpoints in M ∪ {±∞} and a finite set X0 such that

X = X 0 ∪ I 1 ∪ … ∪ I r .

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 [0,1]n, restricted to [0,1]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.

    References

    O-minimal theory Wikipedia
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