o-minimal theory

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.

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.[1] 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.[2] 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,[3] 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

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,[6] Whitney and Verdier stratification theorems and a good notion of dimension and Euler characteristic.

See also

Notes

  1. ↑ Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988).
  2. ↑ Marker (2002) p.81
  3. ↑ The condition that the interpretation of < be dense is not strictly necessary, but it is known that discrete orders lead to essentially trivial o-minimal structures, see, for example, MR0899083 and MR0943306.
  4. ↑ Marker (2002) p.99
  5. ↑ Patrick Speisseger, Pfaffian sets and o-minimality, in: Lecture notes on o-minimal structures and real analytic geometry, C. Miller, J.-P. Rolin, and P. Speissegger (eds.), Fields Institute Communications vol. 62, 2012, pp. 179–218. doi:10.1007/978-1-4614-4042-0_5
  6. ↑ Marker (2002) p.103

References

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