Weakly o-minimal structure

In model theory, a weakly o-minimal structure is a model theoretic structure whose definable sets in the domain are just finite unions of convex sets.

Definition

A linearly ordered structure, M, with language L including an ordering relation <, is called weakly o-minimal if every parametrically definable subset of M is a finite union of convex (definable) subsets. A theory is weakly o-minimal if all its models are weakly o-minimal.

Note that, in contrast to o-minimality, it is possible for a theory to have models which are weakly o-minimal and to have other models which are not weakly o-minimal.[1]

Difference from o-minimality

In an o-minimal structure (M,<,...) the definable sets in M are finite unions of points and intervals, an interval here being a set of the form

I=\{r\in M\,:\,a<r<b\}

for some (possibly infinite) constants a and b in M.[2] For weakly o-minimal structures (m,<,...) this is relaxed so that the definable sets in m are finite unions of convex sets. A set C is convex if whenever a and b are in C with a < b and c ∈ m satisfies a < c < b, then c is in C.

On the face of it, convex sets are just intervals, in which case weakly o-minimal structures would just be o-minimal structures. And indeed, if we have a weakly o-minimal structure expanding (R,<), the real ordered field, then this structure will be o-minimal. The two notions are different in other settings though. For example, let R be the ordered field of real algebraic numbers with the usual ordering < inherited from R. Take a transcendental number, say π, and add a unary relation S to the structure given by the subset (−π,π) ∩ R. Now consider the subset A of R defined by the formula

0<a \,\wedge\, S(a)

so that the set consists of all strictly positive real algebraic numbers that are less than π. The set is clearly convex, but cannot be written as a finite union of points and intervals whose endpoints are in R. To write it as an interval one would either have to include the endpoint π, which isn't in R, or one would require infinitely many intervals, such as the union

\bigcup_{\alpha<\pi}(0,\alpha).

Since we have a definable set that isn't a finite union of points and intervals, this structure is not o-minimal. However, it is known that the structure is weakly o-minimal, and in fact the theory of this structure is weakly o-minimal.[3]

Notes

  1. ↑ M.A.Dickmann, Elimination of Quantifiers for Ordered Valuation Rings, The Journal of symbolic Logic, Vol. 52, No. 1 (Mar., 1987), pp 116-128.
  2. ↑ "Infinite" here simply refers to the two extra points added to M that act as lower and upper bounds, usually denoted −∞ and +∞.
  3. ↑ D. Macpherson, D. Marker, C. Steinhorn, Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 (2000), no. 12, pp.5435–5483, MR 1781273.