nLab
o-minimal structure

Contents

Definition

Let (R,<)(R, \lt) be a dense? linear order without endpoints?. An order-minimal or o-minimal structure on RR is a structure 𝒮\mathcal{S} on RR such that

  • The relation <\lt belongs to 𝒮 2\mathcal{S}_2;

  • The elements of 𝒮 1\mathcal{S}_1 are precisely finite unions of points and intervals in RR.

Here an interval can mean a set of the form I a,b={xR:a<x<b}I_{a, b} = \{x \in R: a \lt x \lt b\}, or I a={xR:x<a}I_{\downarrow a} = \{x \in R: x \lt a\}, or I a={xR:a<x}I_{\uparrow a} = \{x \in R: a \lt x\}.

Commentary

A structure on a set RR can be thought of as the collection 𝒮= n𝒮 n\mathcal{S} = \bigcup_n \mathcal{S}_n of sets that are definable with respect to a one-sorted first-order language LL with a given interpretation in RR. Thus 𝒮 n\mathcal{S}_n is the collection of subsets of R nR^n which are defined by nn-ary predicates in LL. The definition of o-minimal structure supposes that LL contains a relation symbol <\lt, and that <\lt is interpreted in RR as a dense linear order without endpoints.

The o-minimality condition places a sharp restriction on which subsets of RR can be defined in the language. Essentially, it means that the only definable subsets of RR are those which are definable in terms of constants and the predicates <\lt and ==.

The archetypal example of an o-minimal structure is that of semi-algebraic sets defined over \mathbb{R} (which form a structure due to the Tarski-Seidenberg theorem).

Remarkably, quite a lot can be said about the structure of definable sets in an o-minimal structure over \mathbb{R}, and this is a very active area of model theory. The notion of o-minimal structure has been proposed as a reasonable candidate for an axiomatic approach to Grothendieck’s hoped-for “tame topology” ( topologie modérée ).

O-minimal theories

A theory is o-minimal if every model MM of TT is an o-minimal structure.

References

  • Michel Coste, An Introduction to O-Minimal Geometry , Lecture notes Pisa 1999. (pdf)

  • Lou van den Dries, Exponential rings, exponential polynomials and exponential functions , Pacific Journal of Mathematics 113 no.1 (1984) pp.51–66. (pdf)

  • Lou van den Dries, Tame topology and O-minimal structures, London Math. Soc. Lecture Notes Series 248, Cambridge U. Press 1998.

  • Alexandre Grothendieck, Esquisse d’un Programme, section 5. English translation available in Geometric Galois Actions I (edited by L. Schneps and P. Lochak), LMS Lecture Notes Ser. 242, CUP 1997.

  • Alessandro Berarducci, Definable groups in o-minimal structures, pdf; Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup, J. Symbolic Logic 74:3 (2009), 891-900, MR2548466, euclid, doi.

  • Alessandro Berarducci, O-minimal spectra, infinitesimal subgroups and cohomology, J. Symbolic Logic 72 (2007), no. 4, pp. 1177–1193, MR2371198, euclid, doi.

  • M. Edmundo, G. O. Jones, N. J. Peatfield, Sheaf cohomology in o-minimal structures, J. Math. Logic 6 (2006), no. 2, pp. 163–179, MR2317425, doi

  • Mario J. Edmundo, Luca Prelli, Invariance of o-minimal cohomology with definably compact supports, arxiv/1205.6124

  • Olivier Le Gal, Jean-Philippe Rolin, An o-minimal structure which does not admit C C^{\infty } cellular decomposition, Annales de l’institut Fourier 59:2 (2009), p. 543-562, MR2521427 Zbl 1193.03065 numdam

Revised on September 2, 2014 07:55:18 by Thomas Holder (82.113.99.180)