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

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