O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure (M,&lt;,...) which is totally ordered by < is called an o-minimal structure if and only if every subset X &sub; 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. One can show 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 &isin; Sn then M &times; A and A &times;M are in Sn+1
 * 3) the set {(x1,...,xn) &isin; Mn : x1 = xn} is in Sn
 * 4) if A &isin; Sn+1 and π : Mn+1 &rarr; Mn is the projection map on the first n coordinates, then π(A) &isin; Mn.

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

the set {(x,y) &isin; M2 : x < y} is in S2 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 &mdash; but equivalent &mdash; 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 &sube; M there are finitely many intervals I1,...,Ir with endpoints in M &cup; {&plusmn;∞} and a finite set X0 such that
 * $$X=X_0\cup I_1\cup\ldots\cup I_r.$$

Examples
Examples of o-minimal theories are:
 * RCOF, the axioms for the real closed fields;
 * The complete theory of the real field with a symbol for the exponential function by Wilkie's theorem;
 * The complete theory of the real numbers 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 a o-minimal structure.)
 * Intermediate to the previous two examples is the complete theory of the real numbers with restricted Pfaffian functions added.
 * The complete theory of dense linear orders in the language with just the ordering.

In the first example, 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.