A structure $ (M,<,\ldots) $ is said to be o-minimal if every subset $ X \subset M^1 $ definable with parameters from $ M $ can be written as a finite union of points and intervals, i.e., as a boolean combination of sets of the form $ \{x \in M : x \le a\} $ and $ \{x \in M : x \ge a\} $. Note that this is an assertion about subsets of $ M^1 $, not definable sets in higher dimensions.

This notion is analogous to minimality. In minimality, one assumes that the definable (one-dimensional) sets are quantifier-free definable using nothing but equality. Here, one assumes that the (one-dimensional) sets are quantifier-free definable using nothing but the ordering.

A theory $ T $ with a predicate $ < $ is said to be o-minimal if every model of $ T $ is o-minimal. Unlike the case of minimality vs. strong minimality, there is no notion of strong o-minimality. It turns out that any elementary extension of an o-minimal structure is o-minimal. Consequently, the true theory of any o-minimal structure is an o-minimal theory. The proof of this is rather non-trivial, and uses the cell decomposition result for o-minimal theories.

Often one restricts to the class of o-minimal structures/theories in which the ordering $ (M,<) $ is dense, i.e., a model of DLO. Most o-minimal theories of interest have this property, and many proofs can be simplified with this assumption.

Examples Edit

Some relatively elementary examples:

  • DLO, the theory of dense linear orders. This is the true theory of $ (\mathbb{Q},<) $.
  • RCF, the theory of real closed fields. This is the true theory of $ \mathbb{R} $ as an ordered field.
  • DOAG or ODAG, the theory of divisible ordered abelian groups. This is the true theory of $ (\mathbb{R},+,<) $.

By hard theorems of Alex Wilkie and other people, certain expansions of the ordered field $ \mathbb{R} $ are known to be o-minimal.

  • The structure $ \mathbb{R}_{exp} := (\mathbb{R},+,\cdot,\exp) $ was proven to be o-minimal by Alex Wilkie. This structure consists of the ordered field $ \mathbb{R} $ expanded by adding in a predicate for the exponentiation map. This example is somewhat surprising, given that we lack a recursive axiomatization of this structure.
  • The structure $ \mathbb{R}_{an} $, consisting of the ordered field $ \mathbb{R} $ with restricted analytic functions, is o-minimal. For each real-analytic function $ f $ on an open neighborhood of $ [0,1]^n $, one adds a function symbol for $ f $ restricted to $ [0,1] $. This does not subsume $ \mathbb{R}_{exp} $, since $ \exp $ turns out to not be definable in $ \mathbb{R}_{an} $. In fact the o-minimality of $ \mathbb{R}_{an} $ is a more basic result. It is essentially Gabrielov's theorem.
  • More generally, $ \mathbb{R}_{an,exp} $ is o-minimal. This is the expansion of $ \mathbb{R} $ obtained by adding in both the exponential map and the restricted analytic functions.
  • More generally, one can add all Pfaffian functions. The most general result in this direction is due to Speissegger, maybe.

Properties Edit

O-minimal theories are NIP, but never stable or simple, as they have the order property. O-minimal theories are also superrosy, of finite rank.

In any o-minimal theory, definable closure and algebraic closure agree (on account of the ordering), and these operations define a pregeometry on the home sort. This yields a well-defined notion of dimension of definable sets.

Not all o-minimal theories eliminate imaginaries, even after naming all parameters from a model. However, o-minimal expansions of RCF always eliminate imaginaries, and in fact have definable choice (which includes definable Skolem functions). The same holds for o-minimal expansions of DOAG after naming at least one non-zero element.

Definable functions and definable sets have many nice structural properties. For simplicity assume that the order is dense. Then one has the following results:

  • Every definable function $ f : M^1 \to M^n $ is piecewise continuous: the domain of $ f $ can be written as a finite union of intervals, such that on each interval, $ f $ is continuous. If $ n = 1 $, then one can also arrange that on each interval, $ f $ is either constant, or strictly increasing, or strictly decreasing.
  • Every definable subset of $ M^n $ has finitely many definably connected components. In the presence of definable Skolem functions, each piece is definably path-connected.
  • More precisely, every definable subset $ X \subset M^n $ has a cell-decomposition: it can be written as disjoint union of sets that are "cells" in a certain sense. Each cell is definably connected, and in the case of o-minimal expansions of RCF, is definably homeomorphic to a ball.
  • If $ f : M^n \to M^m $ is a definable function, then the domain of $ f $ has a cell decomposition such that the restriction of $ f $ to each cell is continuous.
  • If $ X \subset M^n $, the topological closure $ \overline{X} $ of $ X $ has dimension no bigger than $ X $, and the frontier $ \overline{X} \setminus X $ has strictly lower dimension than $ X $.

Many of the topological pathologies that are common in pointset topology and real analysis don't occur when working with definable sets in o-minimal expansions of the reals. For example, every definable set is locally path-connected, every connected component is path connected, every set without interior is nowhere dense, and every definable set is homotopy equivalent to a finite simplicial complex. Moreover, every continuous definable function is piecewise differentiable, and in fact piecewise $ C^k $ for every $ k < \infty $. One also knows that if $ f $ and $ g $ are two definable functions $ \mathbb{R} \to \mathbb{R} $, then $ f $ and $ g $ are asymptotically comparable. Limits always exist (possibly taking values $ \pm \infty $).

These results apply in particular to, e.g., the structure $ (\mathbb{R},+,\cdot,\exp) $. In sharp contrast, the definable sets in $ (\mathbb{R},+,\cdot,\sin) $ are exactly the sets in the projective hierarchy, so e.g. there are definable sets which are not Borel.

O-minimal trichotomy Edit

Some analog of the Zilber trichotomy holds in the o-minimal setting.

Applications Edit

Real algebraic geometry, Pila-Wilkie…