Ominimality

Ominimality, often written o-minimality, is a concept in model theory used to study tame expansions of ordered structures, especially real closed fields. An o-minimal structure on a real closed field provides a framework in which the definable sets in every one-dimensional coordinate are simple: every definable subset of the line is a finite union of points and open intervals. More generally, the theory imposes that higher-dimensional definable sets admit tame geometric descriptions, leading to robust dimension theory and topological regularity.

A central feature of o-minimality is the cell decomposition theorem: every definable set in R^n can be partitioned into finitely many definable cells, each of which is a simple, well-behaved geometric object. This yields tame topology, monotonicity of definable functions, and a well-behaved notion of dimension. Definable sets and maps in o-minimal structures avoid pathological fractal behavior common in more general settings.

Key examples and milestones include: the real field (R, +, ×, <) as the basic o-minimal structure; expansions by restricted analytic functions (R_an) which are o-minimal; the real exponential field (R_exp) shown to be o-minimal by Wilkie; and various combinations such as R_an_exp, which are also o-minimal. These results provide powerful tools for real algebraic geometry and tame topology, linking model-theoretic methods with geometric and analytic phenomena.

Ominimality thus offers a rigorous framework for understanding when expansions of ordered fields behave in a controlled, “tame” manner, enabling precise descriptions of the geometry of definable sets and functions.