In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) that is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊆ 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. It is known that the complete theory T of an o-minimal structure is an o-minimal theory.[1] This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure that is not minimal.
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
For a subset A of M, we consider the smallest structure S(A) containing S such that every finite subset of A is contained in S1. A subset D of Mn is called A-definable if it is contained in Sn(A); in that case A is called a set of parameters for D. A subset is called definable if it is A-definable for some A.
If M has a dense linear order without endpoints on it, say <, then a structure S on M is called o-minimal (respect to <) if it satisfies the extra axioms
The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.
O-minimal structures originated in model theory and so have a simpler — but equivalent — definition using the language of model theory.[2] 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,[3] then (M,<,...) is called an o-minimal structure if for any definable set X ⊆ M there are finitely many open intervals I1,..., Ir in M ∪ {±∞} and a finite set X0 such that
Examples of o-minimal theories are:
In the case of RCF, 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,[6] Whitney and Verdier stratification theorems and a good notion of dimension and Euler characteristic.
Moreover, continuously differentiable definable functions in a o-minimal structure satisfy a generalization of Łojasiewicz inequality,[7] a property that has been used to guarantee the convergence of some non-smooth optimization methods, such as the stochastic subgradient method (under some mild assumptions).[8][9][10]