O-minimality
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In mathematical logic, and more specifically in model theory, a theory T is o-minimal if for every model
- M = (<,...)
of T, and every definable set X (with parameters) in M, X can be realized as a finite union of intervals and points. Some kind of ordering concept is implicit in the notion of an "interval." In other words, any set definable in M by an arbitrary formula is also definable using nothing but the ordering. In strong minimality the definable sets in M are precisely those definable by equality.
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;
- 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.)
- Any model of the axioms for a linear order in the language L = {<}.
A major line of current research is based on discovering expansions of the real ordered field that are o-minimal.
[edit] References
- L. van den Dries, Tame Topology and o-minimal Structures, Cambridge University Press, Cambridge, UK, 1998
- Definable sets in ordered structures, A Pillay, C Steinhorn - Trans. Amer. Math. Soc, 1986
- Definable sets in ordered structures, II, JF Knight, A Pillay, C Steinhorn - Trans. Amer. Math. Soc, 1986
- Definable sets in ordered structures. III, A Pillay, C Steinhorn - Trans. Amer. Math. Soc, 1988
- Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, A. J. Wilkie - J. Amer. Math. Soc., 9 (1996)
- P-adic and real subanalytic sets, J. Denef and L. van den Dries - Ann. Math., 54 (1989)
[edit] See also
- d-minimality
- semialgebraic set
- strongly minimal set
- weakly o-minimal structure
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