Strongly minimal theory

Strongly minimal theory

In model theory—a branch of mathematical logic—a minimal structure is an infinite one-sorted structure such that every subset of its domain that is definable with parameters is either finite or cofinite. A strongly minimal theory is a complete theory all models of that are minimal. A strongly minimal structure is a structure whose theory is strongly minimal.

Thus a structure is minimal only if the parametrically definable subsets of its domain cannot be avoided, because they are already parametrically definable in the pure language of equality. Strong minimality was one of the early notions in the new field of classification theory and stability theory that was opened up by Morley's theorem on totally categorical structures.

The nontrivial standard examples for strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACFp of algebraically closed fields. As the example ACFp shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("curves").

More generally, a subset of a structure that is defined as the set of realizations of a formula φ('x') is called a minimal set if every parametrically definable subset of it is either finite or cofinite. It is called a strongly minimal set if this is true even in all elementary extensions.

A strongly minimal set, equipped with the closure operator given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem. Boris Zilber conjectured that the only pregeometries that can arise from strongly minimal sets are those that arise in vector spaces, projective spaces, or algebraically closed fields. This conjecture was refuted by Ehud Hrushovski, who developed a method known as the "Hrushovski construction" to build new strongly minimal structures from finite structures.

See also

References

Baldwin, John T.; Lachlan, Alistair H. (1971), "On Strongly Minimal Sets", The Journal of Symbolic Logic (The Journal of Symbolic Logic, Vol. 36, No. 1) 36 (1): 79–96, doi:10.2307/2271517, JSTOR 2271517 

Hrushovski, Ehud (1993), "A new strongly minimal set", Annals of Pure and Applied Logic 62 (2): 147, doi:10.1016/0168-0072(93)90171-9 


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • o-minimal theory — In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by < is called an o minimal structure if and only if every definable subset X ⊂ M (with parameters taken from… …   Wikipedia

  • O-minimal theory — In mathematical logic, and more specifically in model theory, a totally ordered structure ( M , lt;,...) is o minimal if and only if every definable set X sub; M (with parameters) can be realized as a finite union of intervals and points.A theory …   Wikipedia

  • C-minimal theory — In model theory, a branch of mathematical logic, a C minimal theory is a theory that is minimal with respect to a ternary relation C with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important… …   Wikipedia

  • Minimal Supersymmetric Standard Model — Beyond the Standard Model Standard Model …   Wikipedia

  • Glossary of graph theory — Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with… …   Wikipedia

  • Constructivism (learning theory) — Jean Piaget: founder of Constructivism Constructivism is a theory of knowledge (epistemology)[1] that argues that humans generate knowledge and meaning from an interaction between their experiences and their ideas. During infancy, it was an… …   Wikipedia

  • Info-gap decision theory — is a non probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty,[1][2] in particular applying sensitivity analysis of the stability radius type[3] to perturbations in… …   Wikipedia

  • Density functional theory — Electronic structure methods Tight binding Nearly free electron model Hartree–Fock method Modern valence bond Generalized valence bond Møller–Plesset perturbation theory …   Wikipedia

  • Glossary of order theory — This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be …   Wikipedia

  • Moral influence theory of atonement — Part of a series on Atonement in Christianity Moral influence Recapitulation Substitutionary …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”