Classification theorem: Difference between revisions

Revision as of 00:00, 7 March 2007

Revision as of 03:35, 7 March 2007

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In [[mathematics]], a '''classification theorem''' answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.

==Definition==

In [[mathematics]], a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.

A few related issues to classification are the following.

==Related issues==

There are several related issues:

*The isomorphism problem is "given two objects, determine if they are equivalent"

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*A computable [[complete set of invariants]] (together with which invariants are realizable) solves both the classification problem and the isomorphism problem.

There exist many '''classification theorems''' in [[mathematics]]:, as described below.

==Examples==

There exist many '''classification theorems''' in [[mathematics]]:

===Geometry===

* '''Classification theorem of surfaces'''

** [[Classification of two-dimensional closed manifolds]]

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* Thurston's eight model geometries, and the [[geometrization conjecture]]

===Algebra===

* [[Classification of finite simple groups]]

* [[Artin–Wedderburn theorem]] — a classification theorem for semisimple rings

====[[Linear algebra]]====

* [[Finite-dimensional vector space]]s (by dimension)

* [[rank-nullity theorem]] (by rank and nullity)

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* [[Jordan normal form]]

* [[Sylvester's law of inertia]]

[[Category:Mathematical theorems]]