Classification theorem

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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.

A few related issues to classification are the following.

  • The equivalence problem is "given two objects, determine if they are equivalent".
  • A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it.
  • A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
  • A canonical form solves the classification problem, and is more data: it not only classifies every class, but gives a distinguished (canonical) element of each class.

There exist many classification theorems in mathematics, as described below.

Geometry[edit]

Algebra[edit]

Linear algebra[edit]

Complex analysis[edit]