Classification theorem

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 issues related 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 solves the classification problem, and is often a step in solving it.
 * A (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 provides a distinguished (canonical) element of each class.

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

Geometry

 * Classification of Platonic solids
 * Classification theorems of surfaces
 * of algebraic surfaces (complex dimension two, real dimension four)
 * which characterizes homeomorphisms of a compact surface
 * Thurston's eight model geometries, and the
 * which characterizes homeomorphisms of a compact surface
 * Thurston's eight model geometries, and the

Algebra

 * &mdash; a classification theorem for semisimple rings
 * Classification of Simple Lie algebras and groups
 * &mdash; a classification theorem for semisimple rings
 * Classification of Simple Lie algebras and groups
 * &mdash; a classification theorem for semisimple rings
 * Classification of Simple Lie algebras and groups
 * Classification of Simple Lie algebras and groups
 * Classification of Simple Lie algebras and groups
 * Classification of Simple Lie algebras and groups

Linear algebra

 * s (by dimension)
 * (by rank and nullity)
 * (rational canonical form)
 * (rational canonical form)
 * (rational canonical form)

Dynamical systems

 * Ratner classification theorem
 * Ratner classification theorem