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 realizable, solves the classification problem, and is often a step in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values) 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

Algebra

— a classification theorem for semisimple rings

Classification of Simple Lie algebras and groups

Linear algebra s (by dimension) (by rank and nullity)

(rational canonical form)

Analysis

Dynamical systems

Ratner classification theorem

Mathematical physics

See also

List of theorems

References