Classification theorem

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.

Related issues

There are several related issues:

• The isomorphism 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 isomorphism problem.

Examples

There exist many classification theorems in mathematics:

Geometry

• Classification theorem of surfaces
  • Classification of two-dimensional closed manifolds
  • Enriques-Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
  • Nielsen-Thurston classification which characterizes homeomorphisms of a compact surface
• 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 spaces (by dimension)
• rank-nullity theorem (by rank and nullity)
• Structure theorem for finitely generated modules over a principal ideal domain
• Jordan normal form
• Sylvester's law of inertia