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 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 provides a distinguished (canonical) element of each class.
There exist many classification theorems in mathematics, as described below.
Geometry
- Classification of Euclidean plane isometries
- 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