In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a classification theorem answers the
classification
Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identif ...
problem: "What are the objects of a given type, up to some
equivalence?". It gives a non-redundant
enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the element (mathematics), elements of a Set (mathematics), set. The pre ...
: 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 In mathematics, a complete set of invariants for a classification problem is a collection of maps
:f_i : X \to Y_i
(where X is the collection of objects being classified, up to some equivalence relation \sim, and the Y_i are some sets), such that ...
, 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
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
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
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
{{DEFAULTSORT:Classification Theorem
Mathematical theorems
Mathematical classification systems