View source for Classification theorem

{{short description|Describes the objects of a given type, up to some equivalence}}
{{Unreferenced|date=December 2009}}
In [[mathematics]], a '''classification theorem''' answers the [[classification]] problem: "What are the objects of a given type, up to some [[Equivalence relation|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 {{clarify span|computable complete set of invariants|reason=Shouldn't this be "finite set of computable invariants"? Computability (whatever this is supposed to mean on a set of functions) is of no help if infinitely many functions must be evaluated or if an uncomputable function must be evaluated.|date=October 2020}} (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==
* {{annotated link|Euclidean plane isometry#Classification of Euclidean plane isometries|Classification of Euclidean plane isometries}}
* [[Platonic solid#Classification|Classification of Platonic solids]]
* Classification theorems of surfaces
** {{annotated link|Classification of two-dimensional closed manifolds}}
** {{annotated link|Enriques–Kodaira classification}} of [[algebraic surfaces]] (complex dimension two, real dimension four)
** {{annotated link|Nielsen–Thurston classification}} which characterizes homeomorphisms of a compact surface
* Thurston's eight model geometries, and the {{annotated link|geometrization conjecture}}
* {{annotated link|Holonomy#The Berger classification|Berger classification}}
* {{annotated link|Symmetric space#Classification result|Classification of Riemannian symmetric spaces}}
* {{annotated link|Lens space#Classification of 3-dimensional lens spaces|Classification of 3-dimensional lens spaces}}
* {{annotated link|Classification of manifolds}}

==Algebra==
* {{annotated link|Classification of finite simple groups}}
** {{annotated link|Abelian group#Classification|Classification of Abelian groups}}
** {{annotated link|Finitely generated abelian group#Classification|Classification of Finitely generated abelian group}}
** {{annotated link|Multiple transitivity|Classification of Rank 3 permutation group}}
** {{annotated link|Rank 3 permutation group#Classification|Classification of 2-transitive permutation groups}}
* {{annotated link|Artin–Wedderburn theorem}} &mdash; a classification theorem for semisimple rings
* {{annotated link|Classification of Clifford algebras}}
* {{annotated link|Classification of low-dimensional real Lie algebras}}
* Classification of Simple Lie algebras and groups
** {{annotated link|Semisimple Lie algebra#Classification|Classification of simple complex Lie algebras}}
** {{annotated link|Satake diagram|Classification of simple real Lie algebras}}
** {{annotated link|Simple Lie group#Full classification|Classification of centerless simple Lie groups}}
** {{annotated link|List of simple Lie groups|Classification of simple Lie groups}}
* {{annotated link|Bianchi classification}}
* {{annotated link|ADE classification}}
*{{annotated link|Langlands classification}}

==Linear algebra==
* {{annotated link|Finite-dimensional vector space}}s (by dimension)
* {{annotated link|Rank–nullity theorem}} (by rank and nullity)
* {{annotated link|Structure theorem for finitely generated modules over a principal ideal domain}}
* {{annotated link|Jordan normal form}}
* {{annotated link|Frobenius normal form}} (rational canonical form)
* {{annotated link|Sylvester's law of inertia}}

==Analysis==
* {{annotated link|Classification of discontinuities}}

==Dynamical systems==
* {{annotated link|Classification of Fatou components}}
* [[Ratner's theorems#Short description|Ratner classification theorem]]

==Mathematical physics==
* {{annotated link|Classification of electromagnetic fields}}
* {{annotated link|Petrov classification}}
* {{annotated link|Segre classification}}
* {{annotated link|Wigner's classification}}

==See also==
* {{annotated link|Representation theorem}}
* {{annotated link|Comparison theorem}}
* {{annotated link|List of manifolds}}
* [[List of theorems]]

==References==
{{reflist}}

{{DEFAULTSORT:Classification Theorem}}
[[Category:Mathematical theorems]]
[[Category:Mathematical classification systems]]