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?". 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}}
- 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}} — 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 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