87 lines
5.2 KiB
Markdown
87 lines
5.2 KiB
Markdown
---
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title: "Classification theorem"
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chunk: 1/1
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source: "https://en.wikipedia.org/wiki/Classification_theorem"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T09:08:23.500509+00:00"
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instance: "kb-cron"
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---
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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.
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A few issues related to classification are the following.
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The equivalence problem is "given two objects, determine if they are equivalent".
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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)
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A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
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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.
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There exist many classification theorems in mathematics, as described below.
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== Geometry ==
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Classification of Euclidean plane isometries – Isometry of the Eluclidean plane
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Classification of Platonic solids
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Classification theorems of surfaces
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Classification of two-dimensional closed manifolds – Two-dimensional manifoldPages displaying short descriptions of redirect targets
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Enriques–Kodaira classification – Mathematical classification of surfaces of algebraic surfaces (complex dimension two, real dimension four)
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Nielsen–Thurston classification – Characterizes homeomorphisms of a compact orientable surface which characterizes homeomorphisms of a compact surface
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Thurston's eight model geometries, and the geometrization conjecture – Three dimensional analogue of uniformization conjecture
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Berger classification – Concept in differential geometry
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Classification of Riemannian symmetric spaces – (pseudo-)Riemannian manifold whose geodesics are reversible
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Classification of 3-dimensional lens spaces – Class of topological space
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Classification of manifolds – Basic question in geometry and topology
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== Algebra ==
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Classification of finite simple groups – Theorem classifying finite simple groups
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Classification of Abelian groups – Commutative group (mathematics)
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Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset
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Classification of Rank 3 permutation group – Five sporadic simple groupsPages displaying short descriptions of redirect targets
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Classification of 2-transitive permutation groups
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Artin–Wedderburn theorem – Classification of semi-simple rings and algebrasPages displaying short descriptions of redirect targets — a classification theorem for semisimple rings
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Classification of Clifford algebras – Classification in abstract algebra
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Classification of low-dimensional real Lie algebras
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Classification of Simple Lie algebras and groups
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Classification of simple complex Lie algebras – Direct sum of simple Lie algebras
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Classification of simple real Lie algebras – Term in mathematics
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Classification of centerless simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroups
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Classification of simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroupsPages displaying short descriptions of redirect targets
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Bianchi classification – Lie algebra classification
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ADE classification – Mathematical classification
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Langlands classification – Mathematical theory
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== Linear algebra ==
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Finite-dimensional vector space – Number of vectors in any basis of the vector spacePages displaying short descriptions of redirect targetss (by dimension)
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Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity)
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Structure theorem for finitely generated modules over a principal ideal domain – Statement in abstract algebra
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Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities
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Frobenius normal form – Canonical form of matrices over a field (rational canonical form)
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Sylvester's law of inertia – Theorem of matrix algebra of invariance properties under basis transformations
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== Analysis ==
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Classification of discontinuities – Mathematical analysis of discontinuous points
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== Dynamical systems ==
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Classification of Fatou components – Components of the Fatou set
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Ratner classification theorem
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== Mathematical physics ==
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Classification of electromagnetic fields
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Petrov classification – Classification used in differential geometry and general relativity
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Segre classification – Algebraic classification of rank two symmetric tensors
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Wigner's classification – Classification of irreducible representations of the Poincaré group
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== See also ==
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Representation theorem – Proof that every structure with certain properties is isomorphic to another structure
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Comparison theorem
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Moduli space – Geometric space whose points represent algebro-geometric objects of some fixed kind
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List of manifolds
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List of theorems
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== References == |