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<p>Explained &quot;realizable&quot;.</p> <p><b>New page</b></p><div>{{short description|Describes the objects of a given type, up to some equivalence}}<br /> {{Unreferenced|date=December 2009}}<br /> In [[mathematics]], a &#039;&#039;&#039;classification theorem&#039;&#039;&#039; answers the [[classification]] problem: &quot;What are the objects of a given type, up to some [[Equivalence relation|equivalence]]?&quot;. It gives a non-redundant [[enumeration]]: each object is equivalent to exactly one class.<br /> <br /> A few issues related to classification are the following.<br /> <br /> *The equivalence problem is &quot;given two objects, determine if they are equivalent&quot;.<br /> *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)<br /> *A {{clarify span|computable complete set of invariants|reason=Shouldn&#039;t this be &quot;finite set of computable invariants&quot;? 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.<br /> * 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.<br /> <br /> There exist many &#039;&#039;&#039;classification theorems&#039;&#039;&#039; in [[mathematics]], as described below.<br /> <br /> ==Geometry==<br /> * {{annotated link|Euclidean plane isometry#Classification of Euclidean plane isometries|Classification of Euclidean plane isometries}}<br /> * [[Platonic solid#Classification|Classification of Platonic solids]]<br /> * Classification theorems of surfaces<br /> ** {{annotated link|Classification of two-dimensional closed manifolds}}<br /> ** {{annotated link|Enriques–Kodaira classification}} of [[algebraic surfaces]] (complex dimension two, real dimension four)<br /> ** {{annotated link|Nielsen–Thurston classification}} which characterizes homeomorphisms of a compact surface<br /> * Thurston&#039;s eight model geometries, and the {{annotated link|geometrization conjecture}}<br /> * {{annotated link|Holonomy#The Berger classification|Berger classification}}<br /> * {{annotated link|Symmetric space#Classification result|Classification of Riemannian symmetric spaces}}<br /> * {{annotated link|Lens space#Classification of 3-dimensional lens spaces|Classification of 3-dimensional lens spaces}}<br /> * {{annotated link|Classification of manifolds}}<br /> <br /> ==Algebra==<br /> * {{annotated link|Classification of finite simple groups}}<br /> ** {{annotated link|Abelian group#Classification|Classification of Abelian groups}}<br /> ** {{annotated link|Finitely generated abelian group#Classification|Classification of Finitely generated abelian group}}<br /> ** {{annotated link|Multiple transitivity|Classification of Rank 3 permutation group}}<br /> ** {{annotated link|Rank 3 permutation group#Classification|Classification of 2-transitive permutation groups}}<br /> * {{annotated link|Artin–Wedderburn theorem}} &amp;mdash; a classification theorem for semisimple rings<br /> * {{annotated link|Classification of Clifford algebras}}<br /> * {{annotated link|Classification of low-dimensional real Lie algebras}}<br /> * Classification of Simple Lie algebras and groups<br /> ** {{annotated link|Semisimple Lie algebra#Classification|Classification of simple complex Lie algebras}}<br /> ** {{annotated link|Satake diagram|Classification of simple real Lie algebras}}<br /> ** {{annotated link|Simple Lie group#Full classification|Classification of centerless simple Lie groups}}<br /> ** {{annotated link|List of simple Lie groups|Classification of simple Lie groups}}<br /> * {{annotated link|Bianchi classification}}<br /> * {{annotated link|ADE classification}}<br /> *{{annotated link|Langlands classification}}<br /> <br /> ==Linear algebra==<br /> * {{annotated link|Finite-dimensional vector space}}s (by dimension)<br /> * {{annotated link|Rank–nullity theorem}} (by rank and nullity)<br /> * {{annotated link|Structure theorem for finitely generated modules over a principal ideal domain}}<br /> * {{annotated link|Jordan normal form}}<br /> * {{annotated link|Frobenius normal form}} (rational canonical form)<br /> * {{annotated link|Sylvester&#039;s law of inertia}}<br /> <br /> ==Analysis==<br /> * {{annotated link|Classification of discontinuities}}<br /> <br /> ==Dynamical systems==<br /> * {{annotated link|Classification of Fatou components}}<br /> * [[Ratner&#039;s theorems#Short description|Ratner classification theorem]]<br /> <br /> ==Mathematical physics==<br /> * {{annotated link|Classification of electromagnetic fields}}<br /> * {{annotated link|Petrov classification}}<br /> * {{annotated link|Segre classification}}<br /> * {{annotated link|Wigner&#039;s classification}}<br /> <br /> ==See also==<br /> * {{annotated link|Representation theorem}}<br /> * {{annotated link|Comparison theorem}}<br /> * {{annotated link|List of manifolds}}<br /> * [[List of theorems]]<br /> <br /> ==References==<br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Classification Theorem}}<br /> [[Category:Mathematical theorems]]<br /> [[Category:Mathematical classification systems]]</div>

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