Classification theorem
Template:Short description Script error: No such module "Unsubst". 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 Template:Clarify span (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
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- Classification of Platonic solids
- Classification theorems of surfaces
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- Template:Annotated link of algebraic surfaces (complex dimension two, real dimension four)
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- Thurston's eight model geometries, and the Template:Annotated link
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Algebra
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- Template:Annotated link — a classification theorem for semisimple rings
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- Classification of Simple Lie algebras and groups
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Linear algebra
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- Template:Annotated link (by rank and nullity)
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- Template:Annotated link (rational canonical form)
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