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,[clarify] solves the classification problem, and is often a step in solving it.
A computable complete set of invariants[clarify] (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.
Artin–Wedderburn theorem – Classification of semi-simple rings and algebrasPages displaying short descriptions of redirect targets — a classification theorem for semisimple rings
Classification of simple real Lie algebras – graph encoding the structure of a reductive group over a non-algebraically-closed field, in which vertices are colored black or white according to whether they vanish on a maximal split torus, and the white vertices are acted upon by the Galois groupPages displaying wikidata descriptions as a fallback
Classification of simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroupsPages displaying short descriptions of redirect targets
Finite-dimensional vector space – Number of vectors in any basis of the vector spacePages displaying short descriptions of redirect targetss (by dimension)
Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity)