Rahul Sharma (Editor)

Classification theorem

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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.

Contents

A few related issues 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 computable complete set of invariants (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 gives a distinguished (canonical) element of each class.

    • There exist many classification theorems in mathematics, as described below.

    Geometry

  • Classification of Euclidean plane isometries
  • Classification theorem of surfaces
  • Classification of two-dimensional closed manifolds
  • Enriques–Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
  • Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface
  • Thurston's eight model geometries, and the geometrization conjecture

    • Algebra

  • Classification of finite simple groups
  • Artin–Wedderburn theorem — a classification theorem for semisimple rings

    • Linear algebra

  • Finite-dimensional vector spaces (by dimension)
  • rank–nullity theorem (by rank and nullity)
  • Structure theorem for finitely generated modules over a principal ideal domain
  • Jordan normal form
  • Sylvester's law of inertia

    • Complex analysis

  • Classification of Fatou components

    • References

    Classification theorem Wikipedia
    (Text) CC BY-SA