In mathematics, a complete set of invariants for a classification problem is a collection of maps :

f_i : X \to Y_i

(where

X

is the collection of objects being classified, up to some equivalence relation

\sim

, and the

Y_i

are some sets), such that

x \sim x'

if and only if

f_i(x) = f_i(x')

for all

i

. In words, such that two objects are equivalent if and only if all invariants are equal.. See in particula
p. 97
Symbolically, a complete set of invariants is a collection of maps such that :

\left( \prod f_i \right) : (X/\sim) \to \left( \prod Y_i \right)

is injective. As invariants are, by definition, equal on equivalent objects, equality of invariants is a ''necessary'' condition for equivalence; a ''complete'' set of invariants is a set such that equality of these is also ''sufficient'' for equivalence. In the context of a group action, this may be stated as: invariants are functions of

coinvariant In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

s (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).

Examples

* In the

classification of two-dimensional closed manifolds In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...

, Euler characteristic (or

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

) and

orientability In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...

are a complete set of invariants. *

Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...

of a matrix is a complete invariant for matrices up to conjugation, but

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

s (with multiplicities) are not.

Realizability of invariants

A complete set of invariants does not immediately yield a

classification theorem 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 relate ...

: not all combinations of invariants may be realized. Symbolically, one must also determine the image of :

\prod f_i : X \to \prod Y_i.

References

{{DEFAULTSORT:Complete Set Of Invariants Mathematical terminology