mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 coinvariants (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,

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

(or

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

) and orientability are a complete set of invariants. * Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.

Realizability of invariants

A complete set of invariants does not immediately yield a classification theorem: 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