Complete set of invariants

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, and the $Y_{i}$ are some sets), such that $x$ ∼ $x'$ 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.

Symbolically, a complete set of invariants is a collection of maps such that

$\prod f_{i}:(X/\sim )\to \prod Y_{i}$

is injective.

Examples

• In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) 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

$\prod f_{i}:X\to \prod Y_{i}.$