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 $$\sim$$, 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.

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 (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
 * $$\prod f_i : X \to \prod Y_i.$$