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 $x \sim x'$ 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.^[1]

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

$\prod f_i : (X/\sim) \to \prod Y_i$

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 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

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

References

1. Faticoni, Theodore G. (2006), "Modules and point set topological spaces", Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math. 249, Chapman & Hall/CRC, Boca Raton, FL, pp. 87–105, doi:10.1201/9781420010763.ch10, MR 2229105. See in particular p. 97.