Complete set of invariants
(where X is the collection of objects being classified, up to some equivalence relation, and the are some sets), such that if and only if 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
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).
- 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