Complete set of invariants
In mathematics, a complete set of invariants for a classification problem is a collection of maps

(where  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 . 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

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

References

Mathematical terminology