Complete Set of Invariants

Complete Set Of Invariants

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

(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

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).