In mathematics, a complete set of invariants for a classification problem is a collection of maps
formula_1
(where formula_2 is the collection of objects being classified, up to some equivalence relation formula_3, and the formula_4 are some sets), such that formula_5 if and only if formula_6 for all formula_7. 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
formula_8
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).
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
formula_9
References.
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