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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 \sim, 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.{{citation

Symbolically, a complete set of invariants is a collection of maps such that :\left( \prod f_i \right) : (X/\sim) \to \left( \prod Y_i \right) 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 :\prod f_i : X \to \prod Y_i.

References

Info: Wikipedia Source

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mathematical-terminology

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