Complete set of invariants
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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 Yi are some sets), such that x ∼ x' if and only if fi(x) = fi(x') 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.
[edit] 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.
[edit] 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
