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.[1]
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
edit- 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
editA 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
edit- ^ Faticoni, Theodore G. (2006), "Modules and point set topological spaces", Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, Florida, pp. 87–105, doi:10.1201/9781420010763.ch10 (inactive 2024-11-11), MR 2229105
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: CS1 maint: DOI inactive as of November 2024 (link). See in particular p. 97.