Complete set of invariants
In mathematics, a complete set of invariants for a classification problem is a collection of maps
- [math]\displaystyle{ f_i : X \to Y_i }[/math]
(where [math]\displaystyle{ X }[/math] is the collection of objects being classified, up to some equivalence relation [math]\displaystyle{ \sim }[/math], and the [math]\displaystyle{ Y_i }[/math] are some sets), such that [math]\displaystyle{ x \sim x' }[/math] if and only if [math]\displaystyle{ f_i(x) = f_i(x') }[/math] for all [math]\displaystyle{ i }[/math]. 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
- [math]\displaystyle{ \left( \prod f_i \right) : (X/\sim) \to \left( \prod Y_i \right) }[/math]
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
- [math]\displaystyle{ \prod f_i : X \to \prod Y_i. }[/math]
References
- ↑ Faticoni, Theodore G. (2006), "Modules and point set topological spaces", Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, Florida, pp. 87–105, doi:10.1201/9781420010763.ch10. See in particular p. 97.
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