Complete set of invariants

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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, 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.[1]

Symbolically, a complete set of invariants is a collection of maps such that

\prod f_i : (X/\sim) \to \prod Y_i

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 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[edit]

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.


  1. ^ 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, FL, pp. 87–105, doi:10.1201/9781420010763.ch10, MR 2229105 . See in particular p. 97.