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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 → Y i {\displaystyle f_{i}:X\to Y_{i}}

(where X {\displaystyle X} is the collection of objects being classified, up to some equivalence relation ∼ {\displaystyle \sim } , and the Y i {\displaystyle Y_{i}} are some sets), such that x ∼ x ′ {\displaystyle x\sim x'} if and only if f i ( x ) = f i ( x ′ ) {\displaystyle f_{i}(x)=f_{i}(x')} for all i {\displaystyle 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

( ∏ f i ) : ( X / ∼ ) → ( ∏ Y i ) {\displaystyle \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

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

∏ f i : X → ∏ Y i . {\displaystyle \prod f_{i}:X\to \prod Y_{i}.}

References

  1. 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{{citation}}: CS1 maint: DOI inactive as of November 2024 (link). See in particular p. 97.
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