Complete set of invariants

In mathematics, a complete set of invariants for a classification problem is a collection of maps

${\displaystyle f_{i}:X\to Y_{i}}$

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

${\displaystyle \left(\prod f_{i}\right):(X/\sim )\to \left(\prod Y_{i}\right)}$

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

• 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

${\displaystyle \prod f_{i}:X\to \prod Y_{i}.}$

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.