Complete set of invariants

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

:$f\_i\; :\; X\; \backslash to\; Y\_i$

(where $X$ is the collection of objects being classified, up to some equivalence relation $\backslash sim$, and the $Y\_i$ are some sets), such that $x\; \backslash sim\; x\text{'}$ if and only if $f\_i(x)\; =\; f\_i(x\text{'})$ for all $i$. In words, such that two objects are equivalent if and only if all invariants are equal.{{citation

| last = Faticoni | first = Theodore G.

| contribution = Modules and point set topological spaces

| doi = 10.1201/9781420010763.ch10

| mr = 2229105

| pages = 87–105

| publisher = Chapman & Hall/CRC, Boca Raton, Florida

| series = Lect. Notes Pure Appl. Math.

| title = Abelian groups, rings, modules, and homological algebra

| volume = 249

| year = 2006| doi-broken-date = 2024-11-11

}}. See in particular [https://books.google.com/books?id=VOlZoLXQ92kC&pg=PA97 p. 97].

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

:$\backslash left(\; \backslash prod\; f\_i\; \backslash right)\; :\; (X/\backslash sim)\; \backslash to\; \backslash left(\; \backslash prod\; Y\_i\; \backslash right)$

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).