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| In [[mathematics]], a complete set of [[Invariant (mathematics)|invariant]]s for a [[classification theorems|classification problem]] is a collection of maps
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| :<math>f_i : X \to Y_i \,</math>
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| (where ''X'' is the collection of objects being classified, up to some equivalence relation, and the <math>Y_i</math> are some sets), such that <math>x</math> ∼ <math>x'</math> if and only if <math>f_i(x) = f_i(x')</math> for all ''i''. In words, such that two objects are equivalent if and only if all invariants are equal.
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| Symbolically, a complete set of invariants is a collection of maps such that
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| :<math>\prod f_i : (X/\sim) \to \prod Y_i</math>
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| is [[injective]].
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| 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 [[coinvariant]]s (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).
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| ==Examples==
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| * In the [[classification of two-dimensional closed manifolds]], [[Euler characteristic]] (or [[Genus (mathematics)|genus]]) and [[orientability]] are a complete set of invariants.
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| * [[Jordan normal form]] of a matrix is a complete invariant for matrices up to conjugation, but [[eigenvalue]]s (with multiplicities) are not.
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| ==Realizability of invariants==
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| 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
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| :<math>\prod f_i : X \to \prod Y_i.</math>
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| {{DEFAULTSORT:Complete Set Of Invariants}}
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| [[Category:Mathematical terminology]]
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