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| {{Unreferenced|date=March 2007}}
| | The author is known by the title of Figures Lint. My day occupation is a meter reader. South Dakota is her beginning place but she needs to move simply because of her family members. Playing baseball is the hobby he will never quit doing.<br><br>my blog post ... [http://xrambo.com/blog/191590 xrambo.com] |
| 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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The author is known by the title of Figures Lint. My day occupation is a meter reader. South Dakota is her beginning place but she needs to move simply because of her family members. Playing baseball is the hobby he will never quit doing.
my blog post ... xrambo.com