mirror of https://github.com/CGAL/cgal
44 lines
1.4 KiB
TeX
44 lines
1.4 KiB
TeX
\begin{ccRefConcept}{AlgebraicStructureTraits::UnitPart}
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\ccDefinition
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This \ccc{AdaptableUnaryFunction} computes the unit part of a given ring
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element.
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The mathematical definition of unit part is as follows: Two ring elements $a$
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and $b$ are said to be associate if there exists an invertible ring element
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(i.e. a unit) $u$ such that $a = ub$. This defines an equivalence relation.
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We can distinguish exactly one element of every equivalence class as being
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unit normal. Then each element of a ring possesses a factorization into a unit
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(called its unit part) and a unit-normal ring element
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(called its unit normal associate).
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For the integers, the non-negative numbers are by convention unit normal,
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hence the unit-part of a non-zero integer is its sign. For a \ccc{Field}, every
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non-zero element is a unit and is its own unit part, its unit normal
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associate being one. The unit part of zero is, by convention, one.
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\ccRefines
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\ccc{AdaptableUnaryFunction}
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\ccSetThreeColumns{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}{xxx}{}
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\ccTypes
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\ccTypedef{typedef AlgebraicStructureTraits::AS result_type;}{}\ccGlue
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\ccTypedef{typedef AlgebraicStructureTraits::AS first_argument_type;}{}
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%\ccCreation
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%\ccCreationVariable{functor}
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%\ccConstructor{UnitPart();}{Default constructor.}
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\ccOperations
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\ccMethod{result_type operator()(first_argument_type& x);}{ returns the unit part of $x$.}
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%\ccHasModels
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\ccSeeAlso
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\ccRefIdfierPage{AlgebraicStructureTraits}
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\end{ccRefConcept}
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