\begin{ccRefFunctionObjectConcept}{AlgebraicStructureTraits::UnitPart}

\ccDefinition

This \ccc{AdaptableUnaryFunction} computes the unit part of a given ring 
element. 

The mathematical definition of unit part is as follows: Two ring elements $a$ 
and $b$ are said to be associate if there exists an invertible ring element 
(i.e. a unit) $u$ such that $a = ub$. This defines an equivalence relation. 
We can distinguish exactly one element of every equivalence class as being 
unit normal. Then each element of a ring possesses a factorization into a unit 
(called its unit part) and a unit-normal ring element 
(called its unit normal associate).

For the integers, the non-negative numbers are by convention unit normal, 
hence the unit-part of a non-zero integer is its sign. For a \ccc{Field}, every 
non-zero element is a unit and is its own unit part, its unit normal 
associate being one. The unit part of zero is, by convention, one.

\ccRefines 

\ccc{AdaptableUnaryFunction} 

\ccTypes
\ccNestedType{result_type} 
        { Is \ccc{AlgebraicStructureTraits::Type}.}
\ccNestedType{argument_type}
        { Is \ccc{AlgebraicStructureTraits::Type}.}

\ccOperations
\ccThree{xxxxxxxxxxx}{xxxxxxxxxxx}{}
\ccCreationVariable{unit_part}

\ccMethod{result_type operator()(argument_type x);}
{ returns the unit part of $x$.}

%\ccHasModels

\ccSeeAlso

\ccRefIdfierPage{AlgebraicStructureTraits}

\end{ccRefFunctionObjectConcept}