AlgebraicStructureTraits_::UnitPart Class Reference

#include <Concepts/AlgebraicStructureTraits--UnitPart.h>

Definition

This 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 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.

Refines:

AdaptableUnaryFunction

See also

AlgebraicStructureTraits

Types
typedef unspecified_type	result_type
	Is AlgebraicStructureTraits::Type. More...
typedef unspecified_type	argument_type
	Is AlgebraicStructureTraits::Type. More...

Operations
result_type	operator() (argument_type x)
	returns the unit part of \( x\). More...

Member Typedef Documentation

argument_type

typedef unspecified_type AlgebraicStructureTraits_::UnitPart::argument_type

Is AlgebraicStructureTraits::Type.

result_type

typedef unspecified_type AlgebraicStructureTraits_::UnitPart::result_type

Is AlgebraicStructureTraits::Type.

Member Function Documentation

operator()()

result_type AlgebraicStructureTraits_::UnitPart::operator()

(

argument_type

x

)

returns the unit part of \( x\).