CGAL 4.7 - Algebraic Foundations
AlgebraicStructureTraits Concept Reference

## Definition

A model of AlgebraicStructureTraits reflects the algebraic structure of an associated type Type.

Depending on the concepts that Type fulfills, it contains various functors and descriptive tags. Moreover it gives access to the several possible algebraic operations within that structure.

IntegralDomainWithoutDivision
IntegralDomain
UniqueFactorizationDomain
EuclideanRing
Field
FieldWithSqrt
FieldWithKthRoot
FieldWithRootOf
CGAL::Integral_domain_without_division_tag
CGAL::Integral_domain_tag
CGAL::Unique_factorization_domain_tag
CGAL::Euclidean_ring_tag
CGAL::Field_tag
CGAL::Field_with_sqrt_tag
CGAL::Field_with_kth_root_tag
CGAL::Field_with_root_of_tag
Has Models:
CGAL::Algebraic_structure_traits<T>

## Types

A model of AlgebraicStructureTraits is supposed to provide:

typedef unspecified_type Type
The associated type.

typedef unspecified_type Algebraic_category
Tag indicating the algebraic structure of the associated type. More...

typedef unspecified_type Is_exact
Tag indicating whether Type is exact. More...

typedef unspecified_type Is_numerical_sensitive
Tag indicating whether Type is numerical sensitive. More...

typedef unspecified_type Boolean
This type specifies the return type of the predicates provided by this traits. More...

## Functors

In case a functor is not provided, it is set to CGAL::Null_functor.

typedef unspecified_type Is_zero
A model of AlgebraicStructureTraits_::IsZero. More...

typedef unspecified_type Is_one
A model of AlgebraicStructureTraits_::IsOne. More...

typedef unspecified_type Square
A model of AlgebraicStructureTraits_::Square. More...

typedef unspecified_type Simplify
A model of AlgebraicStructureTraits_::Simplify. More...

typedef unspecified_type Unit_part
A model of AlgebraicStructureTraits_::UnitPart. More...

typedef unspecified_type Integral_division
A model of AlgebraicStructureTraits_::IntegralDivision. More...

typedef unspecified_type Divides
A model of AlgebraicStructureTraits_::Divides. More...

typedef unspecified_type Is_square
A model of AlgebraicStructureTraits_::IsSquare. More...

typedef unspecified_type Gcd
A model of AlgebraicStructureTraits_::Gcd. More...

typedef unspecified_type Mod
A model of AlgebraicStructureTraits_::Mod. More...

typedef unspecified_type Div
A model of AlgebraicStructureTraits_::Div. More...

typedef unspecified_type Div_mod
A model of AlgebraicStructureTraits_::DivMod. More...

typedef unspecified_type Inverse
A model of AlgebraicStructureTraits_::Inverse. More...

typedef unspecified_type Sqrt
A model of AlgebraicStructureTraits_::Sqrt. More...

typedef unspecified_type Kth_root
A model of AlgebraicStructureTraits_::KthRoot. More...

typedef unspecified_type Root_of
A model of AlgebraicStructureTraits_::RootOf. More...

## Member Typedef Documentation

Tag indicating the algebraic structure of the associated type.

 Tag is: Type is model of: CGAL::Null_tag no algebraic concept CGAL::Integral_domain_without_division_tag IntegralDomainWithoutDivision CGAL::Integral_domain_tag IntegralDomain CGAL::Unique_factorization_domain_tag UniqueFactorizationDomain CGAL::Euclidean_ring_tag EuclideanRing CGAL::Field_tag Field CGAL::Field_with_sqrt_tag FieldWithSqrt CGAL::Field_with_kth_root_tag FieldWithKthRoot CGAL::Field_with_root_of_tag FieldWithRootOf

This type specifies the return type of the predicates provided by this traits.

The type must be convertible to bool and typically the type indeed maps to bool. However, there are also cases such as interval arithmetic, in which it is Uncertain<bool> or some similar type.

A model of AlgebraicStructureTraits_::Div.

Required by the concept EuclideanRing.

A model of AlgebraicStructureTraits_::DivMod.

Required by the concept EuclideanRing.

A model of AlgebraicStructureTraits_::Divides.

Required by the concept IntegralDomain.

A model of AlgebraicStructureTraits_::Gcd.

Required by the concept UniqueFactorizationDomain.

A model of AlgebraicStructureTraits_::IntegralDivision.

Required by the concept IntegralDomain.

A model of AlgebraicStructureTraits_::Inverse.

Required by the concept Field.

Tag indicating whether Type is exact.

This is either CGAL::Tag_true or CGAL::Tag_false.

An algebraic structure is considered exact, if all operations required by its concept are computed such that a comparison of two algebraic expressions is always correct. The exactness covers only those operations that are required by the algebraic structure concept.

e.g. an exact Field may have a Sqrt functor that is not exact.

Tag indicating whether Type is numerical sensitive.

This is either CGAL::Tag_true or CGAL::Tag_false.

An algebraic structure is considered as numerically sensitive, if the performance of the type is sensitive to the condition number of an algorithm.

A model of AlgebraicStructureTraits_::IsOne.

Required by the concept IntegralDomainWithoutDivision.

A model of AlgebraicStructureTraits_::IsSquare.

Required by the concept IntegralDomainWithoutDivision.

A model of AlgebraicStructureTraits_::IsZero.

Required by the concept IntegralDomainWithoutDivision. In case Type is also model of RealEmbeddable this is a model of RealEmbeddableTraits_::IsZero.

A model of AlgebraicStructureTraits_::KthRoot.

Required by the concept FieldWithKthRoot.

A model of AlgebraicStructureTraits_::Mod.

Required by the concept EuclideanRing.

A model of AlgebraicStructureTraits_::RootOf.

Required by the concept FieldWithRootOf.

A model of AlgebraicStructureTraits_::Simplify.

Required by the concept IntegralDomainWithoutDivision.

A model of AlgebraicStructureTraits_::Sqrt.

Required by the concept FieldWithSqrt.

A model of AlgebraicStructureTraits_::Square.

Required by the concept IntegralDomainWithoutDivision.

A model of AlgebraicStructureTraits_::UnitPart.

Required by the concept IntegralDomainWithoutDivision.