Contents
.. MODULEAUTHOR::
Simon King,
Friedrich\--Schiller\--Universität Jena,
<simon.king@uni-jena.de>
© 2011/2013
The aim of this tutorial is to explain how one can benefit from Sage's category framework and coercion model when implementing new algebraic structures. It is based on a worksheet created in 2011.
We illustrate the concepts of Sage's category framework and coercion model by means of a detailed example, namely a toy implementation of fraction fields. The code is developed step by step, so that the reader can focus on one detail in each part of this tutorial. The complete code can be found in the appendix.
Use existing base classes
For using Sage's coercion system, it is essential to work with sub--classes of :class:`sage.structure.parent.Parent` or :class:`sage.structure.element.Element`, respectively. They provide default implementations of many "magical" double-underscore Python methods, which must not be overridden. Instead, the actual implementation should be in single underscore methods, such as
_add_or_mul_.Turn your parent structure into an object of a category
Declare the category during initialisation---Your parent structure will inherit further useful methods and consistency tests.
Provide your parent structure with an element class
Assign to it an attribute called
Element---The elements will inherit further useful methods from the category. In addition, some basic conversions will immediately work.Implement further conversions
Never override a parent's
__call__method! Provide the method_element_constructor_instead.Declare coercions
If a conversion happens to be a morphism, you may consider to turn it into a coercion. It will then implicitly be used in arithmetic operations.
Advanced coercion: Define construction functors for your parent structure
Sage will automatically create new parents for you when needed, by the so--called :func:`sage.categories.pushout.pushout` construction.
Run the automatic test suites
Each method should be documented and provide a doc test (we are not giving examples here). In addition, any method defined for the objects or elements of a category should be supported by a test method, that is executed when running the test suite.
In Sage, a "Parent" is an object of a category and contains elements. Parents should inherit from :class:`sage.structure.parent.Parent` and their elements from :class:`sage.structure.element.Element`.
Sage provides appropriate sub--classes of :class:`~sage.structure.parent.Parent` and :class:`~sage.structure.element.Element` for a variety of more concrete algebraic structures, such as groups, rings, or fields, and of their elements. But some old stuff in Sage doesn't use it. Volunteers for refactoring are welcome!
Since we wish to implement a special kind of fields, namely fraction fields, it makes sense to build on top of the base class :class:`sage.rings.ring.Field` provided by Sage.
sage: from sage.rings.ring import Field
This base class provides a lot more methods than a general parent:
sage: [p for p in dir(Field) if p not in dir(Parent)] ['__fraction_field', '__ideal_monoid', '__iter__', '__len__', '__pow__', '__rpow__', '__rtruediv__', '__rxor__', '__truediv__', '__xor__', '_an_element_impl', '_coerce_', '_coerce_c', '_coerce_impl', '_coerce_try', '_default_category', '_gens', '_ideal_class_', '_latex_names', '_list', '_one_element', '_pseudo_fraction_field', '_random_nonzero_element', '_unit_ideal', '_zero_element', '_zero_ideal', 'algebraic_closure', 'base_extend', 'class_group', 'content', 'derivation', 'derivation_module', 'divides', 'epsilon', 'extension', 'fraction_field', 'frobenius_endomorphism', 'gcd', 'gen', 'gens', 'ideal', 'ideal_monoid', 'integral_closure', 'is_commutative', 'is_field', 'is_integral_domain', 'is_integrally_closed', 'is_noetherian', 'is_prime_field', 'is_subring', 'krull_dimension', 'localization', 'ngens', 'one', 'order', 'prime_subfield', 'principal_ideal', 'quo', 'quotient', 'quotient_ring', 'random_element', 'unit_ideal', 'zero', 'zero_ideal', 'zeta', 'zeta_order']
The following is a very basic implementation of fraction fields, that needs to be complemented later.
sage: from sage.structure.unique_representation import UniqueRepresentation
sage: class MyFrac(UniqueRepresentation, Field):
....: def __init__(self, base):
....: if base not in IntegralDomains():
....: raise ValueError("%s is no integral domain" % base)
....: Field.__init__(self, base)
....: def _repr_(self):
....: return "NewFrac(%s)"%repr(self.base())
....: def base_ring(self):
....: return self.base().base_ring()
....: def characteristic(self):
....: return self.base().characteristic()
This basic implementation is formed by the following steps:
Any ring in Sage has a base and a base ring. The "usual" fraction field of a ring R has the base R and the base ring
R.base_ring():sage: Frac(QQ['x']).base(), Frac(QQ['x']).base_ring() (Univariate Polynomial Ring in x over Rational Field, Rational Field)
Declaring the base is easy: We just pass it as an argument to the field constructor.
sage: Field(ZZ['x']).base() Univariate Polynomial Ring in x over Integer Ring
We are implementing a separate method returning the base ring.
Python uses double--underscore methods for arithmetic methods and string representations. Sage's base classes often have a default implementation, and it is requested to implement SINGLE underscore methods _repr_, and similarly _add_, _mul_ etc.
You are encouraged to make your parent "unique". That's to say, parents should only evaluate equal if they are identical. Sage provides frameworks to create unique parents. We use here the most easy one: Inheriting from the class :class:`sage.structure.unique_representation.UniqueRepresentation` is enough. Making parents unique can be quite important for an efficient implementation, because the repeated creation of "the same" parent would take a lot of time.
Fraction fields are only defined for integral domains. Hence, we raise an error if the given ring does not belong to the category of integral domains. This is our first use case of categories.
Last, we add a method that returns the characteristic of the field. We don't go into details, but some automated tests that we study below implicitly rely on this method.
We see that our basic implementation correctly refuses a ring that is not an integral domain:
sage: MyFrac(ZZ['x']) NewFrac(Univariate Polynomial Ring in x over Integer Ring) sage: MyFrac(Integers(15)) Traceback (most recent call last): ... ValueError: Ring of integers modulo 15 is no integral domain
Note
Inheritance from :class:`~sage.structure.unique_representation.UniqueRepresentation` automatically provides our class with pickling, preserving the unique parent condition. If we had defined the class in some external module or in an interactive session, pickling would work immediately.
However, for making the following example work in Sage's doctesting
framework, we need to assign our class as an attribute of the __main__
module, so that the class can be looked up during unpickling.
sage: import __main__ sage: __main__.MyFrac = MyFrac sage: loads(dumps(MyFrac(ZZ))) is MyFrac(ZZ) True
Note
In the following sections, we will successively add or change details of
MyFrac. Rather than giving a full class definition in each step, we
define new versions of MyFrac by inheriting from the previously
defined version of MyFrac. We believe this will help the reader to
focus on the single detail that is relevant in each section.
The complete code can be found in the appendix.
We use the base class :class:`sage.structure.element.FieldElement`. Note that in the creation of field elements it is not tested that the given parent is a field:
sage: from sage.structure.element import FieldElement sage: FieldElement(ZZ) Generic element of a structure
Our toy implementation of fraction field elements is based on the following considerations:
A fraction field element is defined by numerator and denominator, which both need to be elements of the base. There should be methods returning numerator resp. denominator.
The denominator must not be zero, and (provided that the base is an ordered ring) we can make it non-negative, without loss of generality. By default, the denominator is one.
The string representation is returned by the single--underscore method
_repr_. In order to make our fraction field elements distinguishable from those already present in Sage, we use a different string representation.Arithmetic is implemented in single--underscore method
_add_,_mul_, etc. We do not override the default double underscore __add__, __mul__, since otherwise, we could not use Sage's coercion model.Comparisons can be implemented using
_richcmp_. This automatically makes the relational operators like==and<work. Inside this method, you can use therichcmpfunctions and related tools provided by sage.Note that
_richcmp_should be provided, since otherwise comparison does not work:sage: class Foo(sage.structure.element.Element): ....: def __init__(self, parent, x): ....: self.x = x ....: def _repr_(self): ....: return "<%s>" % self.x sage: a = Foo(ZZ, 1) sage: b = Foo(ZZ, 2) sage: a <= b Traceback (most recent call last): ... TypeError: '<=' not supported between instances of 'Foo' and 'Foo'
In the single underscore methods, we can assume that both arguments belong to the same parent. This is one benefit of the coercion model.
When constructing new elements as the result of arithmetic operations, we do not directly name our class, but we use
self.__class__. Later, this will come in handy.
This gives rise to the following code:
sage: class MyElement(FieldElement):
....: def __init__(self, parent,n,d=None):
....: B = parent.base()
....: if d is None:
....: d = B.one()
....: if n not in B or d not in B:
....: raise ValueError("Numerator and denominator must be elements of %s"%B)
....: # Numerator and denominator should not just be "in" B,
....: # but should be defined as elements of B
....: d = B(d)
....: n = B(n)
....: if d==0:
....: raise ZeroDivisionError("The denominator must not be zero")
....: if d<0:
....: self.n = -n
....: self.d = -d
....: else:
....: self.n = n
....: self.d = d
....: FieldElement.__init__(self,parent)
....: def numerator(self):
....: return self.n
....: def denominator(self):
....: return self.d
....: def _repr_(self):
....: return "(%s):(%s)"%(self.n,self.d)
....: def _richcmp_(self, other, op):
....: from sage.structure.richcmp import richcmp
....: return richcmp(self.n*other.denominator(), other.numerator()*self.d, op)
....: def _add_(self, other):
....: C = self.__class__
....: D = self.d*other.denominator()
....: return C(self.parent(), self.n*other.denominator()+self.d*other.numerator(), D)
....: def _sub_(self, other):
....: C = self.__class__
....: D = self.d*other.denominator()
....: return C(self.parent(), self.n*other.denominator()-self.d*other.numerator(),D)
....: def _mul_(self, other):
....: C = self.__class__
....: return C(self.parent(), self.n*other.numerator(), self.d*other.denominator())
....: def _div_(self, other):
....: C = self.__class__
....: return C(self.parent(), self.n*other.denominator(), self.d*other.numerator())
Thanks to the single underscore methods, some basic arithmetics works, if we stay inside a single parent structure:
sage: P = MyFrac(ZZ) sage: a = MyElement(P, 3, 4) sage: b = MyElement(P, 1, 2) sage: a+b, a-b, a*b, a/b ((10):(8), (2):(8), (3):(8), (6):(4)) sage: a-b == MyElement(P, 1, 4) True
There is a default implementation of element tests. We can already do
sage: a in P True
since a is defined as an element of P. However, we can not verify yet that the integers are contained in the fraction field of the ring of integers. It does not even give a wrong answer, but results in an error:
sage: 1 in P Traceback (most recent call last): ... NotImplementedError: cannot construct elements of NewFrac(Integer Ring)
We will take care of this later.
Sometimes the base classes do not reflect the mathematics: The set of mtimes n matrices over a field forms, in general, not more than a vector space. Hence, this set (called :class:`~sage.matrix.matrix_space.MatrixSpace`) is not implemented on top of :class:`sage.rings.ring.Ring`. However, if m=n, then the matrix space is an algebra, thus, is a ring.
From the point of view of Python base classes, both cases are the same:
sage: MS1 = MatrixSpace(QQ,2,3) sage: isinstance(MS1, Ring) False sage: MS2 = MatrixSpace(QQ,2) sage: isinstance(MS2, Ring) False
Sage's category framework can differentiate the two cases:
sage: Rings() Category of rings sage: MS1 in Rings() False sage: MS2 in Rings() True
And indeed, MS2 has more methods than MS1:
sage: import inspect sage: L1 = len([s for s in dir(MS1) if inspect.ismethod(getattr(MS1,s,None))]) sage: L2 = len([s for s in dir(MS2) if inspect.ismethod(getattr(MS2,s,None))]) sage: L1 < L2 True
This is because the class of MS2 also inherits from the parent
class for algebras:
sage: MS1.__class__.__bases__ (<class 'sage.matrix.matrix_space.MatrixSpace'>, <class 'sage.categories.category.JoinCategory.parent_class'>) sage: MS2.__class__.__bases__ (<class 'sage.matrix.matrix_space.MatrixSpace'>, <class 'sage.categories.category.JoinCategory.parent_class'>)
Below, we will explain how this can be taken advantage of.
It is no surprise that our parent P defined above knows that it belongs to the category of fields, as it is derived from the base class of fields.
sage: P.category() Category of fields
However, we could choose a smaller category, namely the category of quotient fields.
One can provide default methods for all objects of a category, and for all elements of such objects. Hence, the category framework is a way to inherit useful stuff that is not present in the base classes. These default methods do not rely on implementation details, but on mathematical concepts.
In addition, the categories define test suites for their objects and elements---see the last section. Hence, one also gets basic sanity tests for free.
Abstract base classes for the objects ("parent_class") and the elements of objects ("element_class") are provided by attributes of the category. During initialisation of a parent, the class of the parent is dynamically changed into a sub--class of the category's parent class. Likewise, sub--classes of the category's element class are available for the creation of elements of the parent, as explained below.
A dynamic change of classes does not work in Cython. Nevertheless, method
inheritance still works, by virtue of a __getattr__ method.
Note
It is strongly recommended to use the category framework both in Python and in Cython.
Let us see whether there is any gain in choosing the category of quotient fields instead of the category of fields:
sage: QuotientFields().parent_class, QuotientFields().element_class (<class 'sage.categories.quotient_fields.QuotientFields.parent_class'>, <class 'sage.categories.quotient_fields.QuotientFields.element_class'>) sage: [p for p in dir(QuotientFields().parent_class) if p not in dir(Fields().parent_class)] [] sage: [p for p in dir(QuotientFields().element_class) if p not in dir(Fields().element_class)] ['_derivative', 'denominator', 'derivative', 'numerator', 'partial_fraction_decomposition']
So, there is no immediate gain for our fraction fields, but additional methods become available to our fraction field elements. Note that some of these methods are place-holders: There is no default implementation, but it is required (respectively is optional) to implement these methods:
sage: QuotientFields().element_class.denominator <abstract method denominator at ...> sage: from sage.misc.abstract_method import abstract_methods_of_class sage: abstract_methods_of_class(QuotientFields().element_class)['optional'] ['_add_', '_mul_'] sage: abstract_methods_of_class(QuotientFields().element_class)['required'] # py2 ['__nonzero__', 'denominator', 'numerator'] sage: abstract_methods_of_class(QuotientFields().element_class)['required'] # py3 ['__bool__', 'denominator', 'numerator']
Hence, when implementing elements of a quotient field, it is required to implement methods returning the denominator and the numerator, and a method that tells whether the element is nonzero, and in addition, it is optional (but certainly recommended) to provide some arithmetic methods. If one forgets to implement the required methods, the test suites of the category framework will complain---see below.
We simply need to declare the correct category by an optional argument of the field constructor, where we provide the possibility to override the default category:
sage: from sage.categories.quotient_fields import QuotientFields
sage: class MyFrac(MyFrac):
....: def __init__(self, base, category=None):
....: if base not in IntegralDomains():
....: raise ValueError("%s is no integral domain" % base)
....: Field.__init__(self, base, category=category or QuotientFields())
When constructing instances of MyFrac, their class is dynamically changed
into a new class called MyFrac_with_category. It is a common sub--class of
MyFrac and of the category's parent class:
sage: P = MyFrac(ZZ) sage: type(P) <class '__main__.MyFrac_with_category'> sage: isinstance(P,MyFrac) True sage: isinstance(P,QuotientFields().parent_class) True
The fraction field P inherits additional methods. For example, the base
class :class:`~sage.rings.fields.Field` does not have a method sum. But
P inherits such method from the category of commutative additive
monoids---see
:meth:`~sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.ParentMethods.sum`:
sage: P.sum.__module__ 'sage.categories.additive_monoids'
We have seen above that we can add elements. Nevertheless, the sum method
does not work, yet:
sage: a = MyElement(P, 3, 4) sage: b = MyElement(P, 1, 2) sage: c = MyElement(P, -1, 2) sage: P.sum([a, b, c]) Traceback (most recent call last): ... NotImplementedError: cannot construct elements of NewFrac(Integer Ring)
The reason is that the sum method starts with the return value of
P.zero(), which defaults to P(0)---but the conversion of
integers into P is not implemented, yet.
Similar to what we have seen for parents, a new class is dynamically created that combines the element class of the parent's category with the class that we have implemented above. However, the category framework is implemented in a different way for elements than for parents:
- We provide the parent P (or its class) with an attribute called
"
Element", whose value is a class. - The parent automatically obtains an attribute
P.element_class, that subclasses bothP.ElementandP.category().element_class.
Hence, for providing our fraction fields with their own element classes, we just need to add a single line to our class:
sage: class MyFrac(MyFrac): ....: Element = MyElement
This little change provides several benefits:
We can now create elements by simply calling the parent:
sage: P = MyFrac(ZZ) sage: P(1), P(2,3) ((1):(1), (2):(3))
There is a method
zeroreturning the expected result:sage: P.zero() (0):(1)
The
summethod mentioned above suddenly works:sage: a = MyElement(P, 9, 4) sage: b = MyElement(P, 1, 2) sage: c = MyElement(P, -1, 2) sage: P.sum([a,b,c]) (36):(16)
Exponentiation now works out of the box using the multiplication that we defined:
sage: a^3 (729):(64)
We provided P.Element, and thus obtain P.element_class, which is a
lazy attribute. It provides a dynamic class, which is a sub--class of
both MyElement defined above and of P.category().element_class:
sage: P.__class__.element_class <sage.misc.lazy_attribute.lazy_attribute object at ...> sage: P.element_class <class '__main__.MyFrac_with_category.element_class'> sage: type(P.element_class) <class 'sage.structure.dynamic_class.DynamicInheritComparisonMetaclass'> sage: issubclass(P.element_class, MyElement) True sage: issubclass(P.element_class,P.category().element_class) True
The default __call__ method of P passes the given arguments to
P.element_class, adding the argument parent=P. This is why we are now
able to create elements by calling the parent.
In particular, these elements are instances of that new dynamic class:
sage: type(P(2,3)) <class '__main__.MyFrac_with_category.element_class'>
Note
All elements of P should use the element class. In order to make sure
that this also holds for the result of arithmetic operations, we created
them as instances of self.__class__ in the arithmetic methods of
MyElement.
P.zero() defaults to returning P(0) and thus returns an
instance of P.element_class. Since P.sum([...]) starts the summation with
P.zero() and the class of the sum only depends on the first
summand, by our implementation, we have:
sage: type(a) <class '__main__.MyElement'> sage: isinstance(a,P.element_class) False sage: type(P.sum([a,b,c])) <class '__main__.MyFrac_with_category.element_class'>
The method factor provided by P.category().element_class (see above)
simply works:
sage: a; a.factor(); P(6,4).factor() (9):(4) 2^-2 * 3^2 2^-1 * 3
But that's surprising: The element a is just an instance of MyElement,
but not of P.element_class, and its class does not know about the factor
method. In fact, this is due to a __getattr__ method defined for
:class:`sage.structure.element.Element`.
sage: hasattr(type(a), 'factor') False sage: hasattr(P.element_class, 'factor') True sage: hasattr(a, 'factor') True
The category framework is sometimes blamed for speed regressions, as in :trac:`9138` and :trac:`11900`. But if the category framework is used properly, then it is fast. For illustration, we determine the time needed to access an attribute inherited from the element class. First, we consider an element that uses the class that we implemented above, but does not use the category framework properly:
sage: type(a)
<class '__main__.MyElement'>
sage: timeit('a.factor',number=1000) # random
1000 loops, best of 3: 2 us per loop
Now, we consider an element that is equal to a, but uses the category framework properly:
sage: a2 = P(9,4)
sage: a2 == a
True
sage: type(a2)
<class '__main__.MyFrac_with_category.element_class'>
sage: timeit('a2.factor',number=1000) # random
1000 loops, best of 3: 365 ns per loop
So, don't be afraid of using categories!
"Coercion" in the C programming language means "automatic type conversion". However, in Sage, coercion is involved if one wants to be able to do arithmetic, comparisons, etc. between elements of distinct parents. Hence, coercion is not about a change of types, but about a change of parents.
As an illustration, we show that elements of the same type may very well belong to rather different parents:
sage: P1 = QQ['v,w']; P2 = ZZ['w,v'] sage: type(P1.gen()) == type(P2.gen()) True sage: P1 == P2 False
P_2 naturally is a sub--ring of P_1. So, it makes sense to be able to add elements of the two rings---the result should then live in P_1, and indeed it does:
sage: (P1.gen()+P2.gen()).parent() is P1 True
It would be rather inconvenient if one needed to manually convert an element of P_2 into P_1 before adding. The coercion system does that conversion automatically.
A coercion happens implicitly, without being explicitly requested by the user. Hence, coercion must be based on mathematical rigour. In our example, any element of P_2 can be naturally interpreted as an element of P_1. We thus have:
sage: P1.has_coerce_map_from(P2) True sage: P1.coerce_map_from(P2) Coercion map: From: Multivariate Polynomial Ring in w, v over Integer Ring To: Multivariate Polynomial Ring in v, w over Rational Field
While there is a conversion from P_1 to P_2 (namely restricted to polynomials with integral coefficients), this conversion is not a coercion:
sage: P2.convert_map_from(P1) Conversion map: From: Multivariate Polynomial Ring in v, w over Rational Field To: Multivariate Polynomial Ring in w, v over Integer Ring sage: P2.has_coerce_map_from(P1) False sage: P2.coerce_map_from(P1) is None True
A coercion is a morphism in an appropriate category.
This first axiom has two implications:
A coercion is defined on all elements of a parent.
A polynomial of degree zero over the integers can be interpreted as an integer---but the attempt to convert a polynomial of non-zero degree would result in an error:
sage: ZZ(P2.one()) 1 sage: ZZ(P2.gen(1)) Traceback (most recent call last): ... TypeError: not a constant polynomial
Hence, we only have a partial map. This is fine for a conversion, but a partial map does not qualify as a coercion.
Coercions are structure preserving.
Any real number can be converted to an integer, namely by rounding. However, such a conversion is not useful in arithmetic operations, since the underlying algebraic structure is not preserved:
sage: int(1.6)+int(2.7) == int(1.6+2.7) False
The structure that is to be preserved depends on the category of the involved parents. For example, the coercion from the integers into the rational field is a homomorphism of euclidean domains:
sage: QQ.coerce_map_from(ZZ).category_for() Join of Category of euclidean domains and Category of infinite sets and Category of metric spaces
There is at most one coercion from one parent to another
In addition, if there is a coercion from P_2 to P_1, then a conversion from P_2 to P_1 is defined for all elements of P_2 and coincides with the coercion. Nonetheless, user-exposed maps are copies of the internally used maps whence the lack of identity between different instantiations:
sage: P1.coerce_map_from(P2) is P1.convert_map_from(P2) False
For internally used maps, the maps are identical:
sage: P1._internal_coerce_map_from(P2) is P1._internal_convert_map_from(P2) True
Coercions can be composed
If there is a coercion varphi: P_1 to P_2 and another coercion psi: P_2 to P_3, then the composition of varphi followed by psi must yield the unique coercion from P_1 to P_3.
The identity is a coercion
Together with the two preceding axioms, it follows: If there are coercions from P_1 to P_2 and from P_2 to P_1, then they are mutually inverse.
We have seen above that some conversions into our fraction fields became
available after providing the attribute Element. However, we can not
convert elements of a fraction field into elements of another fraction field,
yet:
sage: P(2/3) Traceback (most recent call last): ... ValueError: Numerator and denominator must be elements of Integer Ring
For implementing a conversion, the default __call__ method should (almost) never be overridden. Instead, we implement the method _element_constructor_, that should return an instance of the parent's element class. Some old parent classes violate that rule---please help to refactor them!
sage: class MyFrac(MyFrac): ....: def _element_constructor_(self, *args, **kwds): ....: if len(args)!=1: ....: return self.element_class(self, *args, **kwds) ....: x = args[0] ....: try: ....: P = x.parent() ....: except AttributeError: ....: return self.element_class(self, x, **kwds) ....: if P in QuotientFields() and P != self.base(): ....: return self.element_class(self, x.numerator(), x.denominator(), **kwds) ....: return self.element_class(self, x, **kwds)
In addition to the conversion from the base ring and from pairs of base ring elements, we now also have a conversion from the rationals to our fraction field of ZZ:
sage: P = MyFrac(ZZ) sage: P(2); P(2,3); P(3/4) (2):(1) (2):(3) (3):(4)
Recall that above, the test 1 in P failed with an error. We try again and find that the error has disappeared. This is because we are now able to convert the integer 1 into P. But the containment test still yields a wrong answer:
sage: 1 in P False
The technical reason: We have a conversion P(1) of 1 into P, but this is not known as a coercion---yet!
sage: P.has_coerce_map_from(ZZ), P.has_coerce_map_from(QQ) (False, False)
There are two main ways to make Sage use a particular conversion as a coercion:
- One can use :meth:`sage.structure.parent.Parent.register_coercion`, normally during initialisation of the parent (see documentation of the method).
- A more flexible way is to provide a method
_coerce_map_from_for the parent.
Let P and R be parents. If P._coerce_map_from_(R) returns False
or None, then there is no coercion from R to P. If it returns a map
with domain R and codomain P, then this map is used for coercion. If it
returns True, then the conversion from R to P is used as coercion.
Note that in the following implementation, we need a special case for the
rational field, since QQ.base() is not the ring of integers.
sage: class MyFrac(MyFrac): ....: def _coerce_map_from_(self, S): ....: if self.base().has_coerce_map_from(S): ....: return True ....: if S in QuotientFields(): ....: if self.base().has_coerce_map_from(S.base()): ....: return True ....: if hasattr(S,'ring_of_integers') and self.base().has_coerce_map_from(S.ring_of_integers()): ....: return True
By the method above, a parent coercing into the base ring will also coerce into the fraction field, and a fraction field coerces into another fraction field if there is a coercion of the corresponding base rings. Now, we have:
sage: P = MyFrac(QQ['x']) sage: P.has_coerce_map_from(ZZ['x']), P.has_coerce_map_from(Frac(ZZ['x'])), P.has_coerce_map_from(QQ) (True, True, True)
We can now use coercion from ZZ[x] and from QQ into P for arithmetic operations between the two rings:
sage: 3/4+P(2)+ZZ['x'].gen(), (P(2)+ZZ['x'].gen()).parent() is P ((4*x + 11):(4), True)
Recall that above, the test 1 in P gave a wrong answer. Let us repeat the test now:
sage: 1 in P True
Why is that?
The default element containment test x in P is based on the interplay of three building blocks: conversion, coercion, and equality test.
- Clearly, if the conversion P(x) raises an error, then x can not be seen as an element of P. On the other hand, a conversion P(x) can generally do very nasty things. So, the fact that P(x) works without error is necessary, but not sufficient for x in P.
- If P is the parent of x, then the conversion P(x) will not change x (at least, that's the default). Hence, we will have x=P(x).
- Sage uses coercion not only for arithmetic operations, but also for comparison: If there is a coercion from the parent of x to P, then the equality test
x==P(x)reduces toP(x)==P(x). Otherwise,x==P(x)will evaluate as false.
That leads to the following default implementation of element containment testing:
Note
x in P holds if and only if the test x==P(x) does not
raise an error and evaluates as true.
If the user is not happy with that behaviour, the "magical" Python method
__contains__ can be overridden.
So far, we are able to add integers and rational numbers to elements of our new implementation of the fraction field of ZZ.
sage: P = MyFrac(ZZ)
sage: 1/2+P(2,3)+1 (13):(6)
Surprisingly, we can even add a polynomial over the integers to an element of P, even though the result lives in a new parent, namely in a polynomial ring over P:
sage: P(1/2) + ZZ['x'].gen(), (P(1/2) + ZZ['x'].gen()).parent() is P['x'] ((1):(1)*x + (1):(2), True)
In the next, seemingly more easy example, there "obviously" is a coercion from the fraction field of ZZ to the fraction field of ZZ[x]. However, Sage does not know enough about our new implementation of fraction fields. Hence, it does not recognise the coercion:
sage: Frac(ZZ['x']).has_coerce_map_from(P) False
Two obvious questions arise:
- How / why has the new ring been constructed in the example above?
- How can we establish a coercion from P to mathrm{Frac}(ZZ[x])?
The key to answering both question is the construction of parents from simpler pieces, that we are studying now. Note that we will answer the second question not by providing a coercion from P to mathrm{Frac}(ZZ[x]), but by teaching Sage to automatically construct mathrm{MyFrac}(ZZ[x]) and coerce both P and mathrm{Frac}(ZZ[x]) into it.
If we are lucky, a parent can tell how it has been constructed:
sage: Poly,R = QQ['x'].construction() sage: Poly,R (Poly[x], Rational Field) sage: Fract,R = QQ.construction() sage: Fract,R (FractionField, Integer Ring)
In both cases, the first value returned by :meth:`~sage.structure.parent.Parent.construction` is a mathematical construction, called construction functor---see :class:`~sage.categories.pushout.ConstructionFunctor`. The second return value is a simpler parent to which the construction functor is applied.
Being functors, the same construction can be applied to different objects of a category:
sage: Poly(QQ) is QQ['x'] True sage: Poly(ZZ) is ZZ['x'] True sage: Poly(P) is P['x'] True sage: Fract(QQ['x']) Fraction Field of Univariate Polynomial Ring in x over Rational Field
Let us see on which categories these construction functors are defined:
sage: Poly.domain() Category of rings sage: Poly.codomain() Category of rings sage: Fract.domain() Category of integral domains sage: Fract.codomain() Category of fields
In particular, the construction functors can be composed:
sage: Poly*Fract Poly[x](FractionField(...)) sage: (Poly*Fract)(ZZ) is QQ['x'] True
In addition, it is often assumed that we have a coercion from input to output of the construction functor:
sage: ((Poly*Fract)(ZZ)).coerce_map_from(ZZ)
Composite map:
From: Integer Ring
To: Univariate Polynomial Ring in x over Rational Field
Defn: Natural morphism:
From: Integer Ring
To: Rational Field
then
Polynomial base injection morphism:
From: Rational Field
To: Univariate Polynomial Ring in x over Rational Field
Construction functors do not necessarily commute:
sage: (Fract*Poly)(ZZ) Fraction Field of Univariate Polynomial Ring in x over Integer Ring
We can now formulate our problem. We have parents P_1, P_2 and R, and construction functors F_1, F_2, such that P_1 = F_1(R) and P_2 = F_2(R). We want to find a new construction functor F_3, such that both P_1 and P_2 coerce into P_3 = F_3(R).
In analogy to a notion of category theory, P_3 is called the :func:`~sage.categories.pushout.pushout` of P_1 and P_2; and similarly F_3 is called the pushout of F_1 and F_2.
sage: from sage.categories.pushout import pushout sage: pushout(Fract(ZZ),Poly(ZZ)) Univariate Polynomial Ring in x over Rational Field
F_1circ F_2 and F_2circ F_1 are natural candidates for the pushout of F_1 and F_2. However, the order of the functors must rely on a canonical choice. "Indecomposable" construction functors have a rank, and this allows to order them canonically:
Note
If F1.rank is smaller than F2.rank, then the pushout is F_2circ
F_1 (hence, F_1 is applied first).
We have
sage: Fract.rank, Poly.rank (5, 9)
and thus the pushout is
sage: Fract.pushout(Poly), Poly.pushout(Fract) (Poly[x](FractionField(...)), Poly[x](FractionField(...)))
This is why the example above has worked.
However, only "elementary" construction functors have a rank:
sage: (Fract*Poly).rank Traceback (most recent call last): ... AttributeError: 'CompositeConstructionFunctor' object has no attribute 'rank'
If composed construction fuctors ...circ F_2circ F_1 and ...circ G_2circ G_1 are given, then Sage determines their pushout by shuffling the constituents:
- If
F1.rank < G1.rankthen we apply F_1 first, and continue with ...circ F_3circ F_2 and ...circ G_2circ G_1. - If
F1.rank > G1.rankthen we apply G_1 first, and continue with ...circ F_2circ F_1 and ...circ G_3circ G_2.
If F1.rank == G1.rank, then the tie needs to be broken by other techniques
(see below).
As an illustration, we first get us some functors and then see how chains of functors are shuffled.
sage: AlgClos, R = CC.construction(); AlgClos AlgebraicClosureFunctor
sage: Compl, R = RR.construction(); Compl Completion[+Infinity, prec=53]
sage: Matr, R = (MatrixSpace(ZZ,3)).construction(); Matr MatrixFunctor
sage: AlgClos.rank, Compl.rank, Fract.rank, Poly.rank, Matr.rank (3, 4, 5, 9, 10)
When we apply Fract, AlgClos, Poly and Fract to the ring of
integers, we obtain:
sage: (Fract*Poly*AlgClos*Fract)(ZZ) Fraction Field of Univariate Polynomial Ring in x over Algebraic Field
When we apply Compl, Matr and Poly to the ring of integers, we
obtain:
sage: (Poly*Matr*Compl)(ZZ) Univariate Polynomial Ring in x over Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
Applying the shuffling procedure yields
sage: (Poly*Matr*Fract*Poly*AlgClos*Fract*Compl)(ZZ) Univariate Polynomial Ring in x over Full MatrixSpace of 3 by 3 dense matrices over Fraction Field of Univariate Polynomial Ring in x over Algebraic Field
and this is indeed equal to the pushout found by Sage:
sage: pushout((Fract*Poly*AlgClos*Fract)(ZZ), (Poly*Matr*Compl)(ZZ)) Univariate Polynomial Ring in x over Full MatrixSpace of 3 by 3 dense matrices over Fraction Field of Univariate Polynomial Ring in x over Algebraic Field
If F1.rank==G1.rank then Sage's pushout constructions offers two ways to
proceed:
- Construction functors have a method :meth:`~sage.categories.pushout.ConstructionFunctor.merge` that either returns
Noneor returns a construction functor---see below. If eitherF1.merge(G1)orG1.merge(F1)returns a construction functor H_1, then we apply H_1 and continue with ...circ F_3circ F_2 and ...circ G_3circ G_2. - Construction functors have a method :meth:`~sage.categories.pushout.ConstructionFunctor.commutes`. If either
F1.commutes(G1)orG1.commutes(F1)returnsTrue, then we apply both F_1 and G_1 in any order, and continue with ...circ F_3circ F_2 and ...circ G_3circ G_2.
By default, F1.merge(G1) returns F1 if F1==G1, and returns
None otherwise. The commutes() method exists, but it seems that so far
nobody has implemented two functors of the same rank that commute.
The typical application of :meth:`~sage.categories.pushout.ConstructionFunctor.merge` is to provide a coercion between different implementations of the same algebraic structure.
Note
If F1(P) and F2(P) are different implementations of the same thing, then F1.merge(F2)(P) should return the default implementation.
We want to boldly turn our toy implementation of fraction fields into the new default implementation. Hence:
- Next, we implement a new version of the "usual" fraction field functor, having the same rank, but returning our new implementation.
- We make our new implementation the default, by virtue of a merge method.
Warning
- Do not override the default
__call__method of :class:`~sage.categories.pushout.ConstructionFunctor`---implement_apply_functorinstead. - Declare domain and codomain of the functor during initialisation.
sage: from sage.categories.pushout import ConstructionFunctor sage: class MyFracFunctor(ConstructionFunctor): ....: rank = 5 ....: def __init__(self): ....: ConstructionFunctor.__init__(self, IntegralDomains(), Fields()) ....: def _apply_functor(self, R): ....: return MyFrac(R) ....: def merge(self, other): ....: if isinstance(other, (type(self), sage.categories.pushout.FractionField)): ....: return self
sage: MyFracFunctor() MyFracFunctor
We verify that our functor can really be used to construct our implementation of fraction fields, and that it can be merged with either itself or the usual fraction field constructor:
sage: MyFracFunctor()(ZZ) NewFrac(Integer Ring)
sage: MyFracFunctor().merge(MyFracFunctor()) MyFracFunctor
sage: MyFracFunctor().merge(Fract) MyFracFunctor
There remains to let our new fraction fields know about the new construction functor:
sage: class MyFrac(MyFrac): ....: def construction(self): ....: return MyFracFunctor(), self.base()
sage: MyFrac(ZZ['x']).construction() (MyFracFunctor, Univariate Polynomial Ring in x over Integer Ring)
Due to merging, we have:
sage: pushout(MyFrac(ZZ['x']), Frac(QQ['x'])) NewFrac(Univariate Polynomial Ring in x over Rational Field)
Being able to do arithmetics involving elements of different parents, with the automatic creation of a pushout to contain the result, is certainly convenient---but one should not rely on it, if speed matters. Simply the conversion of elements into different parents takes time. Moreover, by :trac:`14058`, the pushout may be subject to Python's cyclic garbage collection. Hence, if one does not keep a strong reference to it, the same parent may be created repeatedly, which is a waste of time. In the following example, we illustrate the slow--down resulting from blindly relying on coercion:
sage: ZZxy = ZZ['x','y']
sage: a = ZZxy('x')
sage: b = 1/2
sage: timeit("c = a+b") # random
10000 loops, best of 3: 172 us per loop
sage: QQxy = QQ['x','y']
sage: timeit("c2 = QQxy(a)+QQxy(b)") # random
10000 loops, best of 3: 168 us per loop
sage: a2 = QQxy(a)
sage: b2 = QQxy(b)
sage: timeit("c2 = a2+b2") # random
100000 loops, best of 3: 10.5 us per loop
Hence, if one avoids the explicit or implicit conversion into the pushout, but works in the pushout right away, one can get a more than 10--fold speed--up.
The category framework does not only provide functionality but also a test framework. This section logically belongs to the section on categories, but without the bits that we have implemented in the section on coercion, our implementation of fraction fields would not have passed the tests yet.
We have already seen above that a category can require/suggest certain parent or element methods, that the user must/should implement. This is in order to smoothly blend with the methods that already exist in Sage.
The methods that ought to be provided are called :func:`~sage.misc.abstract_method.abstract_method`. Let us see what methods are needed for quotient fields and their elements:
sage: from sage.misc.abstract_method import abstract_methods_of_class
sage: abstract_methods_of_class(QuotientFields().parent_class)['optional'] [] sage: abstract_methods_of_class(QuotientFields().parent_class)['required'] ['__contains__']
Hence, the only required method (that is actually required for all parents that belong to the category of sets) is an element containment test. That's fine, because the base class :class:`~sage.structure.parent.Parent` provides a default containment test.
The elements have to provide more:
sage: abstract_methods_of_class(QuotientFields().element_class)['optional'] ['_add_', '_mul_'] sage: abstract_methods_of_class(QuotientFields().element_class)['required'] # py2 ['__nonzero__', 'denominator', 'numerator'] sage: abstract_methods_of_class(QuotientFields().element_class)['required'] # py3 ['__bool__', 'denominator', 'numerator']
Hence, the elements must provide denominator() and numerator()
methods, and must be able to tell whether they are zero or not. The base class
:class:`~sage.structure.element.Element` provides a default __nonzero__()
method. In addition, the elements may provide Sage's single underscore
arithmetic methods (actually any ring element should provide them).
If a parent or element method's name start with "_test_", it gives rise to a test in the automatic test suite. For example, it is tested
- whether a parent P actually is an instance of the parent class of the category of P,
- whether the user has implemented the required abstract methods,
- whether some defining structural properties (e.g., commutativity) hold.
For example, if one forgets to implement required methods, one obtains the following error:
sage: class Foo(Parent): ....: Element = sage.structure.element.Element ....: def __init__(self): ....: Parent.__init__(self, category=QuotientFields()) sage: Bar = Foo() sage: bar = Bar.element_class(Bar) sage: bar._test_not_implemented_methods() Traceback (most recent call last): ... AssertionError: Not implemented method: denominator
Here are the tests that form the test suite of quotient fields:
sage: [t for t in dir(QuotientFields().parent_class) if t.startswith('_test_')]
['_test_additive_associativity',
'_test_an_element',
'_test_associativity',
'_test_cardinality',
'_test_characteristic',
'_test_characteristic_fields',
'_test_distributivity',
'_test_divides',
'_test_elements',
'_test_elements_eq_reflexive',
'_test_elements_eq_symmetric',
'_test_elements_eq_transitive',
'_test_elements_neq',
'_test_euclidean_degree',
'_test_fraction_field',
'_test_gcd_vs_xgcd',
'_test_one',
'_test_prod',
'_test_quo_rem',
'_test_some_elements',
'_test_zero',
'_test_zero_divisors']
We have implemented all abstract methods (or inherit them from base classes), we use the category framework, and we have implemented coercions. So, we are confident that the test suite runs without an error. In fact, it does!
Note
The following trick with the __main__ module is only needed in
doctests, not in an interactive session or when defining the classes
externally.
sage: __main__.MyFrac = MyFrac sage: __main__.MyElement = MyElement sage: P = MyFrac(ZZ['x']) sage: TestSuite(P).run()
Let us see what tests are actually performed:
sage: TestSuite(P).run(verbose=True) running ._test_additive_associativity() . . . pass running ._test_an_element() . . . pass running ._test_associativity() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_characteristic() . . . pass running ._test_characteristic_fields() . . . pass running ._test_distributivity() . . . pass running ._test_divides() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_nonzero_equal() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_eq() . . . pass running ._test_euclidean_degree() . . . pass running ._test_fraction_field() . . . pass running ._test_gcd_vs_xgcd() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_one() . . . pass running ._test_pickling() . . . pass running ._test_prod() . . . pass running ._test_quo_rem() . . . pass running ._test_some_elements() . . . pass running ._test_zero() . . . pass running ._test_zero_divisors() . . . pass
As one can see, tests are also performed on elements. There are methods that return one element or a list of some elements, relying on "typical" elements that can be found in most algebraic structures.
sage: P.an_element(); P.some_elements() (2):(1) [(2):(1)]
Unfortunately, the list of elements that is returned by the default method is
of length one, and that single element could also be a bit more interesting.
The method an_element relies on a method _an_element_(), so, we implement
that. We also override the some_elements method.
sage: class MyFrac(MyFrac): ....: def _an_element_(self): ....: a = self.base().an_element() ....: b = self.base_ring().an_element() ....: if (a+b)!=0: ....: return self(a)**2/(self(a+b)**3) ....: if b != 0: ....: return self(a)/self(b)**2 ....: return self(a)**2*self(b)**3 ....: def some_elements(self): ....: return [self.an_element(),self(self.base().an_element()),self(self.base_ring().an_element())]
sage: P = MyFrac(ZZ['x']) sage: P.an_element(); P.some_elements() (x^2):(x^3 + 3*x^2 + 3*x + 1) [(x^2):(x^3 + 3*x^2 + 3*x + 1), (x):(1), (1):(1)]
Now, as we have more interesting elements, we may also add a test for the "factor" method. Recall that the method was inherited from the category, but it appears that it is not tested.
Normally, a test for a method defined by a category should be provided by the
same category. Hence, since factor is defined in the category of quotient
fields, a test should be added there. But we won't change source code here and
will instead create a sub--category.
Apparently, If e is an element of a quotient field, the product of the
factors returned by e.factor() should be equal to e. For forming the
product, we use the prod method, that, no surprise, is inherited from
another category:
sage: P.prod.__module__ 'sage.categories.monoids'
When we want to create a sub--category, we need to provide a method :meth:`~sage.categories.category.Category.super_categories`, that returns a list of all immediate super categories (here: category of quotient fields).
Warning
A sub--category S of a category C is not implemented as a
sub--class of C.__class__! S becomes a sub--category of C only
if S.super_categories() returns (a sub--category of) C!
The parent and element methods of a category are provided as methods of
classes that are the attributes ParentMethods and Element Methods of
the category, as follows:
sage: from sage.categories.category import Category sage: class QuotientFieldsWithTest(Category): # do *not* inherit from QuotientFields, but ... ....: def super_categories(self): ....: return [QuotientFields()] # ... declare QuotientFields as a super category! ....: class ParentMethods: ....: pass ....: class ElementMethods: ....: def _test_factorisation(self, **options): ....: P = self.parent() ....: assert self == P.prod([P(b)**e for b,e in self.factor()])
We provide an instance of our quotient field implementation with that new
category. Note that categories have a default _repr_ method, that guesses
a good string representation from the name of the class:
QuotientFieldsWithTest becomes "quotient fields with test".
Note
The following trick with the __main__ module is only needed in
doctests, not in an interactive session or when defining the classes
externally.
sage: __main__.MyFrac = MyFrac sage: __main__.MyElement = MyElement sage: __main__.QuotientFieldsWithTest = QuotientFieldsWithTest sage: P = MyFrac(ZZ['x'], category=QuotientFieldsWithTest()) sage: P.category() Category of quotient fields with test
The new test is inherited from the category. Since an_element() is returning a
complicated element, _test_factorisation is a serious test:
sage: P.an_element()._test_factorisation <bound method MyFrac_with_category.element_class._test_factorisation of (x^2):(x^3 + 3*x^2 + 3*x + 1)>
sage: P.an_element().factor() (x + 1)^-3 * x^2
Last, we observe that the new test has automatically become part of the test suite. We remark that the existing tests became more serious as well, since we made :meth:`sage.structure.parent.Parent.an_element` return something more interesting.
sage: TestSuite(P).run(verbose=True) running ._test_additive_associativity() . . . pass running ._test_an_element() . . . pass running ._test_associativity() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_characteristic() . . . pass running ._test_characteristic_fields() . . . pass running ._test_distributivity() . . . pass running ._test_divides() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_factorisation() . . . pass running ._test_new() . . . pass running ._test_nonzero_equal() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_eq() . . . pass running ._test_euclidean_degree() . . . pass running ._test_fraction_field() . . . pass running ._test_gcd_vs_xgcd() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_one() . . . pass running ._test_pickling() . . . pass running ._test_prod() . . . pass running ._test_quo_rem() . . . pass running ._test_some_elements() . . . pass running ._test_zero() . . . pass running ._test_zero_divisors() . . . pass
# Importing base classes, ...
import sage
from sage.rings.ring import Field
from sage.structure.element import FieldElement
from sage.categories.category import Category
# ... the UniqueRepresentation tool,
from sage.structure.unique_representation import UniqueRepresentation
# ... some categories, and ...
from sage.categories.fields import Fields
from sage.categories.quotient_fields import QuotientFields
from sage.categories.integral_domains import IntegralDomains
# construction functors
from sage.categories.pushout import ConstructionFunctor
# Fraction field elements
class MyElement(FieldElement):
def __init__(self, parent, n, d=None):
if parent is None:
raise ValueError("The parent must be provided")
B = parent.base()
if d is None:
# The default denominator is one
d = B.one()
# verify that both numerator and denominator belong to the base
if n not in B or d not in B:
raise ValueError("Numerator and denominator must be elements of %s"%B)
# Numerator and denominator should not just be "in" B,
# but should be defined as elements of B
d = B(d)
n = B(n)
# the denominator must not be zero
if d==0:
raise ZeroDivisionError("The denominator must not be zero")
# normalize the denominator: WLOG, it shall be non-negative.
if d<0:
self.n = -n
self.d = -d
else:
self.n = n
self.d = d
FieldElement.__init__(self,parent)
# Methods required by the category of fraction fields:
def numerator(self):
return self.n
def denominator(self):
return self.d
# String representation (single underscore!)
def _repr_(self):
return "(%s):(%s)"%(self.n,self.d)
# Comparison: We can assume that both arguments are coerced
# into the same parent, which is a fraction field. Hence, we
# are allowed to use the denominator() and numerator() methods
# on the second argument.
def _richcmp_(self, other, op):
from sage.structure.richcmp import richcmp
return richcmp(self.n*other.denominator(), other.numerator()*self.d, op)
# Arithmetic methods, single underscore. We can assume that both
# arguments are coerced into the same parent.
# We return instances of self.__class__, because self.__class__ will
# eventually be a sub-class of MyElement.
def _add_(self, other):
C = self.__class__
D = self.d*other.denominator()
return C(self.parent(), self.n*other.denominator()+self.d*other.numerator(),D)
def _sub_(self, other):
C = self.__class__
D = self.d*other.denominator()
return C(self.parent(), self.n*other.denominator()-self.d*other.numerator(),D)
def _mul_(self, other):
C = self.__class__
return C(self.parent(), self.n*other.numerator(), self.d*other.denominator())
def _div_(self, other):
C = self.__class__
return C(self.parent(), self.n*other.denominator(), self.d*other.numerator())
# Inheritance from UniqueRepresentation implements the unique parent
# behaviour. Moreover, it implements pickling (provided that Python
# succeeds to look up the class definition).
class MyFrac(UniqueRepresentation, Field):
# Implement the category framework for elements, which also
# makes some basic conversions work.
Element = MyElement
# Allow to pass to a different category, by an optional argument
def __init__(self, base, category=None):
# Fraction fields only exist for integral domains
if base not in IntegralDomains():
raise ValueError("%s is no integral domain" % base)
# Implement the category framework for the parent
Field.__init__(self, base, category=category or QuotientFields())
# Single-underscore method for string representation
def _repr_(self):
return "NewFrac(%s)"%repr(self.base())
# Two methods that are implicitly used in some tests
def base_ring(self):
return self.base().base_ring()
def characteristic(self):
return self.base().characteristic()
# Implement conversions. Do not override __call__!
def _element_constructor_(self, *args, **kwds):
if len(args)!=1:
return self.element_class(self, *args, **kwds)
x = args[0]
try:
P = x.parent()
except AttributeError:
return self.element_class(self, x, **kwds)
if P in QuotientFields() and P != self.base():
return self.element_class(self, x.numerator(), x.denominator(), **kwds)
return self.element_class(self, x, **kwds)
# Implement coercion from the base and from fraction fields
# over a ring that coerces into the base
def _coerce_map_from_(self, S):
if self.base().has_coerce_map_from(S):
return True
if S in QuotientFields():
if self.base().has_coerce_map_from(S.base()):
return True
if hasattr(S,'ring_of_integers') and self.base().has_coerce_map_from(S.ring_of_integers()):
return True
# Tell how this parent was constructed, in order to enable pushout constructions
def construction(self):
return MyFracFunctor(), self.base()
# return some elements of this parent
def _an_element_(self):
a = self.base().an_element()
b = self.base_ring().an_element()
if (a+b)!=0:
return self(a)**2/(self(a+b)**3)
if b != 0:
return self(a)/self(b)**2
return self(a)**2*self(b)**3
def some_elements(self):
return [self.an_element(),self(self.base().an_element()),self(self.base_ring().an_element())]
# A construction functor for our implementation of fraction fields
class MyFracFunctor(ConstructionFunctor):
# The rank is the same for Sage's original fraction field functor
rank = 5
def __init__(self):
# The fraction field construction is a functor
# from the category of integral domains into the category of
# fields
# NOTE: We could actually narrow the codomain and use the
# category QuotientFields()
ConstructionFunctor.__init__(self, IntegralDomains(), Fields())
# Applying the functor to an object. Do not override __call__!
def _apply_functor(self, R):
return MyFrac(R)
# Note: To apply the functor to morphisms, implement
# _apply_functor_to_morphism
# Make sure that arithmetic involving elements of Frac(R) and
# MyFrac(R) works and yields elements of MyFrac(R)
def merge(self, other):
if isinstance(other, (type(self), sage.categories.pushout.FractionField)):
return self
# A quotient field category with additional tests.
# Notes:
# - Category inherits from UniqueRepresentation. Hence, there
# is only one category for given arguments.
# - Since QuotientFieldsWithTest is a singleton (there is only
# one instance of this class), we could inherit from
# sage.categories.category_singleton.Category_singleton
# rather than from sage.categories.category.Category
class QuotientFieldsWithTest(Category):
# Our category is a sub-category of the category of quotient fields,
# by means of the following method.
def super_categories(self):
return [QuotientFields()]
# Here, we could implement methods that are available for
# all objects in this category.
class ParentMethods:
pass
# Here, we add a new test that is available for all elements
# of any object in this category.
class ElementMethods:
def _test_factorisation(self, **options):
P = self.parent()
# The methods prod() and factor() are inherited from
# some other categories.
assert self == P.prod([P(b)**e for b,e in self.factor()])