Sets¶
-
exception
sage.categories.sets_cat.
EmptySetError
¶ Bases:
exceptions.ValueError
Exception raised when some operation can’t be performed on the empty set.
EXAMPLES:
sage: def first_element(st): ....: if not st: raise EmptySetError("no elements") ....: else: return st[0] sage: first_element(Set((1,2,3))) 1 sage: first_element(Set([])) Traceback (most recent call last): ... EmptySetError: no elements
-
class
sage.categories.sets_cat.
Sets
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
The category of sets.
The base category for collections of elements with = (equality).
This is also the category whose objects are all parents.
EXAMPLES:
sage: Sets() Category of sets sage: Sets().super_categories() [Category of sets with partial maps] sage: Sets().all_super_categories() [Category of sets, Category of sets with partial maps, Category of objects]
Let us consider an example of set:
sage: P = Sets().example("inherits") sage: P Set of prime numbers
See
P??
for the code.P is in the category of sets:
sage: P.category() Category of sets
and therefore gets its methods from the following classes:
sage: for cl in P.__class__.mro(): print(cl) <class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits_with_category'> <class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits'> <class 'sage.categories.examples.sets_cat.PrimeNumbers_Abstract'> <class 'sage.structure.unique_representation.UniqueRepresentation'> <class 'sage.structure.unique_representation.CachedRepresentation'> <type 'sage.misc.fast_methods.WithEqualityById'> <type 'sage.structure.parent.Parent'> <type 'sage.structure.category_object.CategoryObject'> <type 'sage.structure.sage_object.SageObject'> <class 'sage.categories.sets_cat.Sets.parent_class'> <class 'sage.categories.sets_with_partial_maps.SetsWithPartialMaps.parent_class'> <class 'sage.categories.objects.Objects.parent_class'> <type 'object'>
We run some generic checks on P:
sage: TestSuite(P).run(verbose=True) running ._test_an_element() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . 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_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass
Now, we manipulate some elements of P:
sage: P.an_element() 47 sage: x = P(3) sage: x.parent() Set of prime numbers sage: x in P, 4 in P (True, False) sage: x.is_prime() True
They get their methods from the following classes:
sage: for cl in x.__class__.mro(): print(cl) <class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits_with_category.element_class'> <class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits.Element'> <type 'sage.rings.integer.IntegerWrapper'> <type 'sage.rings.integer.Integer'> <type 'sage.structure.element.EuclideanDomainElement'> <type 'sage.structure.element.PrincipalIdealDomainElement'> <type 'sage.structure.element.DedekindDomainElement'> <type 'sage.structure.element.IntegralDomainElement'> <type 'sage.structure.element.CommutativeRingElement'> <type 'sage.structure.element.RingElement'> <type 'sage.structure.element.ModuleElement'> <class 'sage.categories.examples.sets_cat.PrimeNumbers_Abstract.Element'> <type 'sage.structure.element.Element'> <type 'sage.structure.sage_object.SageObject'> <class 'sage.categories.sets_cat.Sets.element_class'> <class 'sage.categories.sets_with_partial_maps.SetsWithPartialMaps.element_class'> <class 'sage.categories.objects.Objects.element_class'> <type 'object'>
FIXME: Objects.element_class is not very meaningful ...
TESTS:
sage: TestSuite(Sets()).run()
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class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ EXAMPLES:
sage: Sets().Algebras(ZZ).super_categories() [Category of modules with basis over Integer Ring] sage: Sets().Algebras(QQ).extra_super_categories() [Category of vector spaces with basis over Rational Field] sage: Sets().example().algebra(ZZ).categories() [Category of set algebras over Integer Ring, Category of modules with basis over Integer Ring, ... Category of objects]
-
-
class
Sets.
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
EXAMPLES:
sage: C = Sets().CartesianProducts().example() sage: C The Cartesian product of (Set of prime numbers (basic implementation), An example of an infinite enumerated set: the non negative integers, An example of a finite enumerated set: {1,2,3}) sage: C.category() Category of Cartesian products of sets sage: C.categories() [Category of Cartesian products of sets, Category of sets, Category of sets with partial maps, Category of objects] sage: TestSuite(C).run()
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class
ElementMethods
¶ -
cartesian_factors
()¶ Return the Cartesian factors of
self
.EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F" sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G" sage: H = CombinatorialFreeModule(ZZ, [4,7]); H.__custom_name = "H" sage: S = cartesian_product([F, G, H]) sage: x = S.monomial((0,4)) + 2 * S.monomial((0,5)) + 3 * S.monomial((1,6)) + 4 * S.monomial((2,4)) + 5 * S.monomial((2,7)) sage: x.cartesian_factors() (B[4] + 2*B[5], 3*B[6], 4*B[4] + 5*B[7]) sage: [s.parent() for s in x.cartesian_factors()] [F, G, H] sage: S.zero().cartesian_factors() (0, 0, 0) sage: [s.parent() for s in S.zero().cartesian_factors()] [F, G, H]
-
cartesian_projection
(i)¶ Return the projection of
self
onto the \(i\)-th factor of the Cartesian product.INPUT:
i
– the index of a factor of the Cartesian product
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.__custom_name = "F" sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.__custom_name = "G" sage: S = cartesian_product([F, G]) sage: x = S.monomial((0,4)) + 2 * S.monomial((0,5)) + 3 * S.monomial((1,6)) sage: x.cartesian_projection(0) B[4] + 2*B[5] sage: x.cartesian_projection(1) 3*B[6]
-
summand_projection
(*args, **kwds)¶ Deprecated: Use
cartesian_projection()
instead. See trac ticket #10963 for details.
-
summand_split
(*args, **kwds)¶ Deprecated: Use
cartesian_factors()
instead. See trac ticket #10963 for details.
-
-
class
Sets.CartesianProducts.
ParentMethods
¶ -
an_element
()¶ EXAMPLES:
sage: C = Sets().CartesianProducts().example(); C The Cartesian product of (Set of prime numbers (basic implementation), An example of an infinite enumerated set: the non negative integers, An example of a finite enumerated set: {1,2,3}) sage: C.an_element() (47, 42, 1)
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cardinality
()¶ Return the cardinality of self.
EXAMPLES:
sage: E = FiniteEnumeratedSet([1,2,3]) sage: C = cartesian_product([E,SymmetricGroup(4)]) sage: C.cardinality() 72 sage: E = FiniteEnumeratedSet([]) sage: C = cartesian_product([E, ZZ, QQ]) sage: C.cardinality() 0 sage: C = cartesian_product([ZZ, QQ]) sage: C.cardinality() +Infinity sage: cartesian_product([GF(5), Permutations(10)]).cardinality() 18144000 sage: cartesian_product([GF(71)]*20).cardinality() == 71**20 True
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cartesian_factors
()¶ Return the Cartesian factors of
self
.EXAMPLES:
sage: cartesian_product([QQ, ZZ, ZZ]).cartesian_factors() (Rational Field, Integer Ring, Integer Ring)
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cartesian_projection
(i)¶ Return the natural projection onto the \(i\)-th Cartesian factor of
self
.INPUT:
i
– the index of a Cartesian factor ofself
EXAMPLES:
sage: C = Sets().CartesianProducts().example(); C The Cartesian product of (Set of prime numbers (basic implementation), An example of an infinite enumerated set: the non negative integers, An example of a finite enumerated set: {1,2,3}) sage: x = C.an_element(); x (47, 42, 1) sage: pi = C.cartesian_projection(1) sage: pi(x) 42
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is_empty
()¶ Return whether this set is empty.
EXAMPLES:
sage: S1 = FiniteEnumeratedSet([1,2,3]) sage: S2 = Set([]) sage: cartesian_product([S1,ZZ]).is_empty() False sage: cartesian_product([S1,S2,S1]).is_empty() True
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is_finite
()¶ Return whether this set is finite.
EXAMPLES:
sage: E = FiniteEnumeratedSet([1,2,3]) sage: C = cartesian_product([E, SymmetricGroup(4)]) sage: C.is_finite() True sage: cartesian_product([ZZ,ZZ]).is_finite() False sage: cartesian_product([ZZ, Set(), ZZ]).is_finite() True
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random_element
(*args)¶ Return a random element of this Cartesian product.
The extra arguments are passed down to each of the factors of the Cartesian product.
EXAMPLES:
sage: C = cartesian_product([Permutations(10)]*5) sage: C.random_element() # random ([2, 9, 4, 7, 1, 8, 6, 10, 5, 3], [8, 6, 5, 7, 1, 4, 9, 3, 10, 2], [5, 10, 3, 8, 2, 9, 1, 4, 7, 6], [9, 6, 10, 3, 2, 1, 5, 8, 7, 4], [8, 5, 2, 9, 10, 3, 7, 1, 4, 6]) sage: C = cartesian_product([ZZ]*10) sage: c1 = C.random_element() sage: c1 # random (3, 1, 4, 1, 1, -3, 0, -4, -17, 2) sage: c2 = C.random_element(4,7) sage: c2 # random (6, 5, 6, 4, 5, 6, 6, 4, 5, 5) sage: all(4 <= i < 7 for i in c2) True
-
-
Sets.CartesianProducts.
example
()¶ EXAMPLES:
sage: Sets().CartesianProducts().example() The Cartesian product of (Set of prime numbers (basic implementation), An example of an infinite enumerated set: the non negative integers, An example of a finite enumerated set: {1,2,3})
-
Sets.CartesianProducts.
extra_super_categories
()¶ A Cartesian product of sets is a set.
EXAMPLES:
sage: Sets().CartesianProducts().extra_super_categories() [Category of sets] sage: Sets().CartesianProducts().super_categories() [Category of sets]
-
class
-
class
Sets.
ElementMethods
¶ -
cartesian_product
(*elements)¶ Return the Cartesian product of its arguments, as an element of the Cartesian product of the parents of those elements.
EXAMPLES:
sage: C = AlgebrasWithBasis(QQ) sage: A = C.example() sage: (a,b,c) = A.algebra_generators() sage: a.cartesian_product(b, c) B[(0, word: a)] + B[(1, word: b)] + B[(2, word: c)]
FIXME: is this a policy that we want to enforce on all parents?
-
-
Sets.
Facade
¶ alias of
FacadeSets
-
Sets.
Finite
¶ alias of
FiniteSets
-
class
Sets.
Infinite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
TESTS:
sage: C = Sets.Finite(); C Category of finite sets sage: type(C) <class 'sage.categories.finite_sets.FiniteSets_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_singleton'> sage: TestSuite(C).run()
-
class
ParentMethods
¶ -
cardinality
()¶ Count the elements of the enumerated set.
EXAMPLES:
sage: NN = InfiniteEnumeratedSets().example() sage: NN.cardinality() +Infinity
-
is_empty
()¶ Return whether this set is empty.
Since this set is infinite this always returns
False
.EXAMPLES:
sage: C = InfiniteEnumeratedSets().example() sage: C.is_empty() False
-
is_finite
()¶ Return whether this set is finite.
Since this set is infinite this always returns
False
.EXAMPLES:
sage: C = InfiniteEnumeratedSets().example() sage: C.is_finite() False
TESTS:
sage: from six import get_method_function as gmf sage: gmf(C.is_finite) is gmf(sage.categories.sets_cat.Sets.Infinite.ParentMethods.is_finite) True
-
-
class
-
class
Sets.
IsomorphicObjects
(category, *args)¶ Bases:
sage.categories.isomorphic_objects.IsomorphicObjectsCategory
A category for isomorphic objects of sets.
EXAMPLES:
sage: Sets().IsomorphicObjects() Category of isomorphic objects of sets sage: Sets().IsomorphicObjects().all_super_categories() [Category of isomorphic objects of sets, Category of subobjects of sets, Category of quotients of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
-
class
ParentMethods
¶
-
class
-
Sets.
Metric
¶ alias of
MetricSpaces
-
class
Sets.
MorphismMethods
¶
-
class
Sets.
ParentMethods
¶ -
CartesianProduct
¶ alias of
CartesianProduct
-
algebra
(base_ring, category=None)¶ Return the algebra of
self
overbase_ring
.INPUT:
self
– a parent \(S\)base_ring
– a ring \(K\)category
– a super category of the category of \(S\), orNone
This returns the \(K\)-free module with basis indexed by \(S\), endowed with whatever structure can be induced from that of \(S\). Note that the
category
keyword needs to be fed with the structure on \(S\) to be used, not the structure that one wants to obtain on the result; see the examples below.EXAMPLES:
If \(S\) is a monoid, the result is the monoid algebra \(KS\):
sage: S = Monoids().example(); S An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') sage: A = S.algebra(QQ); A Free module generated by An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') over Rational Field sage: A.category() Category of monoid algebras over Rational Field
If \(S\) is a group, the result is the group algebra \(KS\):
sage: S = Groups().example(); S General Linear Group of degree 4 over Rational Field sage: A = S.algebra(QQ); A Group algebra of General Linear Group of degree 4 over Rational Field over Rational Field sage: A.category() Category of group algebras over Rational Field
which is actually a Hopf algebra:
sage: A in HopfAlgebras(QQ) True
By Maschke’s theorem, for a finite group whose cardinality does not divide the characteristic of the base field, the algebra is semisimple:
sage: SymmetricGroup(5).algebra(QQ) in Algebras(QQ).Semisimple() True sage: CyclicPermutationGroup(10).algebra(FiniteField(5)) in Algebras.Semisimple False sage: CyclicPermutationGroup(10).algebra(FiniteField(7)) in Algebras.Semisimple True
One may specify for which category one takes the algebra:
sage: A = S.algebra(QQ, category=Sets()); A Free module generated by General Linear Group of degree 4 over Rational Field over Rational Field sage: A.category() Category of set algebras over Rational Field
One may construct as well algebras of additive magmas, semigroups, monoids, or groups:
sage: S = CommutativeAdditiveMonoids().example(); S An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd') sage: U = S.algebra(QQ); U Free module generated by An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd') over Rational Field
Despite saying “free module”, this is really an algebra and its elements can be multiplied:
sage: U in Algebras(QQ) True sage: (a,b,c,d) = S.additive_semigroup_generators() sage: U(a) * U(b) B[a + b]
Constructing the algebra of a set endowed with both an additive and a multiplicative structure is ambiguous:
sage: Z3 = IntegerModRing(3) sage: A = Z3.algebra(QQ) Traceback (most recent call last): ... TypeError: `S = Ring of integers modulo 3` is both an additive and a multiplicative semigroup. Constructing its algebra is ambiguous. Please use, e.g., S.algebra(QQ, category=Semigroups())
The ambiguity can be resolved using the
category
argument:sage: A = Z3.algebra(QQ, category=Monoids()); A Free module generated by Ring of integers modulo 3 over Rational Field sage: A.category() Category of finite dimensional monoid algebras over Rational Field sage: A = Z3.algebra(QQ, category=CommutativeAdditiveGroups()); A Free module generated by Ring of integers modulo 3 over Rational Field sage: A.category() Category of finite dimensional commutative additive group algebras over Rational Field
Similarly, on , we obtain for additive magmas, monoids, groups.
Warning
As we have seen, in most practical use cases, the result is actually an algebra, hence the name of this method. In the other cases this name is misleading:
sage: A = Sets().example().algebra(QQ); A Free module generated by Set of prime numbers (basic implementation) over Rational Field sage: A.category() Category of set algebras over Rational Field sage: A in Algebras(QQ) False
Suggestions for a uniform, meaningful, and non misleading name are welcome!
-
an_element
()¶ Return a (preferably typical) element of this parent.
This is used both for illustration and testing purposes. If the set
self
is empty,an_element()
should raise the exceptionEmptySetError
.This default implementation calls
_an_element_()
and caches the result. Any parent should implement eitheran_element()
or_an_element_()
.EXAMPLES:
sage: CDF.an_element() 1.0*I sage: ZZ[['t']].an_element() t
-
cartesian_product
(*parents, **kwargs)¶ Return the Cartesian product of the parents.
INPUT:
parents
– a list (or other iterable) of parents.category
– (default:None
) the category the Cartesian product belongs to. IfNone
is passed, thencategory_from_parents()
is used to determine the category.extra_category
– (default:None
) a category that is added to the Cartesian product in addition to the categories obtained from the parents.- other keyword arguments will passed on to the class used
for this Cartesian product (see also
CartesianProduct
).
OUTPUT:
The Cartesian product.
EXAMPLES:
sage: C = AlgebrasWithBasis(QQ) sage: A = C.example(); A.rename("A") sage: A.cartesian_product(A,A) A (+) A (+) A sage: ZZ.cartesian_product(GF(2), FiniteEnumeratedSet([1,2,3])) The Cartesian product of (Integer Ring, Finite Field of size 2, {1, 2, 3}) sage: C = ZZ.cartesian_product(A); C The Cartesian product of (Integer Ring, A)
TESTS:
sage: type(C) <class 'sage.sets.cartesian_product.CartesianProduct_with_category'> sage: C.category() Join of Category of rings and ... and Category of Cartesian products of commutative additive groups
sage: cartesian_product([ZZ, ZZ], category=Sets()).category() Category of sets sage: cartesian_product([ZZ, ZZ]).category() Join of Category of Cartesian products of commutative rings and Category of Cartesian products of enumerated sets sage: cartesian_product([ZZ, ZZ], extra_category=Posets()).category() Join of Category of Cartesian products of commutative rings and Category of posets and Category of Cartesian products of enumerated sets
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is_parent_of
(element)¶ Return whether
self
is the parent ofelement
.INPUT:
element
– any object
EXAMPLES:
sage: S = ZZ sage: S.is_parent_of(1) True sage: S.is_parent_of(2/1) False
This method differs from
__contains__()
because it does not attempt any coercion:sage: 2/1 in S, S.is_parent_of(2/1) (True, False) sage: int(1) in S, S.is_parent_of(int(1)) (True, False)
-
some_elements
()¶ Return a list (or iterable) of elements of
self
.This is typically used for running generic tests (see
TestSuite
).This default implementation calls
an_element()
.EXAMPLES:
sage: S = Sets().example(); S Set of prime numbers (basic implementation) sage: S.an_element() 47 sage: S.some_elements() [47] sage: S = Set([]) sage: S.some_elements() []
This method should return an iterable, not an iterator.
-
-
class
Sets.
Quotients
(category, *args)¶ Bases:
sage.categories.quotients.QuotientsCategory
A category for quotients of sets.
See also
Sets().Quotients()
EXAMPLES:
sage: Sets().Quotients() Category of quotients of sets sage: Sets().Quotients().all_super_categories() [Category of quotients of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
-
class
ParentMethods
¶
-
class
-
class
Sets.
Realizations
(category, *args)¶ Bases:
sage.categories.realizations.RealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
-
class
ParentMethods
¶ -
realization_of
()¶ Return the parent this is a realization of.
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: In = A.In(); In The subset algebra of {1, 2, 3} over Rational Field in the In basis sage: In.realization_of() The subset algebra of {1, 2, 3} over Rational Field
-
-
class
-
class
Sets.
SubcategoryMethods
¶ -
Algebras
(base_ring)¶ Return the category of objects constructed as algebras of objects of
self
overbase_ring
.INPUT:
base_ring
– a ring
See
Sets.ParentMethods.algebra()
for the precise meaning in Sage of the algebra of an object.EXAMPLES:
sage: Monoids().Algebras(QQ) Category of monoid algebras over Rational Field sage: Groups().Algebras(QQ) Category of group algebras over Rational Field sage: AdditiveMagmas().AdditiveAssociative().Algebras(QQ) Category of additive semigroup algebras over Rational Field sage: Monoids().Algebras(Rings()) Category of monoid algebras over Category of rings
TESTS:
sage: TestSuite(Groups().Finite().Algebras(QQ)).run()
-
CartesianProducts
()¶ Return the full subcategory of the objects of
self
constructed as Cartesian products.See also
cartesian_product.CartesianProductFunctor
RegressiveCovariantFunctorialConstruction
EXAMPLES:
sage: Sets().CartesianProducts() Category of Cartesian products of sets sage: Semigroups().CartesianProducts() Category of Cartesian products of semigroups sage: EuclideanDomains().CartesianProducts() Category of Cartesian products of commutative rings
-
Facade
()¶ Return the full subcategory of the facade objects of
self
.What is a facade set?
Recall that, in Sage, sets are modelled by *parents*, and their elements know which distinguished set they belong to. For example, the ring of integers \(\ZZ\) is modelled by the parent
ZZ
, and integers know that they belong to this set:sage: ZZ Integer Ring sage: 42.parent() Integer Ring
Sometimes, it is convenient to represent the elements of a parent
P
by elements of some other parent. For example, the elements of the set of prime numbers are represented by plain integers:sage: Primes() Set of all prime numbers: 2, 3, 5, 7, ... sage: p = Primes().an_element(); p 43 sage: p.parent() Integer Ring
In this case,
P
is called a facade set.This feature is advertised through the category of \(P\):
sage: Primes().category() Category of facade infinite enumerated sets sage: Sets().Facade() Category of facade sets
Typical use cases include modeling a subset of an existing parent:
sage: Set([4,6,9]) # random {4, 6, 9} sage: Sets().Facade().example() An example of facade set: the monoid of positive integers
or the union of several parents:
sage: Sets().Facade().example("union") An example of a facade set: the integers completed by +-infinity
or endowing an existing parent with more (or less!) structure:
sage: Posets().example("facade") An example of a facade poset: the positive integers ordered by divisibility
Let us investigate in detail a close variant of this last example: let \(P\) be set of divisors of \(12\) partially ordered by divisibility. There are two options for representing its elements:
as plain integers:
sage: P = Poset((divisors(12), attrcall("divides")), facade=True)
as integers, modified to be aware that their parent is \(P\):
sage: Q = Poset((divisors(12), attrcall("divides")), facade=False)
The advantage of option 1. is that one needs not do conversions back and forth between \(P\) and \(\ZZ\). The disadvantage is that this introduces an ambiguity when writing \(2 < 3\): does this compare \(2\) and \(3\) w.r.t. the natural order on integers or w.r.t. divisibility?:
sage: 2 < 3 True
To raise this ambiguity, one needs to explicitly specify the underlying poset as in \(2 <_P 3\):
sage: P = Posets().example("facade") sage: P.lt(2,3) False
On the other hand, with option 2. and once constructed, the elements know unambiguously how to compare themselves:
sage: Q(2) < Q(3) False sage: Q(2) < Q(6) True
Beware that
P(2)
is still the integer \(2\). ThereforeP(2) < P(3)
still compares \(2\) and \(3\) as integers!:sage: P(2) < P(3) True
In short \(P\) being a facade parent is one of the programmatic counterparts (with e.g. coercions) of the usual mathematical idiom: “for ease of notation, we identify an element of \(P\) with the corresponding integer”. Too many identifications lead to confusion; the lack thereof leads to heavy, if not obfuscated, notations. Finding the right balance is an art, and even though there are common guidelines, it is ultimately up to the writer to choose which identifications to do. This is no different in code.
See also
The following examples illustrate various ways to implement subsets like the set of prime numbers; look at their code for details:
sage: Sets().example("facade") Set of prime numbers (facade implementation) sage: Sets().example("inherits") Set of prime numbers sage: Sets().example("wrapper") Set of prime numbers (wrapper implementation)
Specifications
A parent which is a facade must either:
- call
Parent.__init__()
using thefacade
parameter to specify a parent, or tuple thereof. - overload the method
facade_for()
.
Note
The concept of facade parents was originally introduced in the computer algebra system MuPAD.
TESTS:
Check that multiple categories initialisation works (trac ticket #13801):
sage: class A(Parent): ....: def __init__(self): ....: Parent.__init__(self, category=(FiniteEnumeratedSets(),Monoids()), facade=True) sage: a = A() sage: Posets().Facade() Category of facade posets sage: Posets().Facade().Finite() is Posets().Finite().Facade() True
-
Facades
(*args, **kwds)¶ Deprecated: Use
Facade()
instead. See trac ticket #17073 for details.
-
Finite
()¶ Return the full subcategory of the finite objects of
self
.EXAMPLES:
sage: Sets().Finite() Category of finite sets sage: Rings().Finite() Category of finite rings
TESTS:
sage: TestSuite(Sets().Finite()).run() sage: Rings().Finite.__module__ 'sage.categories.sets_cat'
-
Infinite
()¶ Return the full subcategory of the infinite objects of
self
.EXAMPLES:
sage: Sets().Infinite() Category of infinite sets sage: Rings().Infinite() Category of infinite rings
TESTS:
sage: TestSuite(Sets().Infinite()).run() sage: Rings().Infinite.__module__ 'sage.categories.sets_cat'
-
IsomorphicObjects
()¶ Return the full subcategory of the objects of
self
constructed by isomorphism.Given a concrete category
As()
(i.e. a subcategory ofSets()
),As().IsomorphicObjects()
returns the category of objects ofAs()
endowed with a distinguished description as the image of some other object ofAs()
by an isomorphism in this category.See
Subquotients()
for background.EXAMPLES:
In the following example, \(A\) is defined as the image by \(x\mapsto x^2\) of the finite set \(B = \{1,2,3\}\):
sage: A = FiniteEnumeratedSets().IsomorphicObjects().example(); A The image by some isomorphism of An example of a finite enumerated set: {1,2,3}
Since \(B\) is a finite enumerated set, so is \(A\):
sage: A in FiniteEnumeratedSets() True sage: A.cardinality() 3 sage: A.list() [1, 4, 9]
The isomorphism from \(B\) to \(A\) is available as:
sage: A.retract(3) 9
and its inverse as:
sage: A.lift(9) 3
It often is natural to declare those morphisms as coercions so that one can do
A(b)
andB(a)
to go back and forth between \(A\) and \(B\) (TODO: refer to a category example where the maps are declared as a coercion). This is not done by default. Indeed, in many cases one only wants to transport part of the structure of \(B\) to \(A\). Assume for example, that one wants to construct the set of integers \(B=ZZ\), endowed withmax
as addition, and+
as multiplication instead of the usual+
and*
. One can construct \(A\) as isomorphic to \(B\) as an infinite enumerated set. However \(A\) is not isomorphic to \(B\) as a ring; for example, for \(a\in A\) and \(a\in B\), the expressions \(a+A(b)\) and \(B(a)+b\) give completely different results; hence we would not want the expression \(a+b\) to be implicitly resolved to any one of above two, as the coercion mechanism would do.Coercions also cannot be used with facade parents (see
Sets.Facade
) like in the example above.We now look at a category of isomorphic objects:
sage: C = Sets().IsomorphicObjects(); C Category of isomorphic objects of sets sage: C.super_categories() [Category of subobjects of sets, Category of quotients of sets] sage: C.all_super_categories() [Category of isomorphic objects of sets, Category of subobjects of sets, Category of quotients of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
Unless something specific about isomorphic objects is implemented for this category, one actually get an optimized super category:
sage: C = Semigroups().IsomorphicObjects(); C Join of Category of quotients of semigroups and Category of isomorphic objects of sets
See also
Subquotients()
for backgroundisomorphic_objects.IsomorphicObjectsCategory
RegressiveCovariantFunctorialConstruction
TESTS:
sage: TestSuite(Sets().IsomorphicObjects()).run()
-
Metric
()¶ Return the subcategory of the metric objects of
self
.TESTS:
sage: TestSuite(Sets().Metric()).run()
-
Quotients
()¶ Return the full subcategory of the objects of
self
constructed as quotients.Given a concrete category
As()
(i.e. a subcategory ofSets()
),As().Quotients()
returns the category of objects ofAs()
endowed with a distinguished description as quotient (in fact homomorphic image) of some other object ofAs()
.Implementing an object of
As().Quotients()
is done in the same way as forAs().Subquotients()
; namely by providing an ambient space and a lift and a retract map. SeeSubquotients()
for detailed instructions.See also
Subquotients()
for backgroundquotients.QuotientsCategory
RegressiveCovariantFunctorialConstruction
EXAMPLES:
sage: C = Semigroups().Quotients(); C Category of quotients of semigroups sage: C.super_categories() [Category of subquotients of semigroups, Category of quotients of sets] sage: C.all_super_categories() [Category of quotients of semigroups, Category of subquotients of semigroups, Category of semigroups, Category of subquotients of magmas, Category of magmas, Category of quotients of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
The caller is responsible for checking that the given category admits a well defined category of quotients:
sage: EuclideanDomains().Quotients() Join of Category of euclidean domains and Category of subquotients of monoids and Category of quotients of semigroups
TESTS:
sage: TestSuite(C).run()
-
Subobjects
()¶ Return the full subcategory of the objects of
self
constructed as subobjects.Given a concrete category
As()
(i.e. a subcategory ofSets()
),As().Subobjects()
returns the category of objects ofAs()
endowed with a distinguished embedding into some other object ofAs()
.Implementing an object of
As().Subobjects()
is done in the same way as forAs().Subquotients()
; namely by providing an ambient space and a lift and a retract map. In the case of a trivial embedding, the two maps will typically be identity maps that just change the parent of their argument. SeeSubquotients()
for detailed instructions.See also
Subquotients()
for backgroundsubobjects.SubobjectsCategory
RegressiveCovariantFunctorialConstruction
EXAMPLES:
sage: C = Sets().Subobjects(); C Category of subobjects of sets sage: C.super_categories() [Category of subquotients of sets] sage: C.all_super_categories() [Category of subobjects of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
Unless something specific about subobjects is implemented for this category, one actually gets an optimized super category:
sage: C = Semigroups().Subobjects(); C Join of Category of subquotients of semigroups and Category of subobjects of sets
The caller is responsible for checking that the given category admits a well defined category of subobjects.
TESTS:
sage: Semigroups().Subobjects().is_subcategory(Semigroups().Subquotients()) True sage: TestSuite(C).run()
-
Subquotients
()¶ Return the full subcategory of the objects of
self
constructed as subquotients.Given a concrete category
self == As()
(i.e. a subcategory ofSets()
),As().Subquotients()
returns the category of objects ofAs()
endowed with a distinguished description as subquotient of some other object ofAs()
.EXAMPLES:
sage: Monoids().Subquotients() Category of subquotients of monoids
A parent \(A\) in
As()
is further inAs().Subquotients()
if there is a distinguished parent \(B\) inAs()
, called the ambient set, a subobject \(B'\) of \(B\), and a pair of maps:\[l: A \to B' \text{ and } r: B' \to A\]called respectively the lifting map and retract map such that \(r \circ l\) is the identity of \(A\) and \(r\) is a morphism in
As()
.Todo
Draw the typical commutative diagram.
It follows that, for each operation \(op\) of the category, we have some property like:
\[op_A(e) = r(op_B(l(e))), \text{ for all } e\in A\]This allows for implementing the operations on \(A\) from those on \(B\).
The two most common use cases are:
- homomorphic images (or quotients), when \(B'=B\),
\(r\) is an homomorphism from \(B\) to \(A\) (typically a
canonical quotient map), and \(l\) a section of it (not
necessarily a homomorphism); see
Quotients()
; - subobjects (up to an isomorphism), when \(l\) is an
embedding from \(A\) into \(B\); in this case, \(B'\) is
typically isomorphic to \(A\) through the inverse
isomorphisms \(r\) and \(l\); see
Subobjects()
;
Note
- The usual definition of “subquotient” (Wikipedia article Subquotient) does not involve the lifting map \(l\). This map is required in Sage’s context to make the definition constructive. It is only used in computations and does not affect their results. This is relatively harmless since the category is a concrete category (i.e., its objects are sets and its morphisms are set maps).
- In mathematics, especially in the context of quotients, the retract map \(r\) is often referred to as a projection map instead.
- Since \(B'\) is not specified explicitly, it is possible to abuse the framework with situations where \(B'\) is not quite a subobject and \(r\) not quite a morphism, as long as the lifting and retract maps can be used as above to compute all the operations in \(A\). Use at your own risk!
Assumptions:
For any category
As()
,As().Subquotients()
is a subcategory ofAs()
.Example: a subquotient of a group is a group (e.g., a left or right quotient of a group by a non-normal subgroup is not in this category).
This construction is covariant: if
As()
is a subcategory ofBs()
, thenAs().Subquotients()
is a subcategory ofBs().Subquotients()
.Example: if \(A\) is a subquotient of \(B\) in the category of groups, then it is also a subquotient of \(B\) in the category of monoids.
If the user (or a program) calls
As().Subquotients()
, then it is assumed that subquotients are well defined in this category. This is not checked, and probably never will be. Note that, if a categoryAs()
does not specify anything about its subquotients, then its subquotient category looks like this:sage: EuclideanDomains().Subquotients() Join of Category of euclidean domains and Category of subquotients of monoids
Interface: the ambient set \(B\) of \(A\) is given by
A.ambient()
. The subset \(B'\) needs not be specified, so the retract map is handled as a partial map from \(B\) to \(A\).The lifting and retract map are implemented respectively as methods
A.lift(a)
andA.retract(b)
. As a shorthand for the former, one can use alternativelya.lift()
:sage: S = Semigroups().Subquotients().example(); S An example of a (sub)quotient semigroup: a quotient of the left zero semigroup sage: S.ambient() An example of a semigroup: the left zero semigroup sage: S(3).lift().parent() An example of a semigroup: the left zero semigroup sage: S(3) * S(1) == S.retract( S(3).lift() * S(1).lift() ) True
See
S?
for more.Todo
use a more interesting example, like \(\ZZ/n\ZZ\).
See also
Quotients()
,Subobjects()
,IsomorphicObjects()
subquotients.SubquotientsCategory
RegressiveCovariantFunctorialConstruction
TESTS:
sage: TestSuite(Sets().Subquotients()).run()
- homomorphic images (or quotients), when \(B'=B\),
\(r\) is an homomorphism from \(B\) to \(A\) (typically a
canonical quotient map), and \(l\) a section of it (not
necessarily a homomorphism); see
-
Topological
()¶ Return the subcategory of the topological objects of
self
.TESTS:
sage: TestSuite(Sets().Topological()).run()
-
-
class
Sets.
Subobjects
(category, *args)¶ Bases:
sage.categories.subobjects.SubobjectsCategory
A category for subobjects of sets.
See also
Sets().Subobjects()
EXAMPLES:
sage: Sets().Subobjects() Category of subobjects of sets sage: Sets().Subobjects().all_super_categories() [Category of subobjects of sets, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
-
class
ParentMethods
¶
-
class
-
class
Sets.
Subquotients
(category, *args)¶ Bases:
sage.categories.subquotients.SubquotientsCategory
A category for subquotients of sets.
See also
Sets().Subquotients()
EXAMPLES:
sage: Sets().Subquotients() Category of subquotients of sets sage: Sets().Subquotients().all_super_categories() [Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]
-
class
ElementMethods
¶ -
lift
()¶ Lift
self
to the ambient space for its parent.EXAMPLES:
sage: S = Semigroups().Subquotients().example() sage: s = S.an_element() sage: s, s.parent() (42, An example of a (sub)quotient semigroup: a quotient of the left zero semigroup) sage: S.lift(s), S.lift(s).parent() (42, An example of a semigroup: the left zero semigroup) sage: s.lift(), s.lift().parent() (42, An example of a semigroup: the left zero semigroup)
-
-
class
Sets.Subquotients.
ParentMethods
¶ -
ambient
()¶ Return the ambient space for
self
.EXAMPLES:
sage: Semigroups().Subquotients().example().ambient() An example of a semigroup: the left zero semigroup
See also
Sets.SubcategoryMethods.Subquotients()
for the specifications andlift()
andretract()
.
-
lift
(x)¶ Lift \(x\) to the ambient space for
self
.INPUT:
x
– an element ofself
EXAMPLES:
sage: S = Semigroups().Subquotients().example() sage: s = S.an_element() sage: s, s.parent() (42, An example of a (sub)quotient semigroup: a quotient of the left zero semigroup) sage: S.lift(s), S.lift(s).parent() (42, An example of a semigroup: the left zero semigroup) sage: s.lift(), s.lift().parent() (42, An example of a semigroup: the left zero semigroup)
See also
Sets.SubcategoryMethods.Subquotients
for the specifications,ambient()
,retract()
, and alsoSets.Subquotients.ElementMethods.lift()
.
-
retract
(x)¶ Retract
x
toself
.INPUT:
x
– an element of the ambient space forself
See also
Sets.SubcategoryMethods.Subquotients
for the specifications,ambient()
,retract()
, and alsoSets.Subquotients.ElementMethods.retract()
.EXAMPLES:
sage: S = Semigroups().Subquotients().example() sage: s = S.ambient().an_element() sage: s, s.parent() (42, An example of a semigroup: the left zero semigroup) sage: S.retract(s), S.retract(s).parent() (42, An example of a (sub)quotient semigroup: a quotient of the left zero semigroup)
-
-
class
-
Sets.
Topological
¶ alias of
TopologicalSpaces
-
class
Sets.
WithRealizations
(category, *args)¶ Bases:
sage.categories.with_realizations.WithRealizationsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
-
class
ParentMethods
¶ -
class
Realizations
(parent_with_realization)¶ Bases:
sage.categories.realizations.Category_realization_of_parent
TESTS:
sage: from sage.categories.realizations import Category_realization_of_parent sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: C = A.Realizations(); C Category of realizations of The subset algebra of {1, 2, 3} over Rational Field sage: isinstance(C, Category_realization_of_parent) True sage: C.parent_with_realization The subset algebra of {1, 2, 3} over Rational Field sage: TestSuite(C).run(skip=["_test_category_over_bases"])
Todo
Fix the failing test by making
C
a singleton category. This will require some fiddling with the assertion inCategory_singleton.__classcall__()
-
super_categories
()¶ EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.Realizations().super_categories() [Category of realizations of sets]
-
-
Sets.WithRealizations.ParentMethods.
a_realization
()¶ Return a realization of
self
.EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.a_realization() The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
-
Sets.WithRealizations.ParentMethods.
facade_for
()¶ Return the parents
self
is a facade for, that is the realizations ofself
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.facade_for() [The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis, The subset algebra of {1, 2, 3} over Rational Field in the In basis, The subset algebra of {1, 2, 3} over Rational Field in the Out basis] sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: f = A.F().an_element(); f F[{}] + 2*F[{1}] + 3*F[{2}] + F[{1, 2}] sage: i = A.In().an_element(); i In[{}] + 2*In[{1}] + 3*In[{2}] + In[{1, 2}] sage: o = A.Out().an_element(); o Out[{}] + 2*Out[{1}] + 3*Out[{2}] + Out[{1, 2}] sage: f in A, i in A, o in A (True, True, True)
-
Sets.WithRealizations.ParentMethods.
inject_shorthands
(verbose=True)¶ Import standard shorthands into the global namespace.
INPUT:
verbose
– boolean (defaultTrue
) ifTrue
, prints the defined shorthands
EXAMPLES:
sage: Q = QuasiSymmetricFunctions(ZZ) sage: Q.inject_shorthands() Injecting M as shorthand for Quasisymmetric functions over the Integer Ring in the Monomial basis Injecting F as shorthand for Quasisymmetric functions over the Integer Ring in the Fundamental basis Injecting dI as shorthand for Quasisymmetric functions over the Integer Ring in the dualImmaculate basis Injecting QS as shorthand for Quasisymmetric functions over the Integer Ring in the Quasisymmetric Schur basis sage: F[1,2,1] + 5*M[1,3] + F[2]^2 5*F[1, 1, 1, 1] - 5*F[1, 1, 2] - 3*F[1, 2, 1] + 6*F[1, 3] + 2*F[2, 2] + F[3, 1] + F[4] sage: F Quasisymmetric functions over the Integer Ring in the Fundamental basis sage: M Quasisymmetric functions over the Integer Ring in the Monomial basis
-
Sets.WithRealizations.ParentMethods.
realizations
()¶ Return all the realizations of
self
thatself
is aware of.EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.realizations() [The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis, The subset algebra of {1, 2, 3} over Rational Field in the In basis, The subset algebra of {1, 2, 3} over Rational Field in the Out basis]
Note
Constructing a parent
P
in the categoryA.Realizations()
automatically addsP
to this list by callingA._register_realization(A)
-
class
-
Sets.WithRealizations.
example
(base_ring=None, set=None)¶ Return an example of set with multiple realizations, as per
Category.example()
.EXAMPLES:
sage: Sets().WithRealizations().example() The subset algebra of {1, 2, 3} over Rational Field sage: Sets().WithRealizations().example(ZZ, Set([1,2])) The subset algebra of {1, 2} over Integer Ring
-
Sets.WithRealizations.
extra_super_categories
()¶ A set with multiple realizations is a facade parent.
EXAMPLES:
sage: Sets().WithRealizations().extra_super_categories() [Category of facade sets] sage: Sets().WithRealizations().super_categories() [Category of facade sets]
-
class
-
Sets.
example
(choice=None)¶ Returns examples of objects of
Sets()
, as perCategory.example()
.EXAMPLES:
sage: Sets().example() Set of prime numbers (basic implementation) sage: Sets().example("inherits") Set of prime numbers sage: Sets().example("facade") Set of prime numbers (facade implementation) sage: Sets().example("wrapper") Set of prime numbers (wrapper implementation)
-
Sets.
super_categories
()¶ We include SetsWithPartialMaps between Sets and Objects so that we can define morphisms between sets that are only partially defined. This is also to have the Homset constructor not complain that SetsWithPartialMaps is not a supercategory of Fields, for example.
EXAMPLES:
sage: Sets().super_categories() [Category of sets with partial maps]
-
class
-
sage.categories.sets_cat.
print_compare
(x, y)¶ Helper method used in
Sets.ParentMethods._test_elements_eq_symmetric()
,Sets.ParentMethods._test_elements_eq_tranisitive()
.INPUT:
x
– an elementy
– an element
EXAMPLES:
sage: from sage.categories.sets_cat import print_compare sage: print_compare(1,2) 1 != 2 sage: print_compare(1,1) 1 == 1