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'> <... '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_new() . . . 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_new() . . . 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'> <... 'object'>
FIXME: Objects.element_class is not very meaningful …

class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory

class
ParentMethods
¶ 
construction
()¶ Return the functorial construction of
self
.EXAMPLES:
sage: A = GroupAlgebra(KleinFourGroup(), QQ) sage: A.construction() (GroupAlgebraFunctor, Rational Field)


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

class
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()

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
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)

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

cartesian_factors
()¶ Return the Cartesian factors of
self
.EXAMPLES:
sage: cartesian_product([QQ, ZZ, ZZ]).cartesian_factors() (Rational Field, Integer Ring, Integer Ring)

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

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

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

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


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})

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
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?


Enumerated
¶ alias of
EnumeratedSets

Facade
¶ alias of
FacadeSets

Finite
¶ alias of
FiniteSets

class
Infinite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton

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


class

class
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

Metric
¶ alias of
MetricSpaces

class
MorphismMethods
¶ 
is_injective
()¶ Return whether this map is injective.
EXAMPLES:
sage: f = ZZ.hom(GF(3)); f Natural morphism: From: Integer Ring To: Finite Field of size 3 sage: f.is_injective() False
Note that many maps do not implement this method:
sage: R.<x> = ZZ[] sage: f = R.hom([x]) sage: f.is_injective() Traceback (most recent call last): ... NotImplementedError


class
ParentMethods
¶ 
CartesianProduct
¶ alias of
CartesianProduct

algebra
(base_ring, category=None, **kwds)¶ 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 space of formal linear combinations of elements of \(G\) with coefficients in \(R\), endowed with whatever structure can be induced from that of \(S\). See the documentation of
sage.categories.algebra_functor
for details.EXAMPLES:
If \(S\) is a
group
, the result is its group algebra \(KS\):sage: S = DihedralGroup(4); S Dihedral group of order 8 as a permutation group sage: A = S.algebra(QQ); A Algebra of Dihedral group of order 8 as a permutation group over Rational Field sage: A.category() Category of finite group algebras over Rational Field sage: a = A.an_element(); a () + (1,3) + 2*(1,3)(2,4) + 3*(1,4,3,2)
This space is endowed with an algebra structure, obtained by extending by bilinearity the multiplication of \(G\) to a multiplication on \(RG\):
sage: a * a 6*() + 4*(2,4) + 3*(1,2)(3,4) + 12*(1,2,3,4) + 2*(1,3) + 13*(1,3)(2,4) + 6*(1,4,3,2) + 3*(1,4)(2,3)
If \(S\) is a
monoid
, the result is its 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 Algebra of 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
Similarly, we can construct algebras for additive magmas, monoids, and groups.
One may specify for which category one takes the algebra; here we build the algebra of the additive group \(GF_3\):
sage: from sage.categories.additive_groups import AdditiveGroups sage: S = GF(7) sage: A = S.algebra(QQ, category=AdditiveGroups()); A Algebra of Finite Field of size 7 over Rational Field sage: A.category() Category of finite dimensional additive group algebras over Rational Field sage: a = A(S(1)) sage: a 1 sage: 1 + a * a * a 0 + 3
Note that the
category
keyword needs to be fed with the structure on \(S\) to be used, not the induced structure on the result.

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)
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

construction
()¶ Return a pair
(functor, parent)
such thatfunctor(parent)
returnsself
. Ifself
does not have a functorial construction, returnNone
.EXAMPLES:
sage: QQ.construction() (FractionField, Integer Ring) sage: f, R = QQ['x'].construction() sage: f Poly[x] sage: R Rational Field sage: f(R) Univariate Polynomial Ring in x over Rational Field

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
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
Realizations
(category, *args)¶ Bases:
sage.categories.realizations.RealizationsCategory

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
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

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

Enumerated
()¶ Return the full subcategory of the enumerated objects of
self
.An enumerated object can be iterated to get its elements.
EXAMPLES:
sage: Sets().Enumerated() Category of enumerated sets sage: Rings().Finite().Enumerated() Category of finite enumerated rings sage: Rings().Infinite().Enumerated() Category of infinite enumerated 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.

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

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

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

Metric
()¶ Return the subcategory of the metric objects of
self
.

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

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.

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 nonnormal 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
 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
.


class
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
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
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

Topological
¶ alias of
TopologicalSpaces

class
WithRealizations
(category, *args)¶ Bases:
sage.categories.with_realizations.WithRealizationsCategory

class
ParentMethods
¶ 
class
Realizations
(parent_with_realization)¶ Bases:
sage.categories.realizations.Category_realization_of_parent

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]


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

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)

inject_shorthands
(shorthands=None, verbose=True)¶ Import standard shorthands into the global namespace.
INPUT:
shorthands
– a list (or iterable) of strings (default:self._shorthands
) or"all"
(forself._shorthands_all
)verbose
– boolean (defaultTrue
); whether to print the defined shorthands
EXAMPLES:
When computing with a set with multiple realizations, like
SymmetricFunctions
orSubsetAlgebra
, it is convenient to define shorthands for the various realizations, but cumbersome to do it by hand:sage: S = SymmetricFunctions(ZZ); S Symmetric Functions over Integer Ring sage: s = S.s(); s Symmetric Functions over Integer Ring in the Schur basis sage: e = S.e(); e Symmetric Functions over Integer Ring in the elementary basis
This method automatizes the process:
sage: S.inject_shorthands() Defining e as shorthand for Symmetric Functions over Integer Ring in the elementary basis Defining f as shorthand for Symmetric Functions over Integer Ring in the forgotten basis Defining h as shorthand for Symmetric Functions over Integer Ring in the homogeneous basis Defining m as shorthand for Symmetric Functions over Integer Ring in the monomial basis Defining p as shorthand for Symmetric Functions over Integer Ring in the powersum basis Defining s as shorthand for Symmetric Functions over Integer Ring in the Schur basis sage: s[1] + e[2] * p[1,1] + 2*h[3] + m[2,1] s[1]  2*s[1, 1, 1] + s[1, 1, 1, 1] + s[2, 1] + 2*s[2, 1, 1] + s[2, 2] + 2*s[3] + s[3, 1] sage: e Symmetric Functions over Integer Ring in the elementary basis sage: p Symmetric Functions over Integer Ring in the powersum basis sage: s Symmetric Functions over Integer Ring in the Schur basis
Sometimes, like for symmetric functions, one can request for all shorthands to be defined, including less common ones:
sage: S.inject_shorthands("all") Defining e as shorthand for Symmetric Functions over Integer Ring in the elementary basis Defining f as shorthand for Symmetric Functions over Integer Ring in the forgotten basis Defining h as shorthand for Symmetric Functions over Integer Ring in the homogeneous basis Defining ht as shorthand for Symmetric Functions over Integer Ring in the induced trivial character basis Defining m as shorthand for Symmetric Functions over Integer Ring in the monomial basis Defining o as shorthand for Symmetric Functions over Integer Ring in the orthogonal basis Defining p as shorthand for Symmetric Functions over Integer Ring in the powersum basis Defining s as shorthand for Symmetric Functions over Integer Ring in the Schur basis Defining sp as shorthand for Symmetric Functions over Integer Ring in the symplectic basis Defining st as shorthand for Symmetric Functions over Integer Ring in the irreducible symmetric group character basis Defining w as shorthand for Symmetric Functions over Integer Ring in the Witt basis
The messages can be silenced by setting
verbose=False
:sage: Q = QuasiSymmetricFunctions(ZZ) sage: Q.inject_shorthands(verbose=False) 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
One can also just import a subset of the shorthands:
sage: SQ = SymmetricFunctions(QQ) sage: SQ.inject_shorthands(['p', 's'], verbose=False) sage: p Symmetric Functions over Rational Field in the powersum basis sage: s Symmetric Functions over Rational Field in the Schur basis
Note that
e
is left unchanged:sage: e Symmetric Functions over Integer Ring in the elementary basis

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

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

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

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)

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