Sets#

exception sage.categories.sets_cat.EmptySetError#

Bases: 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: 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'>
<class 'sage.misc.fast_methods.WithEqualityById'>
<class 'sage.structure.parent.Parent'>
<class 'sage.structure.category_object.CategoryObject'>
<class '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_construction() . . . 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'>
<class 'sage.rings.integer.IntegerWrapper'>
<class 'sage.rings.integer.Integer'>
<class 'sage.structure.element.EuclideanDomainElement'>
<class 'sage.structure.element.PrincipalIdealDomainElement'>
<class 'sage.structure.element.DedekindDomainElement'>
<class 'sage.structure.element.IntegralDomainElement'>
<class 'sage.structure.element.CommutativeRingElement'>
<class 'sage.structure.element.RingElement'>
<class 'sage.structure.element.ModuleElement'>
<class 'sage.categories.examples.sets_cat.PrimeNumbers_Abstract.Element'>
<class 'sage.structure.element.Element'>
<class '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: AlgebrasCategory

class ParentMethods#

Bases: object

construction()#

Return the functorial construction of self.

EXAMPLES:

sage: A = GroupAlgebra(KleinFourGroup(), QQ)
sage: F, arg = A.construction(); F, arg
(GroupAlgebraFunctor, Rational Field)
sage: F(arg) is A
True

This also works for structures such as monoid algebras (see trac ticket #27937):

sage: A = FreeAbelianMonoid('x,y').algebra(QQ)
sage: F, arg = A.construction(); F, arg
(The algebra functorial construction,
 Free abelian monoid on 2 generators (x, y))
sage: F(arg) is A
True
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 CartesianProducts(category, *args)#

Bases: 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#

Bases: object

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]
class ParentMethods#

Bases: object

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 of self

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
construction()#

The construction functor and the list of Cartesian factors.

EXAMPLES:

sage: C = cartesian_product([QQ, ZZ, ZZ])
sage: C.construction()
(The cartesian_product functorial construction,
(Rational Field, Integer Ring, Integer Ring))
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 ElementMethods#

Bases: object

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: CategoryWithAxiom_singleton

class ParentMethods#

Bases: object

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 IsomorphicObjects(category, *args)#

Bases: 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#

Bases: object

Metric#

alias of MetricSpaces

class MorphismMethods#

Bases: object

image(domain_subset=None)#

Return the image of the domain or of domain_subset.

EXAMPLES:

sage: P = Partitions(6)
sage: H = Hom(P, ZZ)
sage: f = H(ZZ.sum)
sage: X = f.image()
sage: list(X)
[6]
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
class ParentMethods#

Bases: object

CartesianProduct#

alias of CartesianProduct

algebra(base_ring, category=None, **kwds)#

Return the algebra of self over base_ring.

INPUT:

  • self – a parent \(S\)

  • base_ring – a ring \(K\)

  • category – a super category of the category of \(S\), or None

This returns the space of formal linear combinations of elements of \(S\) with coefficients in \(K\), 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 exception EmptySetError.

This default implementation calls _an_element_() and caches the result. Any parent should implement either an_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. If None is passed, then category_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)
construction()#

Return a pair (functor, parent) such that functor(parent) returns self. If self does not have a functorial construction, return None.

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 of element.

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: list(S.some_elements())
[]

This method should return an iterable, not an iterator.

class Quotients(category, *args)#

Bases: 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#

Bases: object

class Realizations(category, *args)#

Bases: RealizationsCategory

class ParentMethods#

Bases: object

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 SubcategoryMethods#

Bases: object

Algebras(base_ring)#

Return the category of objects constructed as algebras of objects of self over base_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

See also

CartesianProducts()#

Return the full subcategory of the objects of self constructed as Cartesian products.

See also

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:

  1. as plain integers:

    sage: P = Poset((divisors(12), attrcall("divides")), facade=True)

  2. 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 \(2\). Therefore P(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 the facade 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.

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 of Sets()), As().IsomorphicObjects() returns the category of objects of As() endowed with a distinguished description as the image of some other object of As() 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) and B(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 with max 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

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 of Sets()), As().Quotients() returns the category of objects of As() endowed with a distinguished description as quotient (in fact homomorphic image) of some other object of As().

Implementing an object of As().Quotients() is done in the same way as for As().Subquotients(); namely by providing an ambient space and a lift and a retract map. See Subquotients() for detailed instructions.

See also

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 of Sets()), As().Subobjects() returns the category of objects of As() endowed with a distinguished embedding into some other object of As().

Implementing an object of As().Subobjects() is done in the same way as for As().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. See Subquotients() for detailed instructions.

See also

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 of Sets()), As().Subquotients() returns the category of objects of As() endowed with a distinguished description as subquotient of some other object of As().

EXAMPLES:

sage: Monoids().Subquotients()
Category of subquotients of monoids

A parent \(A\) in As() is further in As().Subquotients() if there is a distinguished parent \(B\) in As(), 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 of As().

    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 of Bs(), then As().Subquotients() is a subcategory of Bs().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 category As() 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) and A.retract(b). As a shorthand for the former, one can use alternatively a.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

Topological()#

Return the subcategory of the topological objects of self.

class Subobjects(category, *args)#

Bases: 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#

Bases: object

class Subquotients(category, *args)#

Bases: 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#

Bases: object

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#

Bases: object

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 and lift() and retract().

lift(x)#

Lift \(x\) to the ambient space for self.

INPUT:

  • x – an element of self

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 also Sets.Subquotients.ElementMethods.lift().

retract(x)#

Retract x to self.

INPUT:

  • x – an element of the ambient space for self

See also

Sets.SubcategoryMethods.Subquotients for the specifications, ambient(), retract(), and also Sets.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)
Topological#

alias of TopologicalSpaces

class WithRealizations(category, *args)#

Bases: WithRealizationsCategory

class ParentMethods#

Bases: object

class Realizations(parent_with_realization)#

Bases: 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 of self

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" (for self._shorthands_all)

  • verbose – boolean (default True);

    whether to print the defined shorthands

EXAMPLES:

When computing with a set with multiple realizations, like SymmetricFunctions or SubsetAlgebra, 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 automates 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 symmetric group 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 that self 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 category A.Realizations() automatically adds P to this list by calling A._register_realization(A)

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]
example(choice=None)#

Return examples of objects of Sets(), as per Category.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]
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 element

  • y – 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