# 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)

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.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_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)
<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'>
<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()

class Algebras(category, *args)

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
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()

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)

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.

INPUTS:

• 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)

cardinality()

Return the cardinality of self

EXAMPLES:

sage: C = cartesian_product([GF(3), FiniteEnumeratedSet(['a','b']), GF(5)])
sage: C.cardinality()
30

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

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 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.Finite

alias of FiniteSets

class Sets.HomCategory(category, name=None)

Initializes this HomCategory

INPUT:
• category – the category whose Homsets are the objects of this category.
• name – An optional name for this category.

EXAMPLES:

We need to skip one test, since the hierarchy of hom categories isn’t consistent yet:

sage: C = sage.categories.category.HomCategory(Rings()); C
Category of hom sets in Category of rings
sage: TestSuite(C).run(skip=['_test_category_graph'])

class Sets.Infinite(base_category)

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_finite()

Return False since self is not finite.

EXAMPLES:

sage: C = InfiniteEnumeratedSets().example()
sage: C.is_finite()
False


TESTS:

sage: C.is_finite.im_func is sage.categories.sets_cat.Sets.Infinite.ParentMethods.is_finite.im_func
True

class Sets.IsomorphicObjects(category, *args)

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.ParentMethods
CartesianProduct

alias of CartesianProduct

algebra(base_ring, category=None)

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 $$K$$-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


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: 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 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 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 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

cardinality()

The cardinality of self.

self.cardinality() should return the cardinality of the set self as a sage Integer or as infinity.

This if the default implementation from the category Sets(); it raises a NotImplementedError since one does not know whether the set is finite or not.

EXAMPLES:

sage: class broken(UniqueRepresentation, Parent):
....:     def __init__(self):
....:         Parent.__init__(self, category = Sets())
sage: broken().cardinality()
Traceback (most recent call last):
...
NotImplementedError: unknown cardinality

cartesian_product(*parents)

Return the cartesian product of the parents.

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

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: S.some_elements()
[]


This method should return an iterable, not an iterator.

class Sets.Quotients(category, *args)

A category for quotients of sets.

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 Sets.Realizations(category, *args)

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
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 Sets.SubcategoryMethods
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

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.

EXAMPLES:

sage: Sets().CartesianProducts()
Category of Cartesian products of sets
sage: Semigroups().CartesianProducts()
Category of Cartesian products of semigroups
sage: EuclideanDomains().CartesianProducts()
Join of Category of rings and Category of Cartesian products of ...


Return the full subcategory of the facade objects of self.

A facade set is a parent P whose elements actually belong to some other parent:

sage: P = Sets().example(); P
Set of prime numbers (basic implementation)
sage: p = Sets().example().an_element(); p
47
sage: p in P
True
sage: p.parent()
Integer Ring


Typical use cases include modeling a subset of an existing parent:

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 a parent with more (or less!) structure:

sage: Posets().example("facade")
An example of a facade poset: the positive integers ordered by divisibility


Let us consider one of the examples above in detail: the partially ordered set $$P$$ of positive integers w.r.t. divisibility order. There are two options for representing its elements:

1. as plain integers
2. as integers, modified to be aware that their parent is $$P$$

The advantage of 1. is that one needs not to do conversions back and forth between $$P$$ and $$\ZZ$$. The disadvantage is that this introduces an ambiguity when writing $$2 < 3$$:

sage: 2 < 3
True


To raise this ambiguity, one needs to explicitly specify the order as in $$2 <_P 3$$:

sage: P = Posets().example("facade")
sage: P.lt(2,3)
False


Beware that P(2) is still the integer $$2$$. Therefore P(2) < P(3) still compares $$2$$ and $$3$$ as integers!

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.

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.

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):
sage: a = A()


TESTS:

sage: Posets().Facade()
True


Return the full subcategory of the facade objects of self.

A facade set is a parent P whose elements actually belong to some other parent:

sage: P = Sets().example(); P
Set of prime numbers (basic implementation)
sage: p = Sets().example().an_element(); p
47
sage: p in P
True
sage: p.parent()
Integer Ring


Typical use cases include modeling a subset of an existing parent:

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 a parent with more (or less!) structure:

sage: Posets().example("facade")
An example of a facade poset: the positive integers ordered by divisibility


Let us consider one of the examples above in detail: the partially ordered set $$P$$ of positive integers w.r.t. divisibility order. There are two options for representing its elements:

1. as plain integers
2. as integers, modified to be aware that their parent is $$P$$

The advantage of 1. is that one needs not to do conversions back and forth between $$P$$ and $$\ZZ$$. The disadvantage is that this introduces an ambiguity when writing $$2 < 3$$:

sage: 2 < 3
True


To raise this ambiguity, one needs to explicitly specify the order as in $$2 <_P 3$$:

sage: P = Posets().example("facade")
sage: P.lt(2,3)
False


Beware that P(2) is still the integer $$2$$. Therefore P(2) < P(3) still compares $$2$$ and $$3$$ as integers!

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.

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.

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):
sage: a = A()


TESTS:

sage: Posets().Facade()
True

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 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


TESTS:

sage: TestSuite(Sets().IsomorphicObjects()).run()

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.

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 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.

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 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 $$\ZZ/n\ZZ$$.

TESTS:

sage: TestSuite(Sets().Subquotients()).run()

class Sets.Subobjects(category, *args)

A category for subobjects of sets.

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 Sets.Subquotients(category, *args)

A category for subquotients of sets.

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


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)

retract(x)

Retract x to self.

INPUT:

• x – an element of the ambient space for self

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)

class Sets.WithRealizations(category, *args)

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
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)

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 in Category_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


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
[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)

EXAMPLES:

sage: A = Sets().WithRealizations().example(QQ); A
The subset algebra of {1, 2, 3} over Rational Field
sage: A.inject_shorthands()
Injecting F as shorthand for The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
Injecting In as shorthand for The subset algebra of {1, 2, 3} over Rational Field in the In basis
...

Sets.WithRealizations.ParentMethods.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)

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()
sage: Sets().WithRealizations().super_categories()

Sets.example(choice=None)

Returns 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

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]

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


Semirngs