Cartesian Product Functorial Construction¶
AUTHORS:
Nicolas M. Thiery (2008-2010): initial revision and refactorization
- class sage.categories.cartesian_product.CartesianProductFunctor(category=None)¶
Bases:
sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction
,sage.categories.pushout.MultivariateConstructionFunctor
The Cartesian product functor.
EXAMPLES:
sage: cartesian_product The cartesian_product functorial construction
cartesian_product
takes a finite collection of sets, and constructs the Cartesian product of those sets:sage: A = FiniteEnumeratedSet(['a','b','c']) sage: B = FiniteEnumeratedSet([1,2]) sage: C = cartesian_product([A, B]); C The Cartesian product of ({'a', 'b', 'c'}, {1, 2}) sage: C.an_element() ('a', 1) sage: C.list() # todo: not implemented [['a', 1], ['a', 2], ['b', 1], ['b', 2], ['c', 1], ['c', 2]]
If those sets are endowed with more structure, say they are monoids (hence in the category
Monoids()
), then the result is automatically endowed with its natural monoid structure:sage: M = Monoids().example() sage: M An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') sage: M.rename('M') sage: C = cartesian_product([M, ZZ, QQ]) sage: C The Cartesian product of (M, Integer Ring, Rational Field) sage: C.an_element() ('abcd', 1, 1/2) sage: C.an_element()^2 ('abcdabcd', 1, 1/4) sage: C.category() Category of Cartesian products of monoids sage: Monoids().CartesianProducts() Category of Cartesian products of monoids
The Cartesian product functor is covariant: if
A
is a subcategory ofB
, thenA.CartesianProducts()
is a subcategory ofB.CartesianProducts()
(see alsoCovariantFunctorialConstruction
):sage: C.categories() [Category of Cartesian products of monoids, Category of monoids, Category of Cartesian products of semigroups, Category of semigroups, Category of Cartesian products of unital magmas, Category of Cartesian products of magmas, Category of unital magmas, Category of magmas, Category of Cartesian products of sets, Category of sets, ...] [Category of Cartesian products of monoids, Category of monoids, Category of Cartesian products of semigroups, Category of semigroups, Category of Cartesian products of magmas, Category of unital magmas, Category of magmas, Category of Cartesian products of sets, Category of sets, Category of sets with partial maps, Category of objects]
Hence, the role of
Monoids().CartesianProducts()
is solely to provide mathematical information and algorithms which are relevant to Cartesian product of monoids. For example, it specifies that the result is again a monoid, and that its multiplicative unit is the Cartesian product of the units of the underlying sets:sage: C.one() ('', 1, 1)
Those are implemented in the nested class
Monoids.CartesianProducts
ofMonoids(QQ)
. This nested class is itself a subclass ofCartesianProductsCategory
.
- class sage.categories.cartesian_product.CartesianProductsCategory(category, *args)¶
Bases:
sage.categories.covariant_functorial_construction.CovariantConstructionCategory
An abstract base class for all
CartesianProducts
categories.- CartesianProducts()¶
Return the category of (finite) Cartesian products of objects of
self
.By associativity of Cartesian products, this is
self
(a Cartesian product of Cartesian products of \(A\)’s is a Cartesian product of \(A\)’s).EXAMPLES:
sage: ModulesWithBasis(QQ).CartesianProducts().CartesianProducts() Category of Cartesian products of vector spaces with basis over Rational Field
- base_ring()¶
The base ring of a Cartesian product is the base ring of the underlying category.
EXAMPLES:
sage: Algebras(ZZ).CartesianProducts().base_ring() Integer Ring