AUTHORS:
- Nicolas M. Thiery (2008-2010): initial revision and refactorization
Bases: sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction
A singleton class for the Cartesian product functor
EXAMPLES:
sage: cartesian_product
The cartesian_product functorial construction
cartesian_product takes a 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 of B, then A.CartesianProducts() is a subcategory of B.CartesianProducts() (see also CovariantFunctorialConstruction):
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 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 of Monoids(QQ). This nested class is itself a subclass of CartesianProductsCategory.
INPUT:
- self – a concrete category
Returns the category of parents constructed as cartesian products of parents in self.
See CartesianProductFunctor for more information
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 monoids
Bases: sage.categories.covariant_functorial_construction.CovariantConstructionCategory
An abstract base class for all CartesianProducts’s
TESTS:
sage: C = Sets().CartesianProducts()
sage: C
Category of Cartesian products of sets
sage: C.base_category()
Category of sets
sage: latex(C)
\mathbf{CartesianProducts}(\mathbf{Sets})
Returns the category of 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 modules with basis over Rational Field
The base ring of a cartesian product is the base ring of the underlying category.
EXAMPLES:
sage: Algebras(ZZ).CartesianProducts().base_ring()
Integer Ring
The cartesian product functorial construction
See CartesianProductFunctor for more information
EXAMPLES:
sage: cartesian_product
The cartesian_product functorial construction