Cartesian Product Functorial Construction

AUTHORS:

• Nicolas M. Thiery (2008-2010): initial revision and refactorization

class sage.categories.cartesian_product.CartesianProductFunctor(category=None)

Bases: CovariantFunctorialConstruction, MultivariateConstructionFunctor

The Cartesian product functor.

EXAMPLES:

Sage

sage: cartesian_product
The cartesian_product functorial construction

Python

>>> from sage.all import *
>>> cartesian_product
The cartesian_product functorial construction

cartesian_product takes a finite collection of sets, and constructs the Cartesian product of those sets:

Sage

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]]

Python

>>> from sage.all import *
>>> A = FiniteEnumeratedSet(['a','b','c'])
>>> B = FiniteEnumeratedSet([Integer(1),Integer(2)])
>>> C = cartesian_product([A, B]); C
The Cartesian product of ({'a', 'b', 'c'}, {1, 2})
>>> C.an_element()
('a', 1)
>>> 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

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

Python

>>> from sage.all import *
>>> M = Monoids().example()
>>> M
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
>>> M.rename('M')
>>> C = cartesian_product([M, ZZ, QQ])
>>> C
The Cartesian product of (M, Integer Ring, Rational Field)
>>> C.an_element()
('abcd', 1, 1/2)
>>> C.an_element()**Integer(2)
('abcdabcd', 1, 1/4)
>>> C.category()
Category of Cartesian products of monoids

>>> 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

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]

Python

>>> from sage.all import *
>>> 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

sage: C.one()
('', 1, 1)

Python

>>> from sage.all import *
>>> 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.

symbol = ' (+) '

class sage.categories.cartesian_product.CartesianProductsCategory(category, *args)

Bases: 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

sage: ModulesWithBasis(QQ).CartesianProducts().CartesianProducts()
Category of Cartesian products of vector spaces with basis over Rational Field

Python

>>> from sage.all import *
>>> 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

sage: Algebras(ZZ).CartesianProducts().base_ring()
Integer Ring

Python

>>> from sage.all import *
>>> Algebras(ZZ).CartesianProducts().base_ring()
Integer Ring