Finite sets#

class sage.categories.finite_sets.FiniteSets(base_category)[source]#

Bases: CategoryWithAxiom_singleton

The category of finite sets.

EXAMPLES:

Sage

sage: C = FiniteSets(); C
Category of finite sets
sage: C.super_categories()
[Category of sets]
sage: C.all_super_categories()
[Category of finite sets,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]
sage: C.example()
NotImplemented

Python

>>> from sage.all import *
>>> C = FiniteSets(); C
Category of finite sets
>>> C.super_categories()
[Category of sets]
>>> C.all_super_categories()
[Category of finite sets,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]
>>> C.example()
NotImplemented

class Algebras(category, *args)[source]#

Bases: AlgebrasCategory

extra_super_categories()[source]#

EXAMPLES:

Sage

sage: FiniteSets().Algebras(QQ).extra_super_categories()
[Category of finite dimensional vector spaces with basis over Rational Field]

Python

>>> from sage.all import *
>>> FiniteSets().Algebras(QQ).extra_super_categories()
[Category of finite dimensional vector spaces with basis over Rational Field]

This implements the fact that the algebra of a finite set is finite dimensional:

Sage

sage: FiniteMonoids().Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ).FiniteDimensional())
True

Python

>>> from sage.all import *
>>> FiniteMonoids().Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ).FiniteDimensional())
True

class ParentMethods[source]#

Bases: object

is_finite()[source]#

Return True since self is finite.

EXAMPLES:

Sage

sage: C = FiniteEnumeratedSets().example()
sage: C.is_finite()
True

Python

>>> from sage.all import *
>>> C = FiniteEnumeratedSets().example()
>>> C.is_finite()
True

class Subquotients(category, *args)[source]#

Bases: SubquotientsCategory

extra_super_categories()[source]#

EXAMPLES:

Sage

sage: FiniteSets().Subquotients().extra_super_categories()
[Category of finite sets]

Python

>>> from sage.all import *
>>> FiniteSets().Subquotients().extra_super_categories()
[Category of finite sets]

This implements the fact that a subquotient (and therefore a quotient or subobject) of a finite set is finite:

Sage

sage: FiniteSets().Subquotients().is_subcategory(FiniteSets())
True
sage: FiniteSets().Quotients   ().is_subcategory(FiniteSets())
True
sage: FiniteSets().Subobjects  ().is_subcategory(FiniteSets())
True

Python

>>> from sage.all import *
>>> FiniteSets().Subquotients().is_subcategory(FiniteSets())
True
>>> FiniteSets().Quotients   ().is_subcategory(FiniteSets())
True
>>> FiniteSets().Subobjects  ().is_subcategory(FiniteSets())
True