Finite sets#

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

The category of finite sets.

EXAMPLES:

```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
```
```>>> 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]#
extra_super_categories()[source]#

EXAMPLES:

```sage: FiniteSets().Algebras(QQ).extra_super_categories()
[Category of finite dimensional vector spaces with basis over Rational Field]
```
```>>> 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: FiniteMonoids().Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ).FiniteDimensional())
True
```
```>>> 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: C = FiniteEnumeratedSets().example()
sage: C.is_finite()
True
```
```>>> from sage.all import *
>>> C = FiniteEnumeratedSets().example()
>>> C.is_finite()
True
```
class Subquotients(category, *args)[source]#
extra_super_categories()[source]#

EXAMPLES:

```sage: FiniteSets().Subquotients().extra_super_categories()
[Category of finite sets]
```
```>>> 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: FiniteSets().Subquotients().is_subcategory(FiniteSets())
True
sage: FiniteSets().Quotients   ().is_subcategory(FiniteSets())
True
sage: FiniteSets().Subobjects  ().is_subcategory(FiniteSets())
True
```
```>>> from sage.all import *
>>> FiniteSets().Subquotients().is_subcategory(FiniteSets())
True
>>> FiniteSets().Quotients   ().is_subcategory(FiniteSets())
True
>>> FiniteSets().Subobjects  ().is_subcategory(FiniteSets())
True
```