Finite sets¶
- class sage.categories.finite_sets.FiniteSets(base_category)[source]¶
- Bases: - CategoryWithAxiom_singleton- 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]¶
- Bases: - AlgebrasCategory- 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 Subquotients(category, *args)[source]¶
- Bases: - SubquotientsCategory- 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