Additive monoids

class sage.categories.additive_monoids.AdditiveMonoids(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of additive monoids.

An additive monoid is a unital class:\(additive semigroup <sage.categories.additive_semigroups.AdditiveSemigroups>\), that is a set endowed with a binary operation \(+\) which is associative and admits a zero (see Wikipedia article Monoid).

EXAMPLES:

sage: from sage.categories.additive_monoids import AdditiveMonoids
sage: C = AdditiveMonoids(); C
Category of additive monoids
sage: C.super_categories()
[Category of additive unital additive magmas, Category of additive semigroups]
sage: sorted(C.axioms())
['AdditiveAssociative', 'AdditiveUnital']
sage: from sage.categories.additive_semigroups import AdditiveSemigroups
sage: C is AdditiveSemigroups().AdditiveUnital()
True

TESTS:

sage: C.Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ))
True
sage: TestSuite(C).run()
AdditiveCommutative

alias of CommutativeAdditiveMonoids

AdditiveInverse

alias of AdditiveGroups

class Homsets(category, *args)

Bases: sage.categories.homsets.HomsetsCategory

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
extra_super_categories()

Implement the fact that a homset between two monoids is associative.

EXAMPLES:

sage: from sage.categories.additive_monoids import AdditiveMonoids
sage: AdditiveMonoids().Homsets().extra_super_categories()
[Category of additive semigroups]
sage: AdditiveMonoids().Homsets().super_categories()
[Category of homsets of additive unital additive magmas, Category of additive monoids]

Todo

This could be deduced from AdditiveSemigroups.Homsets.extra_super_categories(). See comment in Objects.SubcategoryMethods.Homsets().

class AdditiveMonoids.ParentMethods
sum(args)

Return the sum of the elements in args, as an element of self.

INPUT:

  • args – a list (or iterable) of elements of self

EXAMPLES:

sage: S = CommutativeAdditiveMonoids().example()
sage: (a,b,c,d) = S.additive_semigroup_generators()
sage: S.sum((a,b,a,c,a,b))
3*a + c + 2*b
sage: S.sum(())
0
sage: S.sum(()).parent() == S
True

Previous topic

Additive Magmas

Next topic

Additive semigroups

This Page