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 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

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