Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton
The category of additive monoids.
An additive monoid is a unital class:, 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()
alias of CommutativeAdditiveMonoids
alias of AdditiveGroups
Return the sum of the elements in args, as an element of self.
INPUT:
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