An additive semigroup is an associative additive magma, that is a set endowed with an operation $$+$$ which is associative.

EXAMPLES:

sage: from sage.categories.additive_semigroups import AdditiveSemigroups
sage: C.super_categories()
sage: C.all_super_categories()
Category of sets,
Category of sets with partial maps,
Category of objects]

sage: C.axioms()
True

class Algebras(category, *args)
class ParentMethods

Bases: object

algebra_generators()

Return the generators of this algebra, as per MagmaticAlgebras.ParentMethods.algebra_generators().

They correspond to the generators of the additive semigroup.

EXAMPLES:

sage: S = CommutativeAdditiveSemigroups().example(); S
An example of a commutative semigroup: the free commutative semigroup generated by ('a', 'b', 'c', 'd')
sage: A = S.algebra(QQ)
sage: A.algebra_generators()
Finite family {0: B[a], 1: B[b], 2: B[c], 3: B[d]}

product_on_basis(g1, g2)

Product, on basis elements, as per MagmaticAlgebras.WithBasis.ParentMethods.product_on_basis().

The product of two basis elements is induced by the addition of the corresponding elements of the group.

EXAMPLES:

sage: S = CommutativeAdditiveSemigroups().example(); S
An example of a commutative semigroup: the free commutative semigroup generated by ('a', 'b', 'c', 'd')
sage: A = S.algebra(QQ)
sage: a,b,c,d = A.algebra_generators()
sage: b * d * c
B[b + c + d]

extra_super_categories()

EXAMPLES:

sage: from sage.categories.additive_semigroups import AdditiveSemigroups
[Category of semigroups]
[Category of additive semigroup algebras over Rational Field,

class CartesianProducts(category, *args)
extra_super_categories()

Implement the fact that a Cartesian product of additive semigroups is an additive semigroup.

EXAMPLES:

sage: from sage.categories.additive_semigroups import AdditiveSemigroups
sage: C.extra_super_categories()
sage: C.axioms()

class Homsets(category, *args)
extra_super_categories()

Implement the fact that a homset between two semigroups is a semigroup.

EXAMPLES:

sage: from sage.categories.additive_semigroups import AdditiveSemigroups

Bases: object