Semirngs#

class sage.categories.semirings.Semirings(base_category)[source]#

Bases: CategoryWithAxiom_singleton

The category of semirings.

A semiring \((S,+,*)\) is similar to a ring, but without the requirement that each element must have an additive inverse. In other words, it is a combination of a commutative additive monoid \((S,+)\) and a multiplicative monoid \((S,*)\), where \(*\) distributes over \(+\).

See also

Wikipedia article Semiring

EXAMPLES:

Sage

sage: Semirings()
Category of semirings
sage: Semirings().super_categories()
[Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of monoids]

sage: sorted(Semirings().axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveUnital', 'Associative', 'Distributive', 'Unital']

sage: Semirings() is (CommutativeAdditiveMonoids() & Monoids()).Distributive()
True

sage: Semirings().AdditiveInverse()
Category of rings

Python

>>> from sage.all import *
>>> Semirings()
Category of semirings
>>> Semirings().super_categories()
[Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of monoids]

>>> sorted(Semirings().axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveUnital', 'Associative', 'Distributive', 'Unital']

>>> Semirings() is (CommutativeAdditiveMonoids() & Monoids()).Distributive()
True

>>> Semirings().AdditiveInverse()
Category of rings