Semirings¶
- 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 - EXAMPLES: - 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 - >>> 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