Distributive Magmas and Additive Magmas

class sage.categories.distributive_magmas_and_additive_magmas.DistributiveMagmasAndAdditiveMagmas(base_category)

Bases: CategoryWithAxiom_singleton

The category of sets \((S,+,*)\) with \(*\) distributing on \(+\).

This is similar to a ring, but \(+\) and \(*\) are only required to be (additive) magmas.

EXAMPLES:

Sage

sage: from sage.categories.distributive_magmas_and_additive_magmas import DistributiveMagmasAndAdditiveMagmas
sage: C = DistributiveMagmasAndAdditiveMagmas(); C
Category of distributive magmas and additive magmas
sage: C.super_categories()
[Category of magmas and additive magmas]

Python

>>> from sage.all import *
>>> from sage.categories.distributive_magmas_and_additive_magmas import DistributiveMagmasAndAdditiveMagmas
>>> C = DistributiveMagmasAndAdditiveMagmas(); C
Category of distributive magmas and additive magmas
>>> C.super_categories()
[Category of magmas and additive magmas]

class AdditiveAssociative(base_category)

Bases: CategoryWithAxiom_singleton

class AdditiveCommutative(base_category)

Bases: CategoryWithAxiom_singleton

class AdditiveUnital(base_category)

Bases: CategoryWithAxiom_singleton

class Associative(base_category)

Bases: CategoryWithAxiom_singleton

AdditiveInverse

alias of Rngs

Unital

alias of Semirings

class CartesianProducts(category, *args)

Bases: CartesianProductsCategory

extra_super_categories()

Implement the fact that a Cartesian product of magmas distributing over additive magmas is a magma distributing over an additive magma.

EXAMPLES:

Sage

sage: C = (Magmas() & AdditiveMagmas()).Distributive().CartesianProducts()
sage: C.extra_super_categories()
[Category of distributive magmas and additive magmas]
sage: C.axioms()
frozenset({'Distributive'})

Python

>>> from sage.all import *
>>> C = (Magmas() & AdditiveMagmas()).Distributive().CartesianProducts()
>>> C.extra_super_categories()
[Category of distributive magmas and additive magmas]
>>> C.axioms()
frozenset({'Distributive'})

class ParentMethods

Bases: object