Magmas and Additive Magmas#

class sage.categories.magmas_and_additive_magmas.MagmasAndAdditiveMagmas#

Bases: Category_singleton

The category of sets \((S,+,*)\) with an additive operation ‘+’ and a multiplicative operation \(*\)

EXAMPLES:

sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
sage: C = MagmasAndAdditiveMagmas(); C
Category of magmas and additive magmas

This is the base category for the categories of rings and their variants:

sage: C.Distributive()
Category of distributive magmas and additive magmas
sage: C.Distributive().Associative().AdditiveAssociative().AdditiveCommutative().AdditiveUnital().AdditiveInverse()
Category of rngs
sage: C.Distributive().Associative().AdditiveAssociative().AdditiveCommutative().AdditiveUnital().Unital()
Category of semirings
sage: C.Distributive().Associative().AdditiveAssociative().AdditiveCommutative().AdditiveUnital().AdditiveInverse().Unital()
Category of rings

This category is really meant to represent the intersection of the categories of Magmas and AdditiveMagmas; however Sage’s infrastructure does not allow yet to model this:

sage: Magmas() & AdditiveMagmas()
Join of Category of magmas and Category of additive magmas

sage: Magmas() & AdditiveMagmas()        # todo: not implemented
Category of magmas and additive magmas

class CartesianProducts(category, *args)#

Bases: CartesianProductsCategory

extra_super_categories()#: Implement the fact that this structure is stable under Cartesian products.

Distributive#: alias of DistributiveMagmasAndAdditiveMagmas

class SubcategoryMethods#

Bases: object

Distributive()#

Return the full subcategory of the objects of self where \(*\) is distributive on \(+\).

A magma and additive magma \(M\) is distributive if, for all \(x,y,z \in M\),

\[x * (y+z) = x*y + x*z \text{ and } (x+y) * z = x*z + y*z\]

EXAMPLES:

sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
sage: C = MagmasAndAdditiveMagmas().Distributive(); C
Category of distributive magmas and additive magmas

Note

Given that Sage does not know that MagmasAndAdditiveMagmas is the intersection of Magmas and AdditiveMagmas, this method is not available for:

sage: Magmas() & AdditiveMagmas()
Join of Category of magmas and Category of additive magmas

Still, the natural syntax works:

sage: (Magmas() & AdditiveMagmas()).Distributive()
Category of distributive magmas and additive magmas

thanks to a workaround implemented in Magmas.SubcategoryMethods.Distributive():

sage: (Magmas() & AdditiveMagmas()).Distributive.__module__
'sage.categories.magmas'

additional_structure()#

Return None.

Indeed, this category is meant to represent the join of AdditiveMagmas and Magmas. As such, it defines no additional structure.