Commutative additive groups

class sage.categories.commutative_additive_groups.CommutativeAdditiveGroups(base_category)

Bases: CategoryWithAxiom_singleton, AbelianCategory

The category of abelian groups, i.e. additive abelian monoids where each element has an inverse.

EXAMPLES:

sage: C = CommutativeAdditiveGroups(); C
Category of commutative additive groups
sage: C.super_categories()
[Category of additive groups, Category of commutative additive monoids]
sage: sorted(C.axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital']
sage: C is CommutativeAdditiveMonoids().AdditiveInverse()
True
sage: from sage.categories.additive_groups import AdditiveGroups
sage: C is AdditiveGroups().AdditiveCommutative()
True

Note

This category is currently empty. It’s left there for backward compatibility and because it is likely to grow in the future.

class Algebras(category, *args)

Bases: AlgebrasCategory

class CartesianProducts(category, *args)

Bases: CartesianProductsCategory

class ElementMethods

Bases: object

additive_order()

Return the additive order of this element.

EXAMPLES:

sage: G = cartesian_product([Zmod(3), Zmod(6), Zmod(5)])
sage: G((1,1,1)).additive_order()
30
sage: any((i * G((1,1,1))).is_zero() for i in range(1,30))
False
sage: 30 * G((1,1,1))
(0, 0, 0)

sage: G = cartesian_product([ZZ, ZZ])
sage: G((0,0)).additive_order()
1
sage: G((0,1)).additive_order()
+Infinity

sage: K = GF(9)                                                     # optional - sage.rings.finite_rings
sage: H = cartesian_product([                                       # optional - sage.rings.finite_rings
....:     cartesian_product([Zmod(2), Zmod(9)]), K])
sage: z = H(((1,2), K.gen()))                                       # optional - sage.rings.finite_rings
sage: z.additive_order()                                            # optional - sage.rings.finite_rings
18