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