Commutative additive groups¶
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class
sage.categories.commutative_additive_groups.
CommutativeAdditiveGroups
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
,sage.categories.category_types.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.
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class
Algebras
(category, *args)¶
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
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class
ElementMethods
¶ Bases:
object
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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) sage: H = cartesian_product([cartesian_product([Zmod(2),Zmod(9)]), K]) sage: z = H(((1,2), K.gen())) sage: z.additive_order() 18
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class
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