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

EXAMPLES:

```sage: C = CommutativeAdditiveGroups(); C
sage: C.super_categories()
sage: sorted(C.axioms())
True
True
```
```>>> from sage.all import *
>>> C.super_categories()
>>> sorted(C.axioms())
True
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)[source]#
class CartesianProducts(category, *args)[source]#
class ElementMethods[source]#

Bases: `object`

Return the additive order of this element.

EXAMPLES:

```sage: G = cartesian_product([Zmod(3), Zmod(6), Zmod(5)])
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])
1
+Infinity

sage: # needs sage.rings.finite_rings
sage: K = GF(9)
sage: H = cartesian_product([
....:     cartesian_product([Zmod(2), Zmod(9)]), K])
sage: z = H(((1,2), K.gen()))
18
```
```>>> from sage.all import *
>>> G = cartesian_product([Zmod(Integer(3)), Zmod(Integer(6)), Zmod(Integer(5))])
30
>>> any((i * G((Integer(1),Integer(1),Integer(1)))).is_zero() for i in range(Integer(1),Integer(30)))
False
>>> Integer(30) * G((Integer(1),Integer(1),Integer(1)))
(0, 0, 0)

>>> G = cartesian_product([ZZ, ZZ])