An additive group is a Group whose operation can be thought of as addition in some sense.

For example, the additive group of integers \( (ℤ, 0, +) \).

Infix alias for minus.

Examples:

>>> let x = Sum (3 :: Int)
>>> x - x
Sum {getSum = 0}

Infix alias for additive (<>).

Examples:

>>> Sum (1 :: Int) + Sum (40 :: Int)
Sum {getSum = 41}

Infix alias for copower.

Examples:

>>> let x = Sum (3 :: Int)
>>> 2 × x
Sum {getSum = 6}

Add an element of an additive group to itself n-many times.

This represents ℕ-indexed copowers of an element g of an additive group, i.e. iterated coproducts of group elements. This is representable by the universal property \( C(∐_n g, x) ≅ C(g, x)^n \).

Examples:

>>> copower 2 (Sum (3 :: Int))
Sum {getSum = 6}

An additive abelian group is an Abelian whose operation can be thought of as commutative addition in some sense. Almost all additive groups are abelian.