| Copyright | (c) 2020-2021 Emily Pillmore |
|---|---|
| License | BSD-style |
| Maintainer | Emily Pillmore <emilypi@cohomolo.gy>, Reed Mullanix <reedmullanix@gmail.com> |
| Stability | stable |
| Portability | non-portable |
| Safe Haskell | Safe |
| Language | Haskell2010 |
Data.Group.Additive
Description
This module contains definitions for AdditiveGroup and AdditiveAbelianGroup,
along with the relevant combinators.
Synopsis
- class Group g => AdditiveGroup g
- (-) :: AdditiveGroup a => a -> a -> a
- (+) :: AdditiveGroup g => g -> g -> g
- (×) :: (Integral n, AdditiveGroup a) => n -> a -> a
- copower :: (Integral n, AdditiveGroup g) => n -> g -> g
- class (Abelian g, AdditiveGroup g) => AdditiveAbelianGroup g
Additive groups
class Group g => AdditiveGroup g Source #
An additive group is a Group whose operation can be thought of
as addition in some sense.
For example, the additive group of integers \( (ℤ, 0, +) \).
Instances
Combinators
(-) :: AdditiveGroup a => a -> a -> a infixl 6 Source #
(+) :: AdditiveGroup g => g -> g -> g infixl 6 Source #
(×) :: (Integral n, AdditiveGroup a) => n -> a -> a infixl 7 Source #
copower :: (Integral n, AdditiveGroup g) => n -> g -> g Source #
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}
Additive abelian groups
class (Abelian g, AdditiveGroup g) => AdditiveAbelianGroup g Source #
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.