Data.Group

Documentation

class Monoid m => Group m where

A Group is a Monoid plus a function, invert, such that:

a <> invert a == mempty

invert a <> a == mempty

Minimal complete definition

invert

Methods

invert :: m -> m

pow :: Integral x => m -> x -> m

pow a n == a <> a <> ... <> a

 (n lots of a)

If n is negative, the result is inverted.

Instances

Group ()

Methods

invert :: () -> ()

pow :: Integral x => () -> x -> ()

Group a => Group (Dual a)

Methods

invert :: Dual a -> Dual a

pow :: Integral x => Dual a -> x -> Dual a

Num a => Group (Sum a)

Methods

invert :: Sum a -> Sum a

pow :: Integral x => Sum a -> x -> Sum a

Fractional a => Group (Product a)

Methods

invert :: Product a -> Product a

pow :: Integral x => Product a -> x -> Product a

Group b => Group (a -> b)

Methods

invert :: (a -> b) -> a -> b

pow :: Integral x => (a -> b) -> x -> a -> b

(Group a, Group b) => Group (a, b)

Methods

invert :: (a, b) -> (a, b)

pow :: Integral x => (a, b) -> x -> (a, b)

(Group a, Group b, Group c) => Group (a, b, c)

Methods

invert :: (a, b, c) -> (a, b, c)

pow :: Integral x => (a, b, c) -> x -> (a, b, c)

(Group a, Group b, Group c, Group d) => Group (a, b, c, d)

Methods

invert :: (a, b, c, d) -> (a, b, c, d)

pow :: Integral x => (a, b, c, d) -> x -> (a, b, c, d)

(Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e)

Methods

invert :: (a, b, c, d, e) -> (a, b, c, d, e)

pow :: Integral x => (a, b, c, d, e) -> x -> (a, b, c, d, e)

class Group g => Abelian g

An Abelian group is a Group that follows the rule:

a <> b == b <> a

Instances

Abelian ()

Abelian a => Abelian (Dual a)

Num a => Abelian (Sum a)

Fractional a => Abelian (Product a)

Abelian b => Abelian (a -> b)

(Abelian a, Abelian b) => Abelian (a, b)

(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c)

(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d)

(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e)