Safe Haskell	Safe
Language	Haskell98

Data.Group

Synopsis

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

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) #