Data.Group

Synopsis

# Documentation

class Monoid m => Group m where Source #

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 Source #

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

pow a n == a <> a <> ... <> a
 (n lots of a)

If n is negative, the result is inverted.

Instances

 Group () Source # Methodsinvert :: () -> () Source #pow :: Integral x => () -> x -> () Source # Group a => Group (Dual a) Source # Methodsinvert :: Dual a -> Dual a Source #pow :: Integral x => Dual a -> x -> Dual a Source # Num a => Group (Sum a) Source # Methodsinvert :: Sum a -> Sum a Source #pow :: Integral x => Sum a -> x -> Sum a Source # Fractional a => Group (Product a) Source # Methodsinvert :: Product a -> Product a Source #pow :: Integral x => Product a -> x -> Product a Source # Group b => Group (a -> b) Source # Methodsinvert :: (a -> b) -> a -> b Source #pow :: Integral x => (a -> b) -> x -> a -> b Source # (Group a, Group b) => Group (a, b) Source # Methodsinvert :: (a, b) -> (a, b) Source #pow :: Integral x => (a, b) -> x -> (a, b) Source # (Group a, Group b, Group c) => Group (a, b, c) Source # Methodsinvert :: (a, b, c) -> (a, b, c) Source #pow :: Integral x => (a, b, c) -> x -> (a, b, c) Source # (Group a, Group b, Group c, Group d) => Group (a, b, c, d) Source # Methodsinvert :: (a, b, c, d) -> (a, b, c, d) Source #pow :: Integral x => (a, b, c, d) -> x -> (a, b, c, d) Source # (Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e) Source # Methodsinvert :: (a, b, c, d, e) -> (a, b, c, d, e) Source #pow :: Integral x => (a, b, c, d, e) -> x -> (a, b, c, d, e) Source #

class Group g => Abelian g Source #

An Abelian group is a Group that follows the rule:

a <> b == b <> a

Instances

 Abelian () Source # Abelian a => Abelian (Dual a) Source # Num a => Abelian (Sum a) Source # Fractional a => Abelian (Product a) Source # Abelian b => Abelian (a -> b) Source # (Abelian a, Abelian b) => Abelian (a, b) Source # (Abelian a, Abelian b, Abelian c) => Abelian (a, b, c) Source # (Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d) Source # (Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e) Source #