Safe Haskell	Safe
Language	Haskell98

Data.Group

Synopsis

class Monoid m => Group m where
- invert :: m -> m
- pow :: Integral x => m -> x -> m
class Group g => Abelian g

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

Instances details

Group () Source #
Instance details Defined in Data.Group Methods invert :: () -> () Source # pow :: Integral x => () -> x -> () Source #
Group a => Group (Dual a) Source #
Instance details Defined in Data.Group Methods invert :: Dual a -> Dual a Source # pow :: Integral x => Dual a -> x -> Dual a Source #
Fractional a => Group (Product a) Source #
Instance details Defined in Data.Group Methods invert :: Product a -> Product a Source # pow :: Integral x => Product a -> x -> Product a Source #
Num a => Group (Sum a) Source #
Instance details Defined in Data.Group Methods invert :: Sum a -> Sum a Source # pow :: Integral x => Sum a -> x -> Sum a Source #
Group b => Group (a -> b) Source #
Instance details Defined in Data.Group Methods invert :: (a -> b) -> a -> b Source # pow :: Integral x => (a -> b) -> x -> a -> b Source #
(Group a, Group b) => Group (a, b) Source #
Instance details Defined in Data.Group Methods invert :: (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 #
Instance details Defined in Data.Group Methods invert :: (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 #
Instance details Defined in Data.Group Methods invert :: (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 #
Instance details Defined in Data.Group Methods invert :: (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

Instances details

Abelian () Source #
Instance details Defined in Data.Group
Abelian a => Abelian (Dual a) Source #
Instance details Defined in Data.Group
Fractional a => Abelian (Product a) Source #
Instance details Defined in Data.Group
Num a => Abelian (Sum a) Source #
Instance details Defined in Data.Group
Abelian b => Abelian (a -> b) Source #
Instance details Defined in Data.Group
(Abelian a, Abelian b) => Abelian (a, b) Source #
Instance details Defined in Data.Group
(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c) Source #
Instance details Defined in Data.Group
(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d) Source #
Instance details Defined in Data.Group
(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e) Source #
Instance details Defined in Data.Group