Modulegroups-0.5.3Haskell2010

Data.Group

Synopsis

classclass Monoid m => Group m where

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

a <> invert a == mempty

invert a <> a == mempty

Instances16Group, …

Group ()Defined in groups-0.5.3 · Data.Group
Num a => Group (Sum a)Defined in groups-0.5.3 · Data.Group
Fractional a => Group (Product a)Defined in groups-0.5.3 · Data.Group
Group a => Group (Identity a)Defined in groups-0.5.3 · Data.Group
Identity lifts groups pointwise (at only one point).
Group a => Group (Down a)Defined in groups-0.5.3 · Data.Group
Group a => Group (Dual a)Defined in groups-0.5.3 · Data.Group
Group (Proxy x)Defined in groups-0.5.3 · Data.Group
Trivial group, Functor style.
Group a => Group (Op a b)Defined in groups-0.5.3 · Data.Group
Group b => Group (a -> b)Defined in groups-0.5.3 · Data.Group
(Group a, Group b) => Group (a, b)Defined in groups-0.5.3 · Data.Group
Group a => Group (Const a x)Defined in groups-0.5.3 · Data.Group
Const lifts groups into a functor.
(Group a, Group b, Group c) => Group (a, b, c)Defined in groups-0.5.3 · Data.Group
(Group (f a), Group (g a)) => Group ((:*:) f g a)Defined in groups-0.5.3 · Data.Group
Product of groups, Functor style.
(Group a, Group b, Group c, Group d) => Group (a, b, c, d)Defined in groups-0.5.3 · Data.Group
Group (f (g a)) => Group ((:.:) f g a)Defined in groups-0.5.3 · Data.Group
(Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e)Defined in groups-0.5.3 · Data.Group

classclass Group g => Abelian g

An Abelian group is a Group that follows the rule:

a <> b == b <> a

Instances16Abelian, …

Abelian ()Defined in groups-0.5.3 · Data.Group
Num a => Abelian (Sum a)Defined in groups-0.5.3 · Data.Group
Fractional a => Abelian (Product a)Defined in groups-0.5.3 · Data.Group
Abelian a => Abelian (Identity a)Defined in groups-0.5.3 · Data.Group
Abelian a => Abelian (Down a)Defined in groups-0.5.3 · Data.Group
Abelian a => Abelian (Dual a)Defined in groups-0.5.3 · Data.Group
Abelian (Proxy x)Defined in groups-0.5.3 · Data.Group
Abelian a => Abelian (Op a b)Defined in groups-0.5.3 · Data.Group
Abelian b => Abelian (a -> b)Defined in groups-0.5.3 · Data.Group
(Abelian a, Abelian b) => Abelian (a, b)Defined in groups-0.5.3 · Data.Group
Abelian a => Abelian (Const a x)Defined in groups-0.5.3 · Data.Group
(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c)Defined in groups-0.5.3 · Data.Group
(Abelian (f a), Abelian (g a)) => Abelian ((:*:) f g a)Defined in groups-0.5.3 · Data.Group
(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d)Defined in groups-0.5.3 · Data.Group
Abelian (f (g a)) => Abelian ((:.:) f g a)Defined in groups-0.5.3 · Data.Group
(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e)Defined in groups-0.5.3 · Data.Group

classclass Group a => Cyclic a where

A Group G is Cyclic if there exists an element x of G such that for all y in G, there exists an n, such that

y = pow x n

Instances6Cyclic

valuegenerated :: Cyclic a => [a]

Generate all elements of a Cyclic group using its generator.

Note: Fuses, does not terminate even for finite groups.

valuegenerated' :: (Eq a, Cyclic a) => [a]

Lazily generate all elements of a Cyclic group using its generator.

Note: Fuses, terminates if the underlying group is finite.