Data.Group.Permutation

This module provides definitions for Permutations along with useful combinators.

Permutation groups

data Permutation a

Isomorphism of a finite set onto itself. Each entry consists of one half of the isomorphism.

Note: It is the responsibility of the user to provide inverse proofs for to and from. Be responsible!

Constructors

Permutation

Fields

• to :: a -> a

  The forward half of the bijection

• from :: a -> a

  The inverse half of the bijection

Instances

Semigroup a => Semigroup (Permutation a)

Defined in Data.Group.Permutation

Methods

(<>) :: Permutation a -> Permutation a -> Permutation a

sconcat :: NonEmpty (Permutation a) -> Permutation a

stimes :: Integral b => b -> Permutation a -> Permutation a

Monoid a => Monoid (Permutation a)

Defined in Data.Group.Permutation

Methods

mempty :: Permutation a

mappend :: Permutation a -> Permutation a -> Permutation a

mconcat :: [Permutation a] -> Permutation a

AbelianGroup a => AbelianGroup (Permutation a)

Defined in Data.Group.Permutation

Group a => Group (Permutation a)

Defined in Data.Group.Permutation

Methods

invert :: Permutation a -> Permutation a

gtimes :: Integral n => n -> Permutation a -> Permutation a

minus :: Permutation a -> Permutation a -> Permutation a

AdditiveAbelianGroup a => AdditiveAbelianGroup (Permutation a)

Defined in Data.Group.Permutation

AdditiveGroup a => AdditiveGroup (Permutation a)

Defined in Data.Group.Permutation

MultiplicativeGroup a => MultiplicativeGroup (Permutation a)

Defined in Data.Group.Permutation

Permutation group combinators

permute :: (a -> a) -> (a -> a) -> Permutation a

Build a Permutation from a bijective pair.

pairwise :: Permutation a -> (a -> a, a -> a)

Destroy a Permutation, producing the underlying pair of bijections.

(-$) :: Permutation a -> a -> a infixr 0

Infix alias for the to half of Permutation bijection

($-) :: Permutation a -> a -> a infixr 0

Infix alias for the from half of Permutation bijection

embed :: Group g => g -> Permutation g

Embed a Group into the Permutation group on it's underlying set.

retract :: Group g => Permutation g -> g

Get a group element out of the permutation group. This is a left inverse to embed, i.e.

   retract . embed = id

Permutation patterns

pattern Permute :: Group g => Permutation g -> g

Bidirectional pattern synonym for embedding/retraction of groups into their permutation groups.