Data.Group.Permutation

This module provides definitions for Permutations along with useful combinators.

Permutation groups

data Permutation a

Isomorphism of a type 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!

Examples:

>>> p1 = permute succ pred :: Permutation Integer
>>> p2 = permute negate negate :: Permutation Integer
>>> to (p1 <> p2) 2
-1
>>> from (p1 <> p2) (-1)
2
>>> to (p2 <> p1) 2
-3

Permutations on a finite set a (, indicated by satisfying (Bounded a, Enum a) constraint,) can be tested their equality and computed their orders.

>>> c1 = permute not not :: Permutation Bool
>>> c1 <> c1 == mempty
True
>>> order c1
Finite 2

Constructors

Permutation

Fields

• to :: a -> a

  The forward half of the bijection

• from :: a -> a

  The inverse half of the bijection

Instances

(Enum a, Bounded a) => Eq (Permutation a)

Defined in Data.Group.Permutation

Methods

(==) :: Permutation a -> Permutation a -> Bool

(/=) :: Permutation a -> Permutation a -> Bool

(Enum a, Bounded a) => Ord (Permutation a)

Defined in Data.Group.Permutation

Methods

compare :: Permutation a -> Permutation a -> Ordering

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

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

(>) :: Permutation a -> Permutation a -> Bool

(>=) :: Permutation a -> Permutation a -> Bool

max :: Permutation a -> Permutation a -> Permutation a

min :: Permutation a -> Permutation a -> Permutation 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 (Permutation a)

Defined in Data.Group.Permutation

Methods

mempty :: Permutation a

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

mconcat :: [Permutation a] -> Permutation a

Group (Permutation a)

Defined in Data.Group.Permutation

Methods

invert :: Permutation a -> Permutation a

(~~) :: Permutation a -> Permutation a -> Permutation a

pow :: Integral x => Permutation a -> x -> Permutation a

(Enum a, Bounded a) => GroupOrder (Permutation a)

Defined in Data.Group.Permutation

Methods

order :: Permutation a -> Order

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.