Data.Group.Free.Internal

This module exposes internals of FreeAbelianGroup.

Free abelian groups

newtype FreeAbelianGroup a

A representation of a free abelian group over an alphabet a.

The intuition here is group elements correspond with their positive or negative multiplicities, and as such are simplified by construction.

Examples:

>>> let single a = MkFreeAbelianGroup $ Map.singleton a 1
>>> a = single 'a'
>>> b = single 'b'
>>> a
FreeAbelianGroup $ fromList [('a',1)]
>>> a <> b
FreeAbelianGroup $ fromList [('a',1),('b',1)]
>>> a <> b == b <> a
True
>>> invert a
FreeAbelianGroup $ fromList [('a',-1)]
>>> a <> b <> invert a
FreeAbelianGroup $ fromList [('b',1)]
>>> gtimes 5 (a <> b)
FreeAbelianGroup $ fromList [('a',5),('b',5)]

Constructors

MkFreeAbelianGroup (Map a Integer)

Unsafe "raw" constructor, which does not do normalization work. Please use mkFreeAbelianGroup as it normalizes.

Instances

Eq a => Eq (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

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

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

Ord a => Ord (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

compare :: FreeAbelianGroup a -> FreeAbelianGroup a -> Ordering

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

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

(>) :: FreeAbelianGroup a -> FreeAbelianGroup a -> Bool

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

max :: FreeAbelianGroup a -> FreeAbelianGroup a -> FreeAbelianGroup a

min :: FreeAbelianGroup a -> FreeAbelianGroup a -> FreeAbelianGroup a

Show a => Show (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

showsPrec :: Int -> FreeAbelianGroup a -> ShowS

show :: FreeAbelianGroup a -> String

showList :: [FreeAbelianGroup a] -> ShowS

Ord a => Semigroup (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

(<>) :: FreeAbelianGroup a -> FreeAbelianGroup a -> FreeAbelianGroup a

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

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

Ord a => Monoid (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

mempty :: FreeAbelianGroup a

mappend :: FreeAbelianGroup a -> FreeAbelianGroup a -> FreeAbelianGroup a

mconcat :: [FreeAbelianGroup a] -> FreeAbelianGroup a

Ord a => Abelian (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Ord a => Group (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

invert :: FreeAbelianGroup a -> FreeAbelianGroup a

(~~) :: FreeAbelianGroup a -> FreeAbelianGroup a -> FreeAbelianGroup a

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

Ord a => GroupOrder (FreeAbelianGroup a)

Defined in Data.Group.Free.Internal

Methods

order :: FreeAbelianGroup a -> Order