Data.Group.Free.Product

This module provides definitions for the FreeProduct of two groups, along with useful combinators.

Documentation

newtype FreeProduct g h

The free product on two alphabets

Note: This does not perform simplification upon multiplication or construction. To do this, one should use simplify.

Constructors

FreeProduct

Fields

• runFreeProduct :: Seq (Either g h)

Instances

Bifunctor FreeProduct

Defined in Data.Group.Free.Product

Methods

bimap :: (a -> b) -> (c -> d) -> FreeProduct a c -> FreeProduct b d

first :: (a -> b) -> FreeProduct a c -> FreeProduct b c

second :: (b -> c) -> FreeProduct a b -> FreeProduct a c

Functor (FreeProduct g)

Defined in Data.Group.Free.Product

Methods

fmap :: (a -> b) -> FreeProduct g a -> FreeProduct g b

(<$) :: a -> FreeProduct g b -> FreeProduct g a

(Eq g, Eq h) => Eq (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

(==) :: FreeProduct g h -> FreeProduct g h -> Bool

(/=) :: FreeProduct g h -> FreeProduct g h -> Bool

(Ord g, Ord h) => Ord (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

compare :: FreeProduct g h -> FreeProduct g h -> Ordering

(<) :: FreeProduct g h -> FreeProduct g h -> Bool

(<=) :: FreeProduct g h -> FreeProduct g h -> Bool

(>) :: FreeProduct g h -> FreeProduct g h -> Bool

(>=) :: FreeProduct g h -> FreeProduct g h -> Bool

max :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h

min :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h

(Show g, Show h) => Show (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

showsPrec :: Int -> FreeProduct g h -> ShowS

show :: FreeProduct g h -> String

showList :: [FreeProduct g h] -> ShowS

Semigroup (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

(<>) :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h

sconcat :: NonEmpty (FreeProduct g h) -> FreeProduct g h

stimes :: Integral b => b -> FreeProduct g h -> FreeProduct g h

Monoid (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

mempty :: FreeProduct g h

mappend :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h

mconcat :: [FreeProduct g h] -> FreeProduct g h

(Group g, Group h) => Group (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

invert :: FreeProduct g h -> FreeProduct g h

(~~) :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h

pow :: Integral x => FreeProduct g h -> x -> FreeProduct g h

(GroupOrder g, GroupOrder h) => GroupOrder (FreeProduct g h)

Defined in Data.Group.Free.Product

Methods

order :: FreeProduct g h -> Order

simplify :: (Eq g, Eq h, Monoid g, Monoid h) => FreeProduct g h -> FreeProduct g h

O(n) Simplifies a word in a FreeProduct. This means that we get rid of any identity elements, and perform multiplication of neighboring gs and hs.

coproduct :: Monoid m => (a -> m) -> (b -> m) -> FreeProduct a b -> m

The FreeProduct of two Monoids is a coproduct in the category of monoids (and by extension, the category of groups).

injl :: a -> FreeProduct a b

Left injection of an alphabet a into a free product.

injr :: b -> FreeProduct a b

Right injection of an alphabet b into a free product.