Magicloud Magiclouds
I see. Then what is about Dual and Endo? Especially Endo, I completely confused....
It should help to look at the instances:
-- | The dual of a monoid, obtained by swapping the arguments of 'mappend'. newtype Dual a = Dual { getDual :: a } deriving (Eq, Ord, Read, Show, Bounded)
instance Monoid a => Monoid (Dual a) where mempty = Dual mempty Dual x `mappend` Dual y = Dual (y `mappend` x)
You can tag a monoidal value as being "Dual" and then invoking "mappend" will swap the argument order. Re: Endo:
-- | The monoid of endomorphisms under composition. newtype Endo a = Endo { appEndo :: a -> a }
instance Monoid (Endo a) where mempty = Endo id Endo f `mappend` Endo g = Endo (f . g)
It's a way of labelling functions of type a -> a ("endomorphism") as being a monoid under composition (the "." operator). A short example:
GHCi, version 6.10.4: http://www.haskell.org/ghc/ :? for help Loading package ghc-prim ... linking ... done. Loading package integer ... linking ... done. Loading package base ... linking ... done. Prelude> import Data.Monoid Prelude Data.Monoid> let f = ((+2) :: Double -> Double) Prelude Data.Monoid> let g = ((/4) :: Double -> Double) Prelude Data.Monoid> appEndo (Endo f `mappend` Endo g) 4 3.0
same as "(f . g) 4" == 4/4 + 2
Prelude Data.Monoid> appEndo (getDual (Dual (Endo f) `mappend` Dual (Endo g))) 4 1.5
same as "(g . f) 4" == (4+2)/4.
G.
--
Gregory Collins