```-- | A semigroup is a binary associative operation.
module Data.Semigroup (
Semigroup((.++.))
, (<++>)
, Max(..)
, Min(..)
, BoundedMax(..)
, BoundedMin(..)
) where

import Data.Monoid
import Data.Sequence
import qualified Data.Set as S
import qualified Data.Map as M
import qualified Data.IntSet as IS
import qualified Data.IntMap as IM
import qualified Data.ByteString as B
import Control.Applicative

-- | A binary operation that must satisfy associativity. Unlike a @Monoid@, an identity in not essential.
class Semigroup a where
(.++.) :: a -> a -> a

-- | A binary associative operation lifted into an applicative functor.
(<++>) :: (Applicative f, Semigroup a) => f a -> f a -> f a
(<++>) = liftA2 (.++.)

instance Monoid a => Semigroup (Identity a) where
(.++.) = liftM2 mappend

instance Semigroup () where
_ .++. _ = ()

instance Semigroup b => Semigroup (a -> b) where
(.++.) = (<++>)

instance Semigroup a => Semigroup (IO a) where
(.++.) = (<++>)

instance (Semigroup a, Semigroup b) => Semigroup (a, b) where
(a1, b1) .++. (a2, b2) = (a1 .++. a2, b1 .++. b2)

instance (Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c) where
(a1, b1, c1) .++. (a2, b2, c2) = (a1 .++. a2, b1 .++. b2, c1 .++. c2)

instance (Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d) where
(a1, b1, c1, d1) .++. (a2, b2, c2, d2) = (a1 .++. a2, b1 .++. b2, c1 .++. c2, d1 .++. d2)

instance (Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e) where
(a1, b1, c1, d1, e1) .++. (a2, b2, c2, d2, e2) = (a1 .++. a2, b1 .++. b2, c1 .++. c2, d1 .++. d2, e1 .++. e2)

instance Semigroup Ordering where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup All where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup Any where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup a => Semigroup (Dual a) where
Dual a .++. Dual b = Dual (b .++. a)

instance Semigroup (Endo a) where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Num a => Semigroup (Product a) where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Num a => Semigroup (Sum a) where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup a => Semigroup (Maybe a) where
Nothing .++. b = b
a .++. Nothing = a
Just a .++. Just b = Just (a .++. b)

instance Semigroup (First a) where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup (Last a) where
a .++. b = runIdentity (Identity a .++. Identity b)

instance Semigroup [a] where
(.++.) = (++)

instance Semigroup (Seq a) where
(.++.) = mappend

instance Ord a => Semigroup (S.Set a) where
(.++.) = mappend

instance Ord k => Semigroup (M.Map k v) where
(.++.) = mappend

instance Semigroup IS.IntSet where
(.++.) = mappend

instance Semigroup (IM.IntMap v) where
(.++.) = mappend

instance Semigroup B.ByteString where
(.++.) = mappend

newtype Max a = Max {
runMax :: a
} deriving (Eq, Ord, Show, Read)

instance Ord a => Semigroup (Max a) where
Max a .++. Max b = Max (a `max` b)

newtype Min a = Min {
runMin :: a
} deriving (Eq, Ord, Show, Read)

instance Ord a => Semigroup (Min a) where
Min a .++. Min b = Min (a `min` b)

newtype BoundedMax a = BoundedMax {
runBoundedMax :: Maybe a
} deriving (Eq, Ord, Show, Read)

instance Ord a => Monoid (BoundedMax a) where
mempty = BoundedMax Nothing
mappend = max

newtype BoundedMin a = BoundedMin {
runBoundedMin :: Maybe a
} deriving (Eq, Ord, Show, Read)

instance Ord a => Monoid (BoundedMin a) where
mempty = BoundedMin Nothing
mappend = min

```