base-4.2.0.1: Basic librariesSource codeContentsIndex
Data.Monoid
Portabilityportable
Stabilityexperimental
Maintainerlibraries@haskell.org
Contents
Monoid typeclass
Bool wrappers
Num wrappers
Maybe wrappers
Description
A class for monoids (types with an associative binary operation that has an identity) with various general-purpose instances.
Synopsis
class Monoid a where
mempty :: a
mappend :: a -> a -> a
mconcat :: [a] -> a
newtype Dual a = Dual {
getDual :: a
}
newtype Endo a = Endo {
appEndo :: a -> a
}
newtype All = All {
getAll :: Bool
}
newtype Any = Any {
getAny :: Bool
}
newtype Sum a = Sum {
getSum :: a
}
newtype Product a = Product {
getProduct :: a
}
newtype First a = First {
getFirst :: Maybe a
}
newtype Last a = Last {
getLast :: Maybe a
}
Monoid typeclass
class Monoid a whereSource

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:

  • mappend mempty x = x
  • mappend x mempty = x
  • mappend x (mappend y z) = mappend (mappend x y) z
  • mconcat = foldr mappend mempty

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Minimal complete definition: mempty and mappend.

Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid, e.g. Sum and Product.

Methods
mempty :: aSource
Identity of mappend
mappend :: a -> a -> aSource
An associative operation
mconcat :: [a] -> aSource
Fold a list using the monoid. For most types, the default definition for mconcat will be used, but the function is included in the class definition so that an optimized version can be provided for specific types.
show/hide Instances
Monoid Ordering
Monoid ()
Monoid Any
Monoid All
Monoid [a]
Monoid a => Monoid (Maybe a)
Monoid (Last a)
Monoid (First a)
Num a => Monoid (Product a)
Num a => Monoid (Sum a)
Monoid (Endo a)
Monoid a => Monoid (Dual a)
Monoid b => Monoid (a -> b)
(Monoid a, Monoid b) => Monoid (a, b)
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)
newtype Dual a Source
The dual of a monoid, obtained by swapping the arguments of mappend.
Constructors
Dual
getDual :: a
show/hide Instances
Bounded a => Bounded (Dual a)
Eq a => Eq (Dual a)
Ord a => Ord (Dual a)
Read a => Read (Dual a)
Show a => Show (Dual a)
Monoid a => Monoid (Dual a)
newtype Endo a Source
The monoid of endomorphisms under composition.
Constructors
Endo
appEndo :: a -> a
show/hide Instances
Bool wrappers
newtype All Source
Boolean monoid under conjunction.
Constructors
All
getAll :: Bool
show/hide Instances
newtype Any Source
Boolean monoid under disjunction.
Constructors
Any
getAny :: Bool
show/hide Instances
Num wrappers
newtype Sum a Source
Monoid under addition.
Constructors
Sum
getSum :: a
show/hide Instances
Bounded a => Bounded (Sum a)
Eq a => Eq (Sum a)
Ord a => Ord (Sum a)
Read a => Read (Sum a)
Show a => Show (Sum a)
Num a => Monoid (Sum a)
newtype Product a Source
Monoid under multiplication.
Constructors
Product
getProduct :: a
show/hide Instances
Bounded a => Bounded (Product a)
Eq a => Eq (Product a)
Ord a => Ord (Product a)
Read a => Read (Product a)
Show a => Show (Product a)
Num a => Monoid (Product a)
Maybe wrappers

To implement find or findLast on any Foldable:

 findLast :: Foldable t => (a -> Bool) -> t a -> Maybe a
 findLast pred = getLast . foldMap (x -> if pred x
                                            then Last (Just x)
                                            else Last Nothing)

Much of Data.Map's interface can be implemented with Data.Map.alter. Some of the rest can be implemented with a new alterA function and either First or Last:

 alterA :: (Applicative f, Ord k) =>
           (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

 instance Monoid a => Applicative ((,) a)  -- from Control.Applicative
 insertLookupWithKey :: Ord k => (k -> v -> v -> v) -> k -> v
                     -> Map k v -> (Maybe v, Map k v)
 insertLookupWithKey combine key value =
   Arrow.first getFirst . alterA doChange key
   where
   doChange Nothing = (First Nothing, Just value)
   doChange (Just oldValue) =
     (First (Just oldValue),
      Just (combine key value oldValue))
newtype First a Source
Maybe monoid returning the leftmost non-Nothing value.
Constructors
First
getFirst :: Maybe a
show/hide Instances
Eq a => Eq (First a)
Ord a => Ord (First a)
Read a => Read (First a)
Show a => Show (First a)
Monoid (First a)
newtype Last a Source
Maybe monoid returning the rightmost non-Nothing value.
Constructors
Last
getLast :: Maybe a
show/hide Instances
Eq a => Eq (Last a)
Ord a => Ord (Last a)
Read a => Read (Last a)
Show a => Show (Last a)
Monoid (Last a)
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