Portability	portable
Stability	experimental
Maintainer	libraries@haskell.org

Data.Monoid

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.

Instances

Monoid Ordering

Monoid ()

Monoid Any

Monoid All

Monoid Event

Monoid [a]

Monoid a => Monoid (Maybe a)

Lift a semigroup into Maybe forming a Monoid according to http://en.wikipedia.org/wiki/Monoid: "Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e*e = e and e*s = s = s*e for all s S." Since there is no "Semigroup" typeclass providing just mappend , we use Monoid instead.

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

Fields

getDual :: a

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

Fields

appEndo :: a -> a

Instances

Monoid (Endo a)

Bool wrappers

newtype All Source

Boolean monoid under conjunction.

Constructors

All

Fields

getAll :: Bool

Instances

Bounded All

Eq All

Ord All

Read All

Show All

Monoid All

newtype Any Source

Boolean monoid under disjunction.

Constructors

Any

Fields

getAny :: Bool

Instances

Bounded Any

Eq Any

Ord Any

Read Any

Show Any

Monoid Any

Num wrappers

newtype Sum a Source

Monoid under addition.

Constructors

Sum

Fields

getSum :: a

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

Fields

getProduct :: a

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

Fields