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

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 `ee = e` and `es = 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 getFirst :: Maybe a

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
Fields getLast :: Maybe a

Instances

Eq a => Eq (Last a)
Ord a => Ord (Last a)
Read a => Read (Last a)
Show a => Show (Last a)
Monoid (Last a)