Data.Monoid

A class for monoids (types with an associative binary operation that has an identity) with various general-purpose instances.

Monoid typeclass

class Monoid a where

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.

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.

Minimal complete definition

mempty, mappend

Methods

mempty :: a

Identity of mappend

mappend :: a -> a -> a

An associative operation

mconcat :: [a] -> a

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 (Proxy k s)

Monoid a => Monoid (Const a b)

(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)

Alternative f => Monoid (Alt * f a)

(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)

(<>) :: Monoid m => m -> m -> m infixr 6

An infix synonym for mappend.

Since: 4.5.0.0

newtype Dual a

The dual of a Monoid, obtained by swapping the arguments of mappend.

Constructors

Dual

Fields

getDual :: a

Instances

Generic1 Dual

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)

Generic (Dual a)

Monoid a => Monoid (Dual a)

type Rep1 Dual

type Rep (Dual a)

newtype Endo a

The monoid of endomorphisms under composition.

Constructors

Endo

Fields

appEndo :: a -> a

Instances

Generic (Endo a)

Monoid (Endo a)

type Rep (Endo a)

Bool wrappers

newtype All

Boolean monoid under conjunction (&&).

Constructors

All

Fields

getAll :: Bool

Instances

Bounded All

Eq All

Ord All

Read All

Show All

Generic All

Monoid All

type Rep All

newtype Any

Boolean monoid under disjunction (||).

Constructors

Any

Fields

getAny :: Bool

Instances

Bounded Any

Eq Any

Ord Any

Read Any

Show Any

Generic Any

Monoid Any

type Rep Any

Num wrappers

newtype Sum a

Monoid under addition.

Constructors

Sum

Fields

getSum :: a

Instances

Generic1 Sum

Bounded a => Bounded (Sum a)

Eq a => Eq (Sum a)

Num a => Num (Sum a)

Ord a => Ord (Sum a)

Read a => Read (Sum a)

Show a => Show (Sum a)

Generic (Sum a)

Num a => Monoid (Sum a)

type Rep1 Sum

type Rep (Sum a)

newtype Product a

Monoid under multiplication.

Constructors

Product

Fields

getProduct :: a

Instances

Generic1 Product

Bounded a => Bounded (Product a)

Eq a => Eq (Product a)

Num a => Num (Product a)

Ord a => Ord (Product a)

Read a => Read (Product a)

Show a => Show (Product a)

Generic (Product a)

Num a => Monoid (Product a)

type Rep1 Product

type Rep (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

Maybe monoid returning the leftmost non-Nothing value.

First a is isomorphic to Alt Maybe a, but precedes it historically.

Constructors

First

Fields

getFirst :: Maybe a

Instances

Monad First

Functor First

Applicative First

Generic1 First

Eq a => Eq (First a)

Ord a => Ord (First a)

Read a => Read (First a)

Show a => Show (First a)

Generic (First a)

Monoid (First a)

type Rep1 First

type Rep (First a)

newtype Last a

Maybe monoid returning the rightmost non-Nothing value.

Last a is isomorphic to Dual (First a), and thus to Dual (Alt Maybe a)

Constructors

Last

Fields

getLast :: Maybe a

Instances

Monad Last

Functor Last

Applicative Last

Generic1 Last

Eq a => Eq (Last a)

Ord a => Ord (Last a)

Read a => Read (Last a)

Show a => Show (Last a)

Generic (Last a)

Monoid (Last a)

type Rep1 Last

type Rep (Last a)

Alternative wrapper

newtype Alt f a

Monoid under <|>.

Since: 4.8.0.0

Constructors

Alt

Fields

getAlt :: f a

Instances

Monad f => Monad (Alt * f)

Functor f => Functor (Alt * f)

Applicative f => Applicative (Alt * f)

Generic1 (Alt * f)

MonadPlus f => MonadPlus (Alt * f)

Alternative f => Alternative (Alt * f)

Enum (f a) => Enum (Alt k f a)

Eq (f a) => Eq (Alt k f a)

Num (f a) => Num (Alt k f a)

Ord (f a) => Ord (Alt k f a)

Read (f a) => Read (Alt k f a)

Show (f a) => Show (Alt k f a)

Generic (Alt k f a)

Alternative f => Monoid (Alt * f a)

type Rep1 (Alt k f)

type Rep (Alt k f a)