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Showing content from http://hackage.haskell.org/packages/archive/base/4.6.0.0/doc/html/Data-Monoid.html below:

Data.Monoid

Description

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

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.

newtype Endo a Source

The monoid of endomorphisms under composition.

Bool wrappers

newtype All Source

Boolean monoid under conjunction.

newtype Any Source

Boolean monoid under disjunction.

Num wrappers 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.

newtype Last a Source

Maybe monoid returning the rightmost non-Nothing value.

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