Copyright	(c) Andy Gill 2001, (c) Oregon Graduate Institute of Science and Technology, 2001
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	[email protected]
Stability	experimental
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Monoid typeclass

class Monoid a where Source

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.

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

An infix synonym for mappend.

Since: 4.5.0.0

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

newtype Sum a Source

Monoid under addition.

newtype Product a Source

Monoid under multiplication.

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.

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

newtype Last a Source

Maybe monoid returning the rightmost non-Nothing value.

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

Alternative wrapper

newtype Alt f a Source

Monoid under <|>.

Since: 4.8.0.0