Copyright  (c) Andy Gill 2001 (c) Oregon Graduate Institute of Science and Technology 2001 

License  BSDstyle (see the file libraries/base/LICENSE) 
Maintainer  libraries@haskell.org 
Stability  experimental 
Portability  portable 
Safe Haskell  Trustworthy 
Language  Haskell2010 
A class for monoids (types with an associative binary operation that has an identity) with various generalpurpose instances.
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 newtype
s and make those instances of Monoid
, e.g. Sum
and Product
.
Identity of mappend
An associative operation
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.
Monoid Ordering  Since: 2.1 
Monoid ()  Since: 2.1 
Monoid Any  Since: 2.1 
Monoid All  Since: 2.1 
Monoid Lifetime 
Since: 4.8.0.0 
Monoid Event  Since: 4.3.1.0 
Monoid [a]  Since: 2.1 
Monoid a => Monoid (Maybe a) 
Lift a semigroup into Since: 2.1 
Monoid a => Monoid (IO a)  Since: 4.9.0.0 
Monoid (Last a)  Since: 2.1 
Monoid (First a)  Since: 2.1 
Num a => Monoid (Product a)  Since: 2.1 
Num a => Monoid (Sum a)  Since: 2.1 
Monoid (Endo a)  Since: 2.1 
Monoid a => Monoid (Dual a)  Since: 2.1 
Monoid a => Monoid (Identity a)  
Semigroup a => Monoid (Option a)  Since: 4.9.0.0 
Monoid m => Monoid (WrappedMonoid m)  Since: 4.9.0.0 
(Ord a, Bounded a) => Monoid (Max a)  Since: 4.9.0.0 
(Ord a, Bounded a) => Monoid (Min a)  Since: 4.9.0.0 
Monoid b => Monoid (a > b)  Since: 2.1 
(Monoid a, Monoid b) => Monoid (a, b)  Since: 2.1 
Monoid (Proxy k s)  Since: 4.7.0.0 
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)  Since: 2.1 
Alternative f => Monoid (Alt * f a)  Since: 4.8.0.0 
Monoid a => Monoid (Const k a b)  
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)  Since: 2.1 
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)  Since: 2.1 
(<>) :: Monoid m => m > m > m infixr 6 Source
An infix synonym for mappend
.
Since: 4.5.0.0
The dual of a Monoid
, obtained by swapping the arguments of mappend
.
Monad Dual  Since: 4.8.0.0 
Functor Dual  Since: 4.8.0.0 
MonadFix Dual  Since: 4.8.0.0 
Applicative Dual  Since: 4.8.0.0 
Foldable Dual  Since: 4.8.0.0 
Traversable Dual  Since: 4.8.0.0 
MonadZip Dual  Since: 4.8.0.0 
Bounded a => Bounded (Dual a)  
Eq a => Eq (Dual a)  
Data a => Data (Dual a)  Since: 4.8.0.0 
Ord a => Ord (Dual a)  
Read a => Read (Dual a)  
Show a => Show (Dual a)  
Generic (Dual a)  
Semigroup a => Semigroup (Dual a)  Since: 4.9.0.0 
Monoid a => Monoid (Dual a)  Since: 2.1 
Generic1 * Dual  
type Rep (Dual a)  
type Rep1 * Dual  
The monoid of endomorphisms under composition.
Boolean monoid under conjunction (&&
).
Boolean monoid under disjunction (
).
Monoid under addition.
Monad Sum  Since: 4.8.0.0 
Functor Sum  Since: 4.8.0.0 
MonadFix Sum  Since: 4.8.0.0 
Applicative Sum  Since: 4.8.0.0 
Foldable Sum  Since: 4.8.0.0 
Traversable Sum  Since: 4.8.0.0 
MonadZip Sum  Since: 4.8.0.0 
Bounded a => Bounded (Sum a)  
Eq a => Eq (Sum a)  
Data a => Data (Sum a)  Since: 4.8.0.0 
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 => Semigroup (Sum a)  Since: 4.9.0.0 
Num a => Monoid (Sum a)  Since: 2.1 
Generic1 * Sum  
type Rep (Sum a)  
type Rep1 * Sum  
Monoid under multiplication.
Product  
Fields

Monad Product  Since: 4.8.0.0 
Functor Product  Since: 4.8.0.0 
MonadFix Product  Since: 4.8.0.0 
Applicative Product  Since: 4.8.0.0 
Foldable Product  Since: 4.8.0.0 
Traversable Product  Since: 4.8.0.0 
MonadZip Product  Since: 4.8.0.0 
Bounded a => Bounded (Product a)  
Eq a => Eq (Product a)  
Data a => Data (Product a)  Since: 4.8.0.0 
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 => Semigroup (Product a)  Since: 4.9.0.0 
Num a => Monoid (Product a)  Since: 2.1 
Generic1 * Product  
type Rep (Product a)  
type Rep1 * Product  
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))
Maybe monoid returning the leftmost nonNothing value.
First a
is isomorphic to Alt Maybe a
, but precedes it historically.
Monad First  
Functor First  
MonadFix First  Since: 4.8.0.0 
Applicative First  
Foldable First  Since: 4.8.0.0 
Traversable First  Since: 4.8.0.0 
MonadZip First  Since: 4.8.0.0 
Eq a => Eq (First a)  
Data a => Data (First a)  Since: 4.8.0.0 
Ord a => Ord (First a)  
Read a => Read (First a)  
Show a => Show (First a)  
Generic (First a)  
Semigroup (First a)  Since: 4.9.0.0 
Monoid (First a)  Since: 2.1 
Generic1 * First  
type Rep (First a)  
type Rep1 * First  
Maybe monoid returning the rightmost nonNothing value.
Last a
is isomorphic to Dual (First a)
, and thus to Dual (Alt Maybe a)
Monad Last  
Functor Last  
MonadFix Last  Since: 4.8.0.0 
Applicative Last  
Foldable Last  Since: 4.8.0.0 
Traversable Last  Since: 4.8.0.0 
MonadZip Last  Since: 4.8.0.0 
Eq a => Eq (Last a)  
Data a => Data (Last a)  Since: 4.8.0.0 
Ord a => Ord (Last a)  
Read a => Read (Last a)  
Show a => Show (Last a)  
Generic (Last a)  
Semigroup (Last a)  Since: 4.9.0.0 
Monoid (Last a)  Since: 2.1 
Generic1 * Last  
type Rep (Last a)  
type Rep1 * Last  
Monoid under <>
.
Since: 4.8.0.0
Generic1 k (Alt k f)  
Monad f => Monad (Alt * f)  
Functor f => Functor (Alt * f)  
MonadFix f => MonadFix (Alt * f)  Since: 4.8.0.0 
Applicative f => Applicative (Alt * f)  
MonadPlus f => MonadPlus (Alt * f)  
Alternative f => Alternative (Alt * f)  
MonadZip f => MonadZip (Alt * f)  Since: 4.8.0.0 
Enum (f a) => Enum (Alt k f a)  
Eq (f a) => Eq (Alt k f a)  
(Data (f a), Data a, Typeable (* > *) f) => Data (Alt * f a)  Since: 4.8.0.0 
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 => Semigroup (Alt * f a)  Since: 4.9.0.0 
Alternative f => Monoid (Alt * f a)  Since: 4.8.0.0 
type Rep1 k (Alt k f)  
type Rep (Alt k f a)  
© The University of Glasgow and others
Licensed under a BSDstyle license (see top of the page).
https://downloads.haskell.org/~ghc/8.2.1/docs/html/libraries/base4.10.0.0/DataMonoid.html