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  
Monoid ()  
Monoid Any  
Monoid All  
Monoid Lifetime 

Monoid Event  
Monoid [a]  
Monoid a => Monoid (Maybe a)  Lift a semigroup into 
Monoid a => Monoid (IO a)  
Monoid (Last a)  
Monoid (First a)  
Num a => Monoid (Product a)  
Num a => Monoid (Sum a)  
Monoid (Endo a)  
Monoid a => Monoid (Dual a)  
Semigroup a => Monoid (Option a)  
Monoid m => Monoid (WrappedMonoid m)  
(Ord a, Bounded a) => Monoid (Max a)  
(Ord a, Bounded a) => Monoid (Min a)  
Monoid a => Monoid (Identity a)  
Monoid b => Monoid (a > b)  
(Monoid a, Monoid b) => Monoid (a, b)  
Monoid (Proxy k s)  
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)  
Alternative f => Monoid (Alt * f a)  
Monoid a => Monoid (Const k a b)  
(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 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  
Functor Dual  
MonadFix Dual  
Applicative Dual  
Foldable Dual  
Traversable Dual  
Generic1 Dual  
MonadZip Dual  
Bounded a => Bounded (Dual a)  
Eq a => Eq (Dual a)  
Data a => Data (Dual a)  
Ord a => Ord (Dual a)  
Read a => Read (Dual a)  
Show a => Show (Dual a)  
Generic (Dual a)  
Semigroup a => Semigroup (Dual a)  
Monoid a => Monoid (Dual a)  
type Rep1 Dual  
type Rep (Dual a)  
The monoid of endomorphisms under composition.
Boolean monoid under conjunction (&&
).
Boolean monoid under disjunction (
).
Monoid under addition.
Monad Sum  
Functor Sum  
MonadFix Sum  
Applicative Sum  
Foldable Sum  
Traversable Sum  
Generic1 Sum  
MonadZip Sum  
Bounded a => Bounded (Sum a)  
Eq a => Eq (Sum a)  
Data a => Data (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 => Semigroup (Sum a)  
Num a => Monoid (Sum a)  
type Rep1 Sum  
type Rep (Sum a)  
Monoid under multiplication.
Product  
Fields

Monad Product  
Functor Product  
MonadFix Product  
Applicative Product  
Foldable Product  
Traversable Product  
Generic1 Product  
MonadZip Product  
Bounded a => Bounded (Product a)  
Eq a => Eq (Product a)  
Data a => Data (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 => Semigroup (Product a)  
Num a => Monoid (Product a)  
type Rep1 Product  
type Rep (Product a)  
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  
Applicative First  
Foldable First  
Traversable First  
Generic1 First  
MonadZip First  
Eq a => Eq (First a)  
Data a => Data (First a)  
Ord a => Ord (First a)  
Read a => Read (First a)  
Show a => Show (First a)  
Generic (First a)  
Semigroup (First a)  
Monoid (First a)  
type Rep1 First  
type Rep (First a)  
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  
Applicative Last  
Foldable Last  
Traversable Last  
Generic1 Last  
MonadZip Last  
Eq a => Eq (Last a)  
Data a => Data (Last a)  
Ord a => Ord (Last a)  
Read a => Read (Last a)  
Show a => Show (Last a)  
Generic (Last a)  
Semigroup (Last a)  
Monoid (Last a)  
type Rep1 Last  
type Rep (Last a)  
Monoid under <>
.
Since: 4.8.0.0
Monad f => Monad (Alt * f)  
Functor f => Functor (Alt * f)  
MonadFix f => MonadFix (Alt * f)  
Applicative f => Applicative (Alt * f)  
Generic1 (Alt * f)  
MonadPlus f => MonadPlus (Alt * f)  
Alternative f => Alternative (Alt * f)  
MonadZip f => MonadZip (Alt * f)  
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)  
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)  
Alternative f => Monoid (Alt * f a)  
type Rep1 (Alt * 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/8.0.1/docs/html/libraries/base4.9.0.0/DataMonoid.html