Safe Haskell  Safe 

Language  Haskell98 
Synopsis
 class Semigroup a => Monoid a where
 newtype First a = First {}
 newtype Last a = Last {}
 newtype Ap (f :: k > Type) (a :: k) :: forall k. (k > Type) > k > Type = Ap {
 getAp :: f a
 newtype Dual a = Dual {
 getDual :: a
 newtype Endo a = Endo {
 appEndo :: a > a
 newtype All = All {}
 newtype Any = Any {}
 newtype Sum a = Sum {
 getSum :: a
 newtype Product a = Product {
 getProduct :: a
 newtype Alt (f :: k > Type) (a :: k) :: forall k. (k > Type) > k > Type = Alt {
 getAlt :: f a
 (<>) :: Semigroup a => a > a > a
Documentation
class Semigroup a => Monoid a where Source #
The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:
x
<>
mempty
= xmempty
<>
x = xx
(<>
(y<>
z) = (x<>
y)<>
zSemigroup
law)mconcat
=foldr
'(<>)'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
.
NOTE: Semigroup
is a superclass of Monoid
since base4.11.0.0.
Identity of mappend
mappend :: a > a > a Source #
An associative operation
NOTE: This method is redundant and has the default
implementation
since base4.11.0.0.mappend
= '(<>)'
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  Since: base2.1 
Monoid ()  Since: base2.1 
Monoid All  Since: base2.1 
Monoid Any  Since: base2.1 
Monoid [a]  Since: base2.1 
Semigroup a => Monoid (Maybe a)  Lift a semigroup into Since 4.11.0: constraint on inner Since: base2.1 
Monoid a => Monoid (IO a)  Since: base4.9.0.0 
Monoid (Predicate a)  
Monoid (Comparison a)  
Defined in Data.Functor.Contravariant mempty :: Comparison a Source # mappend :: Comparison a > Comparison a > Comparison a Source # mconcat :: [Comparison a] > Comparison a Source #  
Monoid (Equivalence a)  
Defined in Data.Functor.Contravariant mempty :: Equivalence a Source # mappend :: Equivalence a > Equivalence a > Equivalence a Source # mconcat :: [Equivalence a] > Equivalence a Source #  
(Ord a, Bounded a) => Monoid (Min a)  Since: base4.9.0.0 
(Ord a, Bounded a) => Monoid (Max a)  Since: base4.9.0.0 
Monoid m => Monoid (WrappedMonoid m)  Since: base4.9.0.0 
Defined in Data.Semigroup mempty :: WrappedMonoid m Source # mappend :: WrappedMonoid m > WrappedMonoid m > WrappedMonoid m Source # mconcat :: [WrappedMonoid m] > WrappedMonoid m Source #  
Semigroup a => Monoid (Option a)  Since: base4.9.0.0 
Monoid a => Monoid (Identity a)  Since: base4.9.0.0 
Monoid (First a)  Since: base2.1 
Monoid (Last a)  Since: base2.1 
Monoid a => Monoid (Dual a)  Since: base2.1 
Monoid (Endo a)  Since: base2.1 
Num a => Monoid (Sum a)  Since: base2.1 
Num a => Monoid (Product a)  Since: base2.1 
Monoid b => Monoid (a > b)  Since: base2.1 
(Monoid a, Monoid b) => Monoid (a, b)  Since: base2.1 
Monoid a => Monoid (ST s a)  Since: base4.11.0.0 
Monoid a => Monoid (Op a b)  
Monoid (Proxy s)  Since: base4.7.0.0 
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)  Since: base2.1 
Monoid a => Monoid (Const a b)  Since: base4.9.0.0 
(Applicative f, Monoid a) => Monoid (Ap f a)  Since: base4.12.0.0 
Alternative f => Monoid (Alt f a)  Since: base4.8.0.0 
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)  Since: base2.1 
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)  Since: base2.1 
Maybe monoid returning the leftmost nonNothing value.
is isomorphic to First
a
, but precedes it
historically.Alt
Maybe
a
>>>
getFirst (First (Just "hello") <> First Nothing <> First (Just "world"))
Just "hello"
Use of this type is discouraged. Note the following equivalence:
Data.Monoid.First x === Maybe (Data.Semigroup.First x)
In addition to being equivalent in the structural sense, the two
also have Monoid
instances that behave the same. This type will
be marked deprecated in GHC 8.8, and removed in GHC 8.10.
Users are advised to use the variant from Data.Semigroup and wrap
it in Maybe
.
Instances
Monad First  Since: base4.8.0.0 
Functor First  Since: base4.8.0.0 
Applicative First  Since: base4.8.0.0 
Foldable First  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => First m > m Source # foldMap :: Monoid m => (a > m) > First a > m Source # foldr :: (a > b > b) > b > First a > b Source # foldr' :: (a > b > b) > b > First a > b Source # foldl :: (b > a > b) > b > First a > b Source # foldl' :: (b > a > b) > b > First a > b Source # foldr1 :: (a > a > a) > First a > a Source # foldl1 :: (a > a > a) > First a > a Source # toList :: First a > [a] Source # null :: First a > Bool Source # length :: First a > Int Source # elem :: Eq a => a > First a > Bool Source # maximum :: Ord a => First a > a Source # minimum :: Ord a => First a > a Source #  
Traversable First  Since: base4.8.0.0 
Eq a => Eq (First a)  Since: base2.1 
Ord a => Ord (First a)  Since: base2.1 
Read a => Read (First a)  Since: base2.1 
Show a => Show (First a)  Since: base2.1 
Generic (First a)  
Semigroup (First a)  Since: base4.9.0.0 
Monoid (First a)  Since: base2.1 
Generic1 First  
type Rep (First a)  Since: base4.7.0.0 
Defined in Data.Monoid  
type Rep1 First  Since: base4.7.0.0 
Defined in Data.Monoid 
Maybe monoid returning the rightmost nonNothing value.
is isomorphic to Last
a
, and thus to
Dual
(First
a)Dual
(Alt
Maybe
a)
>>>
getLast (Last (Just "hello") <> Last Nothing <> Last (Just "world"))
Just "world"
Use of this type is discouraged. Note the following equivalence:
Data.Monoid.Last x === Maybe (Data.Semigroup.Last x)
In addition to being equivalent in the structural sense, the two
also have Monoid
instances that behave the same. This type will
be marked deprecated in GHC 8.8, and removed in GHC 8.10.
Users are advised to use the variant from Data.Semigroup and wrap
it in Maybe
.
Instances
Monad Last  Since: base4.8.0.0 
Functor Last  Since: base4.8.0.0 
Applicative Last  Since: base4.8.0.0 
Foldable Last  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Last m > m Source # foldMap :: Monoid m => (a > m) > Last a > m Source # foldr :: (a > b > b) > b > Last a > b Source # foldr' :: (a > b > b) > b > Last a > b Source # foldl :: (b > a > b) > b > Last a > b Source # foldl' :: (b > a > b) > b > Last a > b Source # foldr1 :: (a > a > a) > Last a > a Source # foldl1 :: (a > a > a) > Last a > a Source # toList :: Last a > [a] Source # null :: Last a > Bool Source # length :: Last a > Int Source # elem :: Eq a => a > Last a > Bool Source # maximum :: Ord a => Last a > a Source # minimum :: Ord a => Last a > a Source #  
Traversable Last  Since: base4.8.0.0 
Eq a => Eq (Last a)  Since: base2.1 
Ord a => Ord (Last a)  Since: base2.1 
Defined in Data.Monoid  
Read a => Read (Last a)  Since: base2.1 
Show a => Show (Last a)  Since: base2.1 
Generic (Last a)  
Semigroup (Last a)  Since: base4.9.0.0 
Monoid (Last a)  Since: base2.1 
Generic1 Last  
type Rep (Last a)  Since: base4.7.0.0 
Defined in Data.Monoid  
type Rep1 Last  Since: base4.7.0.0 
Defined in Data.Monoid 
newtype Ap (f :: k > Type) (a :: k) :: forall k. (k > Type) > k > Type Source #
This data type witnesses the lifting of a Monoid
into an
Applicative
pointwise.
Since: base4.12.0.0
Instances
Generic1 (Ap f :: k > Type)  
Monad f => Monad (Ap f)  Since: base4.12.0.0 
Functor f => Functor (Ap f)  Since: base4.12.0.0 
MonadFail f => MonadFail (Ap f)  Since: base4.12.0.0 
Applicative f => Applicative (Ap f)  Since: base4.12.0.0 
Foldable f => Foldable (Ap f)  Since: base4.12.0.0 
Defined in Data.Foldable fold :: Monoid m => Ap f m > m Source # foldMap :: Monoid m => (a > m) > Ap f a > m Source # foldr :: (a > b > b) > b > Ap f a > b Source # foldr' :: (a > b > b) > b > Ap f a > b Source # foldl :: (b > a > b) > b > Ap f a > b Source # foldl' :: (b > a > b) > b > Ap f a > b Source # foldr1 :: (a > a > a) > Ap f a > a Source # foldl1 :: (a > a > a) > Ap f a > a Source # toList :: Ap f a > [a] Source # null :: Ap f a > Bool Source # length :: Ap f a > Int Source # elem :: Eq a => a > Ap f a > Bool Source # maximum :: Ord a => Ap f a > a Source # minimum :: Ord a => Ap f a > a Source #  
Traversable f => Traversable (Ap f)  Since: base4.12.0.0 
Alternative f => Alternative (Ap f)  Since: base4.12.0.0 
MonadPlus f => MonadPlus (Ap f)  Since: base4.12.0.0 
(Applicative f, Bounded a) => Bounded (Ap f a)  Since: base4.12.0.0 
Enum (f a) => Enum (Ap f a)  Since: base4.12.0.0 
Defined in Data.Monoid succ :: Ap f a > Ap f a Source # pred :: Ap f a > Ap f a Source # toEnum :: Int > Ap f a Source # fromEnum :: Ap f a > Int Source # enumFrom :: Ap f a > [Ap f a] Source # enumFromThen :: Ap f a > Ap f a > [Ap f a] Source # enumFromTo :: Ap f a > Ap f a > [Ap f a] Source # enumFromThenTo :: Ap f a > Ap f a > Ap f a > [Ap f a] Source #  
Eq (f a) => Eq (Ap f a)  Since: base4.12.0.0 
(Applicative f, Num a) => Num (Ap f a)  Since: base4.12.0.0 
Ord (f a) => Ord (Ap f a)  Since: base4.12.0.0 
Defined in Data.Monoid  
Read (f a) => Read (Ap f a)  Since: base4.12.0.0 
Show (f a) => Show (Ap f a)  Since: base4.12.0.0 
Generic (Ap f a)  
(Applicative f, Semigroup a) => Semigroup (Ap f a)  Since: base4.12.0.0 
(Applicative f, Monoid a) => Monoid (Ap f a)  Since: base4.12.0.0 
type Rep1 (Ap f :: k > Type)  Since: base4.12.0.0 
Defined in Data.Monoid  
type Rep (Ap f a)  Since: base4.12.0.0 
Defined in Data.Monoid 
The dual of a Monoid
, obtained by swapping the arguments of mappend
.
>>>
getDual (mappend (Dual "Hello") (Dual "World"))
"WorldHello"
Instances
Monad Dual  Since: base4.8.0.0 
Functor Dual  Since: base4.8.0.0 
Applicative Dual  Since: base4.8.0.0 
Foldable Dual  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Dual m > m Source # foldMap :: Monoid m => (a > m) > Dual a > m Source # foldr :: (a > b > b) > b > Dual a > b Source # foldr' :: (a > b > b) > b > Dual a > b Source # foldl :: (b > a > b) > b > Dual a > b Source # foldl' :: (b > a > b) > b > Dual a > b Source # foldr1 :: (a > a > a) > Dual a > a Source # foldl1 :: (a > a > a) > Dual a > a Source # toList :: Dual a > [a] Source # null :: Dual a > Bool Source # length :: Dual a > Int Source # elem :: Eq a => a > Dual a > Bool Source # maximum :: Ord a => Dual a > a Source # minimum :: Ord a => Dual a > a Source #  
Traversable Dual  Since: base4.8.0.0 
Bounded a => Bounded (Dual a)  Since: base2.1 
Eq a => Eq (Dual a)  Since: base2.1 
Ord a => Ord (Dual a)  Since: base2.1 
Defined in Data.Semigroup.Internal  
Read a => Read (Dual a)  Since: base2.1 
Show a => Show (Dual a)  Since: base2.1 
Generic (Dual a)  
Semigroup a => Semigroup (Dual a)  Since: base4.9.0.0 
Monoid a => Monoid (Dual a)  Since: base2.1 
Generic1 Dual  
type Rep (Dual a)  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep1 Dual  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
The monoid of endomorphisms under composition.
>>>
let computation = Endo ("Hello, " ++) <> Endo (++ "!")
>>>
appEndo computation "Haskell"
"Hello, Haskell!"
Boolean monoid under conjunction (&&
).
>>>
getAll (All True <> mempty <> All False)
False
>>>
getAll (mconcat (map (\x > All (even x)) [2,4,6,7,8]))
False
Boolean monoid under disjunction (
).
>>>
getAny (Any True <> mempty <> Any False)
True
>>>
getAny (mconcat (map (\x > Any (even x)) [2,4,6,7,8]))
True
Monoid under addition.
>>>
getSum (Sum 1 <> Sum 2 <> mempty)
3
Instances
Monad Sum  Since: base4.8.0.0 
Functor Sum  Since: base4.8.0.0 
Applicative Sum  Since: base4.8.0.0 
Foldable Sum  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Sum m > m Source # foldMap :: Monoid m => (a > m) > Sum a > m Source # foldr :: (a > b > b) > b > Sum a > b Source # foldr' :: (a > b > b) > b > Sum a > b Source # foldl :: (b > a > b) > b > Sum a > b Source # foldl' :: (b > a > b) > b > Sum a > b Source # foldr1 :: (a > a > a) > Sum a > a Source # foldl1 :: (a > a > a) > Sum a > a Source # toList :: Sum a > [a] Source # null :: Sum a > Bool Source # length :: Sum a > Int Source # elem :: Eq a => a > Sum a > Bool Source # maximum :: Ord a => Sum a > a Source # minimum :: Ord a => Sum a > a Source #  
Traversable Sum  Since: base4.8.0.0 
Bounded a => Bounded (Sum a)  Since: base2.1 
Eq a => Eq (Sum a)  Since: base2.1 
Num a => Num (Sum a)  Since: base4.7.0.0 
Ord a => Ord (Sum a)  Since: base2.1 
Defined in Data.Semigroup.Internal  
Read a => Read (Sum a)  Since: base2.1 
Show a => Show (Sum a)  Since: base2.1 
Generic (Sum a)  
Num a => Semigroup (Sum a)  Since: base4.9.0.0 
Num a => Monoid (Sum a)  Since: base2.1 
Generic1 Sum  
type Rep (Sum a)  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep1 Sum  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
Monoid under multiplication.
>>>
getProduct (Product 3 <> Product 4 <> mempty)
12
Product  

Instances
Monad Product  Since: base4.8.0.0 
Functor Product  Since: base4.8.0.0 
Applicative Product  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Foldable Product  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Product m > m Source # foldMap :: Monoid m => (a > m) > Product a > m Source # foldr :: (a > b > b) > b > Product a > b Source # foldr' :: (a > b > b) > b > Product a > b Source # foldl :: (b > a > b) > b > Product a > b Source # foldl' :: (b > a > b) > b > Product a > b Source # foldr1 :: (a > a > a) > Product a > a Source # foldl1 :: (a > a > a) > Product a > a Source # toList :: Product a > [a] Source # null :: Product a > Bool Source # length :: Product a > Int Source # elem :: Eq a => a > Product a > Bool Source # maximum :: Ord a => Product a > a Source # minimum :: Ord a => Product a > a Source #  
Traversable Product  Since: base4.8.0.0 
Defined in Data.Traversable  
Bounded a => Bounded (Product a)  Since: base2.1 
Eq a => Eq (Product a)  Since: base2.1 
Num a => Num (Product a)  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal (+) :: Product a > Product a > Product a Source # () :: Product a > Product a > Product a Source # (*) :: Product a > Product a > Product a Source # negate :: Product a > Product a Source # abs :: Product a > Product a Source # signum :: Product a > Product a Source # fromInteger :: Integer > Product a Source #  
Ord a => Ord (Product a)  Since: base2.1 
Defined in Data.Semigroup.Internal  
Read a => Read (Product a)  Since: base2.1 
Show a => Show (Product a)  Since: base2.1 
Generic (Product a)  
Num a => Semigroup (Product a)  Since: base4.9.0.0 
Num a => Monoid (Product a)  Since: base2.1 
Generic1 Product  
type Rep (Product a)  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep1 Product  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
newtype Alt (f :: k > Type) (a :: k) :: forall k. (k > Type) > k > Type Source #
Monoid under <>
.
Since: base4.8.0.0
Instances
Generic1 (Alt f :: k > Type)  
Monad f => Monad (Alt f)  Since: base4.8.0.0 
Functor f => Functor (Alt f)  Since: base4.8.0.0 
Applicative f => Applicative (Alt f)  Since: base4.8.0.0 
Foldable f => Foldable (Alt f)  Since: base4.12.0.0 
Defined in Data.Foldable fold :: Monoid m => Alt f m > m Source # foldMap :: Monoid m => (a > m) > Alt f a > m Source # foldr :: (a > b > b) > b > Alt f a > b Source # foldr' :: (a > b > b) > b > Alt f a > b Source # foldl :: (b > a > b) > b > Alt f a > b Source # foldl' :: (b > a > b) > b > Alt f a > b Source # foldr1 :: (a > a > a) > Alt f a > a Source # foldl1 :: (a > a > a) > Alt f a > a Source # toList :: Alt f a > [a] Source # null :: Alt f a > Bool Source # length :: Alt f a > Int Source # elem :: Eq a => a > Alt f a > Bool Source # maximum :: Ord a => Alt f a > a Source # minimum :: Ord a => Alt f a > a Source #  
Traversable f => Traversable (Alt f)  Since: base4.12.0.0 
Contravariant f => Contravariant (Alt f)  
Alternative f => Alternative (Alt f)  Since: base4.8.0.0 
MonadPlus f => MonadPlus (Alt f)  Since: base4.8.0.0 
Enum (f a) => Enum (Alt f a)  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal succ :: Alt f a > Alt f a Source # pred :: Alt f a > Alt f a Source # toEnum :: Int > Alt f a Source # fromEnum :: Alt f a > Int Source # enumFrom :: Alt f a > [Alt f a] Source # enumFromThen :: Alt f a > Alt f a > [Alt f a] Source # enumFromTo :: Alt f a > Alt f a > [Alt f a] Source # enumFromThenTo :: Alt f a > Alt f a > Alt f a > [Alt f a] Source #  
Eq (f a) => Eq (Alt f a)  Since: base4.8.0.0 
Num (f a) => Num (Alt f a)  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Ord (f a) => Ord (Alt f a)  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Read (f a) => Read (Alt f a)  Since: base4.8.0.0 
Show (f a) => Show (Alt f a)  Since: base4.8.0.0 
Generic (Alt f a)  
Alternative f => Semigroup (Alt f a)  Since: base4.9.0.0 
Alternative f => Monoid (Alt f a)  Since: base4.8.0.0 
type Rep1 (Alt f :: k > Type)  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
type Rep (Alt f a)  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal 