Copyright  (C) 20112015 Edward Kmett 

License  BSDstyle (see the file LICENSE) 
Maintainer  libraries@haskell.org 
Stability  provisional 
Portability  portable 
Safe Haskell  Trustworthy 
Language  Haskell2010 
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element. It also (originally) generalized a group (a monoid with all inverses) to a type where every element did not have to have an inverse, thus the name semigroup.
The use of (<>)
in this module conflicts with an operator with the same name that is being exported by Data.Monoid. However, this package reexports (most of) the contents of Data.Monoid, so to use semigroups and monoids in the same package just
import Data.Semigroup
Since: 4.9.0.0
class Semigroup a where Source
The class of semigroups (types with an associative binary operation).
Since: 4.9.0.0
(<>) :: a > a > a infixr 6 Source
An associative operation.
(a<>
b)<>
c = a<>
(b<>
c)
If a
is also a Monoid
we further require
(<>
) =mappend
(<>) :: Monoid a => a > a > a infixr 6 Source
An associative operation.
(a<>
b)<>
c = a<>
(b<>
c)
If a
is also a Monoid
we further require
(<>
) =mappend
sconcat :: NonEmpty a > a Source
Reduce a nonempty list with <>
The default definition should be sufficient, but this can be overridden for efficiency.
stimes :: Integral b => b > a > a Source
Repeat a value n
times.
Given that this works on a Semigroup
it is allowed to fail if you request 0 or fewer repetitions, and the default definition will do so.
By making this a member of the class, idempotent semigroups and monoids can upgrade this to execute in O(1) by picking stimes = stimesIdempotent
or stimes = stimesIdempotentMonoid
respectively.
Semigroup Ordering  Since: 4.9.0.0 
Semigroup ()  Since: 4.9.0.0 
Semigroup Any  Since: 4.9.0.0 
Semigroup All  Since: 4.9.0.0 
Semigroup Lifetime  Since: 4.10.0.0 
Semigroup Event  Since: 4.10.0.0 
Semigroup Void  Since: 4.9.0.0 
Semigroup [a]  Since: 4.9.0.0 
Semigroup a => Semigroup (Maybe a)  Since: 4.9.0.0 
Semigroup a => Semigroup (IO a)  Since: 4.10.0.0 
Semigroup (Last a)  Since: 4.9.0.0 
Semigroup (First a)  Since: 4.9.0.0 
Num a => Semigroup (Product a)  Since: 4.9.0.0 
Num a => Semigroup (Sum a)  Since: 4.9.0.0 
Semigroup (Endo a)  Since: 4.9.0.0 
Semigroup a => Semigroup (Dual a)  Since: 4.9.0.0 
Semigroup a => Semigroup (Identity a)  Since: 4.9.0.0 
Semigroup (NonEmpty a)  Since: 4.9.0.0 
Semigroup a => Semigroup (Option a)  Since: 4.9.0.0 
Monoid m => Semigroup (WrappedMonoid m)  Since: 4.9.0.0 
Semigroup (Last a)  Since: 4.9.0.0 
Semigroup (First a)  Since: 4.9.0.0 
Ord a => Semigroup (Max a)  Since: 4.9.0.0 
Ord a => Semigroup (Min a)  Since: 4.9.0.0 
Semigroup b => Semigroup (a > b)  Since: 4.9.0.0 
Semigroup (Either a b)  Since: 4.9.0.0 
(Semigroup a, Semigroup b) => Semigroup (a, b)  Since: 4.9.0.0 
Semigroup (Proxy k s)  Since: 4.9.0.0 
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c)  Since: 4.9.0.0 
Alternative f => Semigroup (Alt * f a)  Since: 4.9.0.0 
Semigroup a => Semigroup (Const k a b)  Since: 4.9.0.0 
(Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d)  Since: 4.9.0.0 
(Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e)  Since: 4.9.0.0 
stimesMonoid :: (Integral b, Monoid a) => b > a > a Source
This is a valid definition of stimes
for a Monoid
.
Unlike the default definition of stimes
, it is defined for 0 and so it should be preferred where possible.
stimesIdempotent :: Integral b => b > a > a Source
This is a valid definition of stimes
for an idempotent Semigroup
.
When x <> x = x
, this definition should be preferred, because it works in O(1) rather than O(log n).
stimesIdempotentMonoid :: (Integral b, Monoid a) => b > a > a Source
This is a valid definition of stimes
for an idempotent Monoid
.
When mappend x x = x
, this definition should be preferred, because it works in O(1) rather than O(log n)
mtimesDefault :: (Integral b, Monoid a) => b > a > a Source
Repeat a value n
times.
mtimesDefault n a = a <> a <> ... <> a  using <> (n1) times
Implemented using stimes
and mempty
.
This is a suitable definition for an mtimes
member of Monoid
.
Monad Min  Since: 4.9.0.0 
Functor Min  Since: 4.9.0.0 
MonadFix Min  Since: 4.9.0.0 
Applicative Min  Since: 4.9.0.0 
Foldable Min  Since: 4.9.0.0 
Traversable Min  Since: 4.9.0.0 
Bounded a => Bounded (Min a)  
Enum a => Enum (Min a)  Since: 4.9.0.0 
Eq a => Eq (Min a)  
Data a => Data (Min a)  
Num a => Num (Min a)  Since: 4.9.0.0 
Ord a => Ord (Min a)  
Read a => Read (Min a)  
Show a => Show (Min a)  
Generic (Min a)  
Ord a => Semigroup (Min a)  Since: 4.9.0.0 
(Ord a, Bounded a) => Monoid (Min a)  Since: 4.9.0.0 
Generic1 * Min  
type Rep (Min a)  
type Rep1 * Min  
Monad Max  Since: 4.9.0.0 
Functor Max  Since: 4.9.0.0 
MonadFix Max  Since: 4.9.0.0 
Applicative Max  Since: 4.9.0.0 
Foldable Max  Since: 4.9.0.0 
Traversable Max  Since: 4.9.0.0 
Bounded a => Bounded (Max a)  
Enum a => Enum (Max a)  Since: 4.9.0.0 
Eq a => Eq (Max a)  
Data a => Data (Max a)  
Num a => Num (Max a)  Since: 4.9.0.0 
Ord a => Ord (Max a)  
Read a => Read (Max a)  
Show a => Show (Max a)  
Generic (Max a)  
Ord a => Semigroup (Max a)  Since: 4.9.0.0 
(Ord a, Bounded a) => Monoid (Max a)  Since: 4.9.0.0 
Generic1 * Max  
type Rep (Max a)  
type Rep1 * Max  
Use Option (First a)
to get the behavior of First
from Data.Monoid.
Monad First  Since: 4.9.0.0 
Functor First  Since: 4.9.0.0 
MonadFix First  Since: 4.9.0.0 
Applicative First  Since: 4.9.0.0 
Foldable First  Since: 4.9.0.0 
Traversable First  Since: 4.9.0.0 
Bounded a => Bounded (First a)  
Enum a => Enum (First a)  Since: 4.9.0.0 
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)  Since: 4.9.0.0 
Generic1 * First  
type Rep (First a)  
type Rep1 * First  
Use Option (Last a)
to get the behavior of Last
from Data.Monoid
Monad Last  Since: 4.9.0.0 
Functor Last  Since: 4.9.0.0 
MonadFix Last  Since: 4.9.0.0 
Applicative Last  Since: 4.9.0.0 
Foldable Last  Since: 4.9.0.0 
Traversable Last  Since: 4.9.0.0 
Bounded a => Bounded (Last a)  
Enum a => Enum (Last a)  Since: 4.9.0.0 
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)  Since: 4.9.0.0 
Generic1 * Last  
type Rep (Last a)  
type Rep1 * Last  
newtype WrappedMonoid m Source
Provide a Semigroup for an arbitrary Monoid.
WrapMonoid  
Fields

Bounded m => Bounded (WrappedMonoid m)  
Enum a => Enum (WrappedMonoid a)  Since: 4.9.0.0 
Eq m => Eq (WrappedMonoid m)  
Data m => Data (WrappedMonoid m)  
Ord m => Ord (WrappedMonoid m)  
Read m => Read (WrappedMonoid m)  
Show m => Show (WrappedMonoid m)  
Generic (WrappedMonoid m)  
Monoid m => Semigroup (WrappedMonoid m)  Since: 4.9.0.0 
Monoid m => Monoid (WrappedMonoid m)  Since: 4.9.0.0 
Generic1 * WrappedMonoid  
type Rep (WrappedMonoid m)  
type Rep1 * WrappedMonoid  
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 
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  
Option
is effectively Maybe
with a better instance of Monoid
, built off of an underlying Semigroup
instead of an underlying Monoid
.
Ideally, this type would not exist at all and we would just fix the Monoid
instance of Maybe
Monad Option  Since: 4.9.0.0 
Functor Option  Since: 4.9.0.0 
MonadFix Option  Since: 4.9.0.0 
Applicative Option  Since: 4.9.0.0 
Foldable Option  Since: 4.9.0.0 
Traversable Option  Since: 4.9.0.0 
MonadPlus Option  Since: 4.9.0.0 
Alternative Option  Since: 4.9.0.0 
Eq a => Eq (Option a)  
Data a => Data (Option a)  
Ord a => Ord (Option a)  
Read a => Read (Option a)  
Show a => Show (Option a)  
Generic (Option a)  
Semigroup a => Semigroup (Option a)  Since: 4.9.0.0 
Semigroup a => Monoid (Option a)  Since: 4.9.0.0 
Generic1 * Option  
type Rep (Option a)  
type Rep1 * Option  
option :: b > (a > b) > Option a > b Source
Fold an Option
casewise, just like maybe
.
diff :: Semigroup m => m > Endo m Source
This lets you use a difference list of a Semigroup
as a Monoid
.
cycle1 :: Semigroup m => m > m Source
A generalization of cycle
to an arbitrary Semigroup
. May fail to terminate for some values in some semigroups.
Arg
isn't itself a Semigroup
in its own right, but it can be placed inside Min
and Max
to compute an arg min or arg max.
Arg a b 
Bifunctor Arg  Since: 4.9.0.0 
Bifoldable Arg  Since: 4.10.0.0 
Bitraversable Arg  Since: 4.10.0.0 
Functor (Arg a)  Since: 4.9.0.0 
Foldable (Arg a)  Since: 4.9.0.0 
Traversable (Arg a)  Since: 4.9.0.0 
Generic1 * (Arg a)  
Eq a => Eq (Arg a b)  Since: 4.9.0.0 
(Data b, Data a) => Data (Arg a b)  
Ord a => Ord (Arg a b)  Since: 4.9.0.0 
(Read b, Read a) => Read (Arg a b)  
(Show b, Show a) => Show (Arg a b)  
Generic (Arg a b)  
type Rep1 * (Arg a)  
type Rep (Arg a b)  
© 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/DataSemigroup.html