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  
Semigroup ()  
Semigroup Any  
Semigroup All  
Semigroup Void  
Semigroup [a]  
Semigroup a => Semigroup (Maybe a)  
Semigroup (Last a)  
Semigroup (First a)  
Num a => Semigroup (Product a)  
Num a => Semigroup (Sum a)  
Semigroup (Endo a)  
Semigroup a => Semigroup (Dual a)  
Semigroup (NonEmpty a)  
Semigroup a => Semigroup (Option a)  
Monoid m => Semigroup (WrappedMonoid m)  
Semigroup (Last a)  
Semigroup (First a)  
Ord a => Semigroup (Max a)  
Ord a => Semigroup (Min a)  
Semigroup a => Semigroup (Identity a)  
Semigroup b => Semigroup (a > b)  
Semigroup (Either a b)  
(Semigroup a, Semigroup b) => Semigroup (a, b)  
Semigroup (Proxy k s)  
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c)  
Alternative f => Semigroup (Alt * f a)  
Semigroup a => Semigroup (Const k a b)  
(Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d)  
(Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e)  
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  
Functor Min  
MonadFix Min  
Applicative Min  
Foldable Min  
Traversable Min  
Generic1 Min  
Bounded a => Bounded (Min a)  
Enum a => Enum (Min a)  
Eq a => Eq (Min a)  
Data a => Data (Min a)  
Num a => Num (Min a)  
Ord a => Ord (Min a)  
Read a => Read (Min a)  
Show a => Show (Min a)  
Generic (Min a)  
Ord a => Semigroup (Min a)  
(Ord a, Bounded a) => Monoid (Min a)  
type Rep1 Min  
type Rep (Min a)  
Monad Max  
Functor Max  
MonadFix Max  
Applicative Max  
Foldable Max  
Traversable Max  
Generic1 Max  
Bounded a => Bounded (Max a)  
Enum a => Enum (Max a)  
Eq a => Eq (Max a)  
Data a => Data (Max a)  
Num a => Num (Max a)  
Ord a => Ord (Max a)  
Read a => Read (Max a)  
Show a => Show (Max a)  
Generic (Max a)  
Ord a => Semigroup (Max a)  
(Ord a, Bounded a) => Monoid (Max a)  
type Rep1 Max  
type Rep (Max a)  
Use Option (First a)
to get the behavior of First
from Data.Monoid.
Monad First  
Functor First  
MonadFix First  
Applicative First  
Foldable First  
Traversable First  
Generic1 First  
Bounded a => Bounded (First a)  
Enum a => Enum (First a)  
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)  
type Rep1 First  
type Rep (First a)  
Use Option (Last a)
to get the behavior of Last
from Data.Monoid
Monad Last  
Functor Last  
MonadFix Last  
Applicative Last  
Foldable Last  
Traversable Last  
Generic1 Last  
Bounded a => Bounded (Last a)  
Enum a => Enum (Last a)  
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)  
type Rep1 Last  
type Rep (Last a)  
newtype WrappedMonoid m Source
Provide a Semigroup for an arbitrary Monoid.
WrapMonoid  
Fields

Generic1 WrappedMonoid  
Bounded a => Bounded (WrappedMonoid a)  
Enum a => Enum (WrappedMonoid a)  
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)  
Monoid m => Monoid (WrappedMonoid m)  
type Rep1 WrappedMonoid  
type Rep (WrappedMonoid m)  
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)  
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)  
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  
Functor Option  
MonadFix Option  
Applicative Option  
Foldable Option  
Traversable Option  
Generic1 Option  
MonadPlus Option  
Alternative Option  
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)  
Semigroup a => Monoid (Option a)  
type Rep1 Option  
type Rep (Option a)  
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 
© 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/DataSemigroup.html