semigroups- Anything that associates

Copyright(C) 2011-2015 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <>
Safe HaskellTrustworthy




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 re-exports (most of) the contents of Data.Monoid, so to use semigroups and monoids in the same package just

import Data.Semigroup



class Semigroup a where Source

Minimal complete definition



(<>) :: 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 non-empty list with <>

The default definition should be sufficient, but this can be overridden for efficiency.

times1p :: Natural -> a -> a Source

Repeat a value (n + 1) times.

times1p n a = a <> a <> ... <> a  -- using <> n times

The default definition uses peasant multiplication, exploiting associativity to only require O(log n) uses of <>.

See also timesN.


newtype Min a Source




getMin :: a


newtype Max a Source




getMax :: a


newtype First a Source

Use Option (First a) to get the behavior of First from Data.Monoid.




getFirst :: a

newtype Last a Source

Use Option (Last a) to get the behavior of Last from Data.Monoid




getLast :: a

timesN :: Monoid a => Natural -> a -> a Source

Repeat a value n times.

timesN n a = a <> a <> ... <> a  -- using <> (n-1) times

Implemented using times1p.

Re-exported monoids from Data.Monoid

class Monoid a where

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 newtypes and make those instances of Monoid, e.g. Sum and Product.

Minimal complete definition

mempty, mappend


mempty :: a

Identity of mappend

mappend :: a -> a -> a

An associative operation

mconcat :: [a] -> a

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 All 
Monoid Any 
Monoid Builder 
Monoid ByteString 
Monoid ShortByteString 
Monoid ByteString 
Monoid IntSet 
Monoid Builder 
Monoid [a] 
Ord a => Monoid (Max a) 
Ord a => Monoid (Min a) 
Monoid a => Monoid (Dual a) 
Monoid (Endo a) 
Num a => Monoid (Sum a) 
Num a => Monoid (Product a) 
Monoid (First a) 
Monoid (Last a) 
Monoid a => Monoid (Maybe a)

Lift a semigroup into Maybe forming a Monoid according to "Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e*e = e and e*s = s = s*e for all s ∈ S." Since there is no "Semigroup" typeclass providing just mappend, we use Monoid instead.

Monoid (IntMap a) 
Ord a => Monoid (Set a) 
Monoid (Seq a) 
(Hashable a, Eq a) => Monoid (HashSet 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 b => Monoid (a -> b) 
(Monoid a, Monoid b) => Monoid (a, b) 
Monoid a => Monoid (Const a b) 
Monoid (Proxy k s) 
Ord k => Monoid (Map k v) 
(Eq k, Hashable k) => Monoid (HashMap k v) 
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) 
Alternative f => Monoid (Alt * f a) 
(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) 

newtype Dual a :: * -> *

The dual of a Monoid, obtained by swapping the arguments of mappend.




getDual :: a


Generic1 Dual 
Bounded a => Bounded (Dual a) 
Eq a => Eq (Dual a) 
Ord a => Ord (Dual a) 
Read a => Read (Dual a) 
Show a => Show (Dual a) 
Generic (Dual a) 
Monoid a => Monoid (Dual a) 
NFData a => NFData (Dual a)


Semigroup a => Semigroup (Dual a) 
type Rep1 Dual = D1 D1Dual (C1 C1_0Dual (S1 S1_0_0Dual Par1)) 
type Rep (Dual a) = D1 D1Dual (C1 C1_0Dual (S1 S1_0_0Dual (Rec0 a))) 

newtype Endo a :: * -> *

The monoid of endomorphisms under composition.




appEndo :: a -> a


Generic (Endo a) 
Monoid (Endo a) 
Semigroup (Endo a) 
type Rep (Endo a) = D1 D1Endo (C1 C1_0Endo (S1 S1_0_0Endo (Rec0 (a -> a)))) 

newtype All :: *

Boolean monoid under conjunction (&&).




getAll :: Bool


Bounded All 
Eq All 
Ord All 
Read All 
Show All 
Generic All 
Monoid All 
NFData All


Semigroup All 
type Rep All = D1 D1All (C1 C1_0All (S1 S1_0_0All (Rec0 Bool))) 

newtype Any :: *

Boolean monoid under disjunction (||).




getAny :: Bool


Bounded Any 
Eq Any 
Ord Any 
Read Any 
Show Any 
Generic Any 
Monoid Any 
NFData Any


Semigroup Any 
type Rep Any = D1 D1Any (C1 C1_0Any (S1 S1_0_0Any (Rec0 Bool))) 

newtype Sum a :: * -> *

Monoid under addition.




getSum :: a


Generic1 Sum 
Bounded a => Bounded (Sum a) 
Eq a => Eq (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 => Monoid (Sum a) 
NFData a => NFData (Sum a)


Num a => Semigroup (Sum a) 
type Rep1 Sum = D1 D1Sum (C1 C1_0Sum (S1 S1_0_0Sum Par1)) 
type Rep (Sum a) = D1 D1Sum (C1 C1_0Sum (S1 S1_0_0Sum (Rec0 a))) 

newtype Product a :: * -> *

Monoid under multiplication.




getProduct :: a


Generic1 Product 
Bounded a => Bounded (Product a) 
Eq a => Eq (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 => Monoid (Product a) 
NFData a => NFData (Product a)


Num a => Semigroup (Product a) 
type Rep1 Product = D1 D1Product (C1 C1_0Product (S1 S1_0_0Product Par1)) 
type Rep (Product a) = D1 D1Product (C1 C1_0Product (S1 S1_0_0Product (Rec0 a))) 

A better monoid for Maybe

newtype Option a Source

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




getOption :: Maybe a

option :: b -> (a -> b) -> Option a -> b Source

Fold an Option case-wise, just like maybe.

Difference lists of a semigroup

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.

ArgMin, ArgMax

data Arg a b Source

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 Source 
Functor (Arg a) Source 
Foldable (Arg a) Source 
Traversable (Arg a) Source 
Generic1 (Arg a) Source 
Eq a => Eq (Arg a b) Source 
(Data a, Data b) => Data (Arg a b) Source 
Ord a => Ord (Arg a b) Source 
(Read a, Read b) => Read (Arg a b) Source 
(Show a, Show b) => Show (Arg a b) Source 
Generic (Arg a b) Source 
(NFData a, NFData b) => NFData (Arg a b) Source 
(Hashable a, Hashable b) => Hashable (Arg a b) Source 
type Rep1 (Arg a) Source 
type Rep (Arg a b) Source 

type ArgMin a b = Min (Arg a b) Source

type ArgMax a b = Max (Arg a b) Source