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
 stimesMonoid :: (Integral b, Monoid a) => b > a > a
 stimesIdempotent :: Integral b => b > a > a
 stimesIdempotentMonoid :: (Integral b, Monoid a) => b > a > a
 mtimesDefault :: (Integral b, Monoid a) => b > a > a
 newtype Min a = Min {
 getMin :: a
 newtype Max a = Max {
 getMax :: a
 newtype First a = First {
 getFirst :: a
 newtype Last a = Last {
 getLast :: a
 newtype WrappedMonoid m = WrapMonoid {
 unwrapMonoid :: m
 class Monoid a where
 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 Option a = Option {}
 option :: b > (a > b) > Option a > b
 diff :: Semigroup m => m > Endo m
 cycle1 :: Semigroup m => m > m
 data Arg a b = Arg a b
 type ArgMin a b = Min (Arg a b)
 type ArgMax a b = Max (Arg a b)
Documentation
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 #
stimesIdempotent :: Integral b => b > a > a Source #
stimesIdempotentMonoid :: (Integral b, Monoid a) => b > a > a Source #
mtimesDefault :: (Integral b, Monoid a) => b > a > a Source #
Semigroups
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
to get the behavior of
Option
(First
a)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
to get the behavior of
Option
(Last
a)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  

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 #  
Reexported monoids from Data.Monoid
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
mappend :: a > a > a Source #
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 
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  

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 #  
A better monoid for Maybe
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 #  
Difference lists of a semigroup
ArgMin, ArgMax
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) #  