Data.Semigroup

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

Documentation

class Semigroup a where

Minimal complete definition

Nothing

Methods

(<>) :: a -> a -> a infixr 6

An associative operation.

(a <> b) <> c = a <> (b <> c)

If a is also a Monoid we further require

(<>) = mappend

sconcat :: NonEmpty a -> a

Reduce a non-empty list with <>

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

stimes :: Integral b => b -> a -> a

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.

Since: 0.17

Instances

Semigroup Ordering

Semigroup ()

Semigroup Void

Semigroup All

Semigroup Any

Semigroup ByteString

Semigroup ByteString

Semigroup Builder

Semigroup ShortByteString

Semigroup IntSet

Semigroup Builder

Semigroup Text

Semigroup Text

Semigroup [a]

Semigroup a => Semigroup (Dual a)

Semigroup (Endo a)

Num a => Semigroup (Sum a)

Num a => Semigroup (Product a)

Semigroup (First a)

Semigroup (Last a)

Semigroup a => Semigroup (Maybe a)

Semigroup (IntMap v)

Ord a => Semigroup (Set a)

Semigroup (Seq a)

(Hashable a, Eq a) => Semigroup (HashSet 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 b => Semigroup (a -> b)

Semigroup (Either a b)

(Semigroup a, Semigroup b) => Semigroup (a, b)

Semigroup a => Semigroup (Const a b)

Semigroup (Proxy k s)

Ord k => Semigroup (Map k v)

(Hashable k, Eq k) => Semigroup (HashMap k a)

(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c)

Alternative f => Semigroup (Alt * f a)

Semigroup a => Semigroup (Tagged k s a)

(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

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

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

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

Repeat a value n times.

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

Implemented using stimes and mempty.

This is a suitable definition for an mtimes member of Monoid.

Since: 0.17

Semigroups

newtype Min a

Constructors

Min

Fields

getMin :: a

Instances

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, Bounded a) => Monoid (Min a)

NFData a => NFData (Min a)

Hashable a => Hashable (Min a)

Ord a => Semigroup (Min a)

type Rep1 Min

type Rep (Min a)

newtype Max a

Constructors

Max

Fields

getMax :: a

Instances

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, Bounded a) => Monoid (Max a)

NFData a => NFData (Max a)

Hashable a => Hashable (Max a)

Ord a => Semigroup (Max a)

type Rep1 Max

type Rep (Max a)

newtype First a

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

Constructors

First

Fields

getFirst :: a

Instances

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)

NFData a => NFData (First a)

Hashable a => Hashable (First a)

Semigroup (First a)

type Rep1 First

type Rep (First a)

newtype Last a

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

Constructors

Last

Fields

getLast :: a

Instances

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)

NFData a => NFData (Last a)

Hashable a => Hashable (Last a)

Semigroup (Last a)

type Rep1 Last

type Rep (Last a)

newtype WrappedMonoid m

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid

Fields

unwrapMonoid :: m

Instances

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 => Monoid (WrappedMonoid m)

NFData m => NFData (WrappedMonoid m)

Hashable a => Hashable (WrappedMonoid a)

Monoid m => Semigroup (WrappedMonoid m)

type Rep1 WrappedMonoid

type Rep (WrappedMonoid m)

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

Methods

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.

Instances

Monoid Ordering

Monoid ()

Monoid All

Monoid Any

Monoid ByteString

Monoid ByteString

Monoid Builder

Monoid ShortByteString

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 http://en.wikipedia.org/wiki/Monoid: "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 (Tagged k s 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.

Constructors

Dual

Fields

getDual :: a

Instances

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)

Since: 1.4.0.0

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.

Constructors

Endo

Fields

appEndo :: a -> a

Instances

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 (&&).

Constructors

All

Fields

getAll :: Bool

Instances

Bounded All

Eq All

Ord All

Read All

Show All

Generic All

Monoid All

NFData All

Since: 1.4.0.0

Semigroup All

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

newtype Any :: *

Boolean monoid under disjunction (||).

Constructors

Any

Fields

getAny :: Bool

Instances

Bounded Any

Eq Any

Ord Any

Read Any

Show Any

Generic Any

Monoid Any

NFData Any

Since: 1.4.0.0

Semigroup Any

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

newtype Sum a :: * -> *

Monoid under addition.

Constructors

Sum

Fields

getSum :: a

Instances

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)

Since: 1.4.0.0

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.

Constructors

Product

Fields

getProduct :: a

Instances

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)

Since: 1.4.0.0

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

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

Constructors

Option

Fields

getOption :: Maybe a

Instances

Monad Option

Functor Option

MonadFix Option

Applicative Option

Foldable Option

Traversable Option

Generic1 Option

Alternative Option

MonadPlus 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 => Monoid (Option a)

NFData a => NFData (Option a)

Hashable a => Hashable (Option a)

Semigroup a => Semigroup (Option a)

type Rep1 Option

type Rep (Option a)

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

Fold an Option case-wise, just like maybe.

Difference lists of a semigroup

diff :: Semigroup m => m -> Endo m

This lets you use a difference list of a Semigroup as a Monoid.

cycle1 :: Semigroup m => m -> m

A generalization of cycle to an arbitrary Semigroup. May fail to terminate for some values in some semigroups.

ArgMin, ArgMax

data Arg a b

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.

Constructors

Arg a b

Instances

Bifunctor Arg

Functor (Arg a)

Foldable (Arg a)

Traversable (Arg a)

Generic1 (Arg a)

Eq a => Eq (Arg a b)

(Data a, Data b) => Data (Arg a b)

Ord a => Ord (Arg a b)

(Read a, Read b) => Read (Arg a b)

(Show a, Show b) => Show (Arg a b)

Generic (Arg a b)

(NFData a, NFData b) => NFData (Arg a b)

(Hashable a, Hashable b) => Hashable (Arg a b)

type Rep1 (Arg a)

type Rep (Arg a b)

type ArgMin a b = Min (Arg a b)

type ArgMax a b = Max (Arg a b)