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

Methods

(<>) :: a -> a -> a

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.

times1p :: Whole n => n -> a -> a

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.

Instances

Semigroup Ordering

Semigroup ()

Semigroup All

Semigroup Any

Semigroup ByteString

Semigroup ByteString

Semigroup IntSet

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 (Seq a)

Semigroup (IntMap v)

Ord a => Semigroup (Set 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)

Ord k => Semigroup (Map k v)

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

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

(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)

Semigroups

newtype Min a

Constructors

Min

Fields

getMin :: a

Instances

Monad Min

Functor Min

Typeable1 Min

MonadFix Min

Applicative Min

Foldable Min

Traversable Min

Bounded a => Bounded (Min a)

Enum a => Enum (Min a)

Eq a => Eq (Min a)

Data a => Data (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)

Hashable a => Hashable (Min a)

Ord a => Semigroup (Min a)

newtype Max a

Constructors

Max

Fields

getMax :: a

Instances

Monad Max

Functor Max

Typeable1 Max

MonadFix Max

Applicative Max

Foldable Max

Traversable Max

Bounded a => Bounded (Max a)

Enum a => Enum (Max a)

Eq a => Eq (Max a)

Data a => Data (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)

Hashable a => Hashable (Max a)

Ord a => Semigroup (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

Typeable1 First

MonadFix First

Applicative First

Foldable First

Traversable 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)

Hashable a => Hashable (First a)

Semigroup (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

Typeable1 Last

MonadFix Last

Applicative Last

Foldable Last

Traversable 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)

Hashable a => Hashable (Last a)

Semigroup (Last a)

newtype WrappedMonoid m

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid

Fields

unwrapMonoid :: m

Instances

Typeable1 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)

Hashable a => Hashable (WrappedMonoid a)

Monoid m => Semigroup (WrappedMonoid m)

timesN :: (Whole n, Monoid a) => n -> a -> a

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.

Minimal complete definition: mempty and mappend.

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.

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 IntSet

Monoid Text

Monoid Text

Monoid [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 (Seq a)

Monoid (IntMap a)

Ord a => Monoid (Set 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)

Ord k => Monoid (Map k v)

(Eq k, Hashable k) => Monoid (HashMap k v)

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

(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

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)

Monoid a => Monoid (Dual a)

Semigroup a => Semigroup (Dual a)

newtype Endo a

The monoid of endomorphisms under composition.

Constructors

Endo

Fields

appEndo :: a -> a

Instances

Monoid (Endo a)

Semigroup (Endo a)

newtype All

Boolean monoid under conjunction.

Constructors

All

Fields

getAll :: Bool

Instances

Bounded All

Eq All

Ord All

Read All

Show All

Monoid All

Semigroup All

newtype Any

Boolean monoid under disjunction.

Constructors

Any

Fields

getAny :: Bool

Instances

Bounded Any

Eq Any

Ord Any

Read Any

Show Any

Monoid Any

Semigroup Any

newtype Sum a

Monoid under addition.

Constructors

Sum

Fields

getSum :: a

Instances

Bounded a => Bounded (Sum a)

Eq a => Eq (Sum a)

Ord a => Ord (Sum a)

Read a => Read (Sum a)

Show a => Show (Sum a)

Num a => Monoid (Sum a)

Num a => Semigroup (Sum a)

newtype Product a

Monoid under multiplication.

Constructors

Product

Fields

getProduct :: a

Instances

Bounded a => Bounded (Product a)

Eq a => Eq (Product a)

Ord a => Ord (Product a)

Read a => Read (Product a)

Show a => Show (Product a)

Num a => Monoid (Product a)

Num a => Semigroup (Product 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

Typeable1 Option

MonadFix Option

MonadPlus Option

Applicative Option

Foldable Option

Traversable 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 => Monoid (Option a)

Hashable a => Hashable (Option a)

Semigroup a => Semigroup (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.