Portability	portable
Stability	provisional
Maintainer	Edward Kmett <ekmett@gmail.com>
Safe Haskell	Safe-Inferred

Data.Semigroup

Contents

Semigroups
Re-exported monoids from Data.Monoid
A better monoid for Maybe
Difference lists of a semigroup

Description

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

Synopsis

Documentation

class Semigroup a whereSource

Methods

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

An associative operation.

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

sconcat :: NonEmpty a -> aSource

Reduce a non-empty list with <>

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

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

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 <>.

Instances

Semigroup Ordering
Semigroup ()
Semigroup All
Semigroup Any
Semigroup IntSet
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)
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)
Ord k => Semigroup (Map k v)
(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 Source

Constructors

Min
Fields getMin :: a

Instances

Typeable1 Min
Bounded a => Bounded (Min a)
Eq a => Eq (Min a)
(Typeable (Min a), Data a) => Data (Min a)
(Eq (Min a), Ord a) => Ord (Min a)
Read a => Read (Min a)
Show a => Show (Min a)
(Ord a, Bounded a) => Monoid (Min a)
Ord a => Semigroup (Min a)

newtype Max a Source

Constructors

Max
Fields getMax :: a

Instances

Typeable1 Max
Bounded a => Bounded (Max a)
Eq a => Eq (Max a)
(Typeable (Max a), Data a) => Data (Max a)
(Eq (Max a), Ord a) => Ord (Max a)
Read a => Read (Max a)
Show a => Show (Max a)
(Ord a, Bounded a) => Monoid (Max a)
Ord a => Semigroup (Max a)

newtype First a Source

Use Option (First a) -- to get the behavior of First

Constructors

First
Fields getFirst :: a

Instances

Typeable1 First
Bounded a => Bounded (First a)
Eq a => Eq (First a)
(Typeable (First a), Data a) => Data (First a)
(Eq (First a), Ord a) => Ord (First a)
Read a => Read (First a)
Show a => Show (First a)
Semigroup (First a)

newtype Last a Source

Use Option (Last a) -- to get the behavior of Last

Constructors

Last
Fields getLast :: a

Instances

Typeable1 Last
Bounded a => Bounded (Last a)
Eq a => Eq (Last a)
(Typeable (Last a), Data a) => Data (Last a)
(Eq (Last a), Ord a) => Ord (Last a)
Read a => Read (Last a)
Show a => Show (Last a)
Semigroup (Last a)

newtype WrappedMonoid m Source

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid
Fields unwrapMonoid :: m

Instances

Typeable1 WrappedMonoid
Bounded m => Bounded (WrappedMonoid m)
Eq m => Eq (WrappedMonoid m)
(Typeable (WrappedMonoid m), Data m) => Data (WrappedMonoid m)
(Eq (WrappedMonoid m), Ord m) => Ord (WrappedMonoid m)
Read m => Read (WrappedMonoid m)
Show m => Show (WrappedMonoid m)
Monoid m => Monoid (WrappedMonoid m)
Monoid m => Semigroup (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.

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 IntSet
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 `ee = e` and `es = 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)
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)
(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)
(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)
(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)
(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 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 intance 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)
(Typeable (Option a), Data a) => Data (Option a)
(Eq (Option a), Ord a) => Ord (Option a)
Read a => Read (Option a)
Show a => Show (Option a)
Semigroup a => Monoid (Option a)
Semigroup a => Semigroup (Option a)

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

Difference lists of a semigroup

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

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

cycle1 :: Semigroup m => m -> mSource

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