Copyright	(C) 2011-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell98

Data.Semigroup

Contents

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

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 where Source

Minimal complete definition

Nothing

Methods

(<>) :: 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.

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.

Since: 0.17

Instances

Semigroup Ordering Source
Semigroup () Source
Semigroup Void Source
Semigroup All Source
Semigroup Any Source
Semigroup ByteString Source
Semigroup ByteString Source
Semigroup Builder Source
Semigroup ShortByteString Source
Semigroup IntSet Source
Semigroup Builder Source
Semigroup Text Source
Semigroup Text Source
Semigroup [a] Source
Semigroup a => Semigroup (Dual a) Source
Semigroup (Endo a) Source
Num a => Semigroup (Sum a) Source
Num a => Semigroup (Product a) Source
Semigroup (First a) Source
Semigroup (Last a) Source
Semigroup a => Semigroup (Maybe a) Source
Semigroup (IntMap v) Source
Ord a => Semigroup (Set a) Source
Semigroup (Seq a) Source
(Hashable a, Eq a) => Semigroup (HashSet a) Source
Semigroup (NonEmpty a) Source
Semigroup a => Semigroup (Option a) Source
Monoid m => Semigroup (WrappedMonoid m) Source
Semigroup (Last a) Source
Semigroup (First a) Source
Ord a => Semigroup (Max a) Source
Ord a => Semigroup (Min a) Source
Semigroup b => Semigroup (a -> b) Source
Semigroup (Either a b) Source
(Semigroup a, Semigroup b) => Semigroup (a, b) Source
Semigroup a => Semigroup (Const a b) Source
Semigroup (Proxy k s) Source
Ord k => Semigroup (Map k v) Source
(Hashable k, Eq k) => Semigroup (HashMap k a) Source
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c) Source
Alternative f => Semigroup (Alt * f a) Source
Semigroup a => Semigroup (Tagged k s a) Source
(Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d) Source
(Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e) Source

stimesMonoid :: (Integral b, Monoid a) => b -> a -> a Source

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 Source

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 Source

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 Source

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 Source

Constructors

Min
Fields getMin :: a

Instances

Monad Min Source
Functor Min Source
MonadFix Min Source
Applicative Min Source
Foldable Min Source
Traversable Min Source
Generic1 Min Source
Bounded a => Bounded (Min a) Source
Enum a => Enum (Min a) Source
Eq a => Eq (Min a) Source
Data a => Data (Min a) Source
Num a => Num (Min a) Source
Ord a => Ord (Min a) Source
Read a => Read (Min a) Source
Show a => Show (Min a) Source
Generic (Min a) Source
(Ord a, Bounded a) => Monoid (Min a) Source
NFData a => NFData (Min a) Source
Hashable a => Hashable (Min a) Source
Ord a => Semigroup (Min a) Source
type Rep1 Min Source
type Rep (Min a) Source

newtype Max a Source

Constructors

Max
Fields getMax :: a

Instances

Monad Max Source
Functor Max Source
MonadFix Max Source
Applicative Max Source
Foldable Max Source
Traversable Max Source
Generic1 Max Source
Bounded a => Bounded (Max a) Source
Enum a => Enum (Max a) Source
Eq a => Eq (Max a) Source
Data a => Data (Max a) Source
Num a => Num (Max a) Source
Ord a => Ord (Max a) Source
Read a => Read (Max a) Source
Show a => Show (Max a) Source
Generic (Max a) Source
(Ord a, Bounded a) => Monoid (Max a) Source
NFData a => NFData (Max a) Source
Hashable a => Hashable (Max a) Source
Ord a => Semigroup (Max a) Source
type Rep1 Max Source
type Rep (Max a) Source

newtype First a Source

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

Constructors

First
Fields getFirst :: a

Instances

Monad First Source
Functor First Source
MonadFix First Source
Applicative First Source
Foldable First Source
Traversable First Source
Generic1 First Source
Bounded a => Bounded (First a) Source
Enum a => Enum (First a) Source
Eq a => Eq (First a) Source
Data a => Data (First a) Source
Ord a => Ord (First a) Source
Read a => Read (First a) Source
Show a => Show (First a) Source
Generic (First a) Source
NFData a => NFData (First a) Source
Hashable a => Hashable (First a) Source
Semigroup (First a) Source
type Rep1 First Source
type Rep (First a) Source

newtype Last a Source

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

Constructors

Last
Fields getLast :: a

Instances

Monad Last Source
Functor Last Source
MonadFix Last Source
Applicative Last Source
Foldable Last Source
Traversable Last Source
Generic1 Last Source
Bounded a => Bounded (Last a) Source
Enum a => Enum (Last a) Source
Eq a => Eq (Last a) Source
Data a => Data (Last a) Source
Ord a => Ord (Last a) Source
Read a => Read (Last a) Source
Show a => Show (Last a) Source
Generic (Last a) Source
NFData a => NFData (Last a) Source
Hashable a => Hashable (Last a) Source
Semigroup (Last a) Source
type Rep1 Last Source
type Rep (Last a) Source

newtype WrappedMonoid m Source

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid
Fields unwrapMonoid :: m

Instances

Generic1 WrappedMonoid Source
Bounded a => Bounded (WrappedMonoid a) Source
Enum a => Enum (WrappedMonoid a) Source
Eq m => Eq (WrappedMonoid m) Source
Data m => Data (WrappedMonoid m) Source
Ord m => Ord (WrappedMonoid m) Source
Read m => Read (WrappedMonoid m) Source
Show m => Show (WrappedMonoid m) Source
Generic (WrappedMonoid m) Source
Monoid m => Monoid (WrappedMonoid m) Source
NFData m => NFData (WrappedMonoid m) Source
Hashable a => Hashable (WrappedMonoid a) Source
Monoid m => Semigroup (WrappedMonoid m) Source
type Rep1 WrappedMonoid Source
type Rep (WrappedMonoid m) Source

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 `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 (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) Source
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) Source
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 Source
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 Source
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) Source
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) Source
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

Constructors

Option
Fields getOption :: Maybe a

Instances

Monad Option Source
Functor Option Source
MonadFix Option Source
Applicative Option Source
Foldable Option Source
Traversable Option Source
Generic1 Option Source
Alternative Option Source
MonadPlus Option Source
Eq a => Eq (Option a) Source
Data a => Data (Option a) Source
Ord a => Ord (Option a) Source
Read a => Read (Option a) Source
Show a => Show (Option a) Source
Generic (Option a) Source
Semigroup a => Monoid (Option a) Source
NFData a => NFData (Option a) Source
Hashable a => Hashable (Option a) Source
Semigroup a => Semigroup (Option a) Source
type Rep1 Option Source
type Rep (Option a) Source

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.

Constructors

Arg a b

Instances

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