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.

times1p :: Natural -> a -> a Source

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 <>.

Semigroups

newtype Min a Source

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)
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)
Typeable (* -> *) Min
type Rep1 Min
type Rep (Min a)

newtype Max a Source

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)
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)
Typeable (* -> *) Max
type Rep1 Max
type Rep (Max a)

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
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)
Typeable (* -> *) First
type Rep1 First
type Rep (First a)

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
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)
Typeable (* -> *) Last
type Rep1 Last
type Rep (Last a)

newtype WrappedMonoid m Source

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)
Typeable (* -> *) WrappedMonoid
type Rep1 WrappedMonoid
type Rep (WrappedMonoid m)

timesN :: Monoid a => Natural -> a -> a Source

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.

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 Builder
Monoid ByteString
Monoid ShortByteString
Monoid ByteString
Monoid IntSet
Monoid Builder
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 `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 * s)
Ord k => Monoid (Map k v)
(Eq k, Hashable k) => Monoid (HashMap k v)
Typeable (* -> Constraint) Monoid
(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

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)
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
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
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)
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)
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 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

Alternative Option
Monad Option
Functor Option
MonadFix Option
MonadPlus Option
Applicative Option
Foldable Option
Traversable Option
Generic1 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)
Typeable (* -> *) Option
type Rep1 Option
type Rep (Option a)

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

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)
Typeable (* -> * -> *) Arg
type Rep1 (Arg a)
type Rep (Arg a b)

type ArgMin a b = Min (Arg a b) Source

type ArgMax a b = Max (Arg a b) Source

Semigroup Ordering
Semigroup ()
Semigroup All
Semigroup Any
Semigroup Builder
Semigroup ByteString
Semigroup ShortByteString
Semigroup ByteString
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)
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)