Copyright	(C) 2011-2014 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

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

class Semigroup a where
- (<>) :: a -> a -> a
- sconcat :: NonEmpty a -> a
- times1p :: Whole n => n -> a -> a
newtype Min a = Min {
- getMin :: a
}
newtype Max a = Max {
- getMax :: a
}
newtype First a = First {
- getFirst :: a
}
newtype Last a = Last {
- getLast :: a
}
newtype WrappedMonoid m = WrapMonoid {
- unwrapMonoid :: m
}
timesN :: (Whole n, Monoid a) => n -> a -> a
class Monoid a where
- mempty :: a
- mappend :: a -> a -> a
- mconcat :: [a] -> a
newtype Dual a :: * -> * = Dual {
- getDual :: a
}
newtype Endo a :: * -> * = Endo {
- appEndo :: a -> a
}
newtype All :: * = All {
- getAll :: Bool
}
newtype Any :: * = Any {
- getAny :: Bool
}
newtype Sum a :: * -> * = Sum {
- getSum :: a
}
newtype Product a :: * -> * = Product {
- getProduct :: a
}
newtype Option a = Option {
- getOption :: Maybe a
}
option :: b -> (a -> b) -> Option a -> b
diff :: Semigroup m => m -> Endo m
cycle1 :: Semigroup m => m -> m

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 :: Whole n => n -> 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

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 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

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 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

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 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

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 Rep (Last a)

newtype WrappedMonoid m Source

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid

Fields

unwrapMonoid :: m

Instances

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 Rep (WrappedMonoid m)

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

Monoid IntSet

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.