semigroups-0.9.2: Haskell 98 semigroups

Copyright(C) 2011 Edward Kmett,
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilityportable
Safe HaskellSafe
LanguageHaskell98

Data.Semigroup

Contents

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

Minimal complete definition

Nothing

Methods

(<>) :: a -> a -> a infixr 6

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 times.

Semigroups

newtype Min a

Constructors

Min 

Fields

getMin :: a
 

Instances

Bounded a => Bounded (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) 
(Ord a, Bounded a) => Monoid (Min a) 
Ord a => Semigroup (Min a) 

newtype Max a

Constructors

Max 

Fields

getMax :: a
 

Instances

Bounded a => Bounded (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) 
(Ord a, Bounded a) => Monoid (Max a) 
Ord a => Semigroup (Max a) 

newtype First a

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

Constructors

First 

Fields

getFirst :: a
 

Instances

Bounded a => Bounded (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) 
Semigroup (First a) 

newtype Last a

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

Constructors

Last 

Fields

getLast :: a
 

Instances

Bounded a => Bounded (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) 
Semigroup (Last a) 

newtype WrappedMonoid m

Provide a Semigroup for an arbitrary Monoid.

Constructors

WrapMonoid 

Fields

unwrapMonoid :: m
 

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

Repeat a value n times.

times 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.

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 IntSet 
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 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 (IntMap a) 
Ord a => Monoid (Set a) 
Monoid (Seq 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) 
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) 
Alternative f => Monoid (Alt * f 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) 
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

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
 

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

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.