| Portability | portable |
|---|---|
| Stability | provisional |
| Maintainer | Edward Kmett <ekmett@gmail.com> |
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
- class Semigroup a where
- 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
- class Monoid a where
- newtype Dual a = Dual {
- getDual :: a
- newtype Endo a = Endo {
- appEndo :: a -> a
- newtype All = All {}
- newtype Any = Any {}
- newtype Sum a = Sum {
- getSum :: a
- newtype Product a = Product {
- getProduct :: a
- newtype Option a = Option {}
- option :: b -> (a -> b) -> Option a -> b
- diff :: Semigroup m => m -> Endo m
- cycle1 :: Semigroup m => m -> m
Documentation
Methods
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 <>.
Semigroups
newtype WrappedMonoid m Source
Provide a Semigroup for an arbitrary Monoid.
Constructors
| WrapMonoid | |
Fields
| |
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 =
foldrmappend 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.
newtype Endo a
The monoid of endomorphisms under composition.