# Difference between revisions of "Data.Semigroup"

(Created page with "The <strong><hask>Semigroup</hask></strong> represents a set with an associative binary operation. This makes a semigroup a superset of monoids. Semigoups have...") |
m (corrected end tag and did math) |
||

Line 23: | Line 23: | ||

== Description == |
== Description == |
||

− | <div>Any datatype <hask>a</hask> which has an associative binary operation will be able to become a member of the <hask>Semigroup</hask> typeclass. An instance of <hask>Monoid a<hask> automatically satisfies the requirements of a <hask>Semigroup</hask> making <hask>Semigroup</hask> a strict superset of <hask>Monoid</hask>. The <hask>Monoid</hask> typeclass however does not enforce it's instances to already be instances of <hask>Semigroup</hask></div> |
+ | <div>Any datatype <hask>a</hask> which has an associative binary operation will be able to become a member of the <hask>Semigroup</hask> typeclass. An instance of <hask>Monoid a</hask> automatically satisfies the requirements of a <hask>Semigroup</hask> making <hask>Semigroup</hask> a strict superset of <hask>Monoid</hask>. The <hask>Monoid</hask> typeclass however does not enforce it's instances to already be instances of <hask>Semigroup</hask></div> |

<div>The <hask>Semigroup</hask> is a particularly forgiving typeclass in it's requirements, and datatypes may have many instances of <hask>Semigroup</hask> as long as they have functions which satisfy the requirements.</div> |
<div>The <hask>Semigroup</hask> is a particularly forgiving typeclass in it's requirements, and datatypes may have many instances of <hask>Semigroup</hask> as long as they have functions which satisfy the requirements.</div> |
||

Line 35: | Line 35: | ||

:As long as you do not change the order of the arguments, you can insert parenthesis anywhere, and the result will be the same. |
:As long as you do not change the order of the arguments, you can insert parenthesis anywhere, and the result will be the same. |
||

− | For example, addition (<hask>a + (b + c) == (a + b) + c</hask>), and multiplication (<hask>a * (b * c) == (a * b) * c</hask>) satisfy this requirement. Therefore <hask><></hask> could be defined as <hask>+</hask> or <hask>*</hask> for instances of class <hask>Num a</hask>. Division (<hask>div</hask>) however, would not be a candidate as it is not associative: <hask> |
+ | For example, addition (<hask>a + (b + c) == (a + b) + c</hask>), and multiplication (<hask>a * (b * c) == (a * b) * c</hask>) satisfy this requirement. Therefore <hask><></hask> could be defined as <hask>+</hask> or <hask>*</hask> for instances of class <hask>Num a</hask>. Division (<hask>div</hask>) however, would not be a candidate as it is not associative: <hask>8 `div` (4 `div` 2) == 8 `div` 2 == 4</hask> is not equal to <hask>(8 `div` 4) `div` 2 == 2 `div` 2 == 1</hask>. |

<div>In essence, the <hask><></hask> function could do anything, as long as it doesn't matter where you put parenthesis.</div> |
<div>In essence, the <hask><></hask> function could do anything, as long as it doesn't matter where you put parenthesis.</div> |

## Revision as of 18:38, 18 August 2018

The ** Semigroup** represents a set with an associative binary operation. This makes a semigroup a superset of monoids. Semigoups have no other restrictions, and are a very general typeclass.

`Semigroup`

with an identity value.## Packages

- (base) Data.Semigroup

## Syntax

class Semigroup a where (<>) :: a -> a -> a sconcat :: [[Data.List.Nonempty|Nonempty]] a -> a stimes :: Integral b => b -> a -> a

### Minimal Complete Definition

(<>)

## Description

`a`

which has an associative binary operation will be able to become a member of the `Semigroup`

typeclass. An instance of `Monoid a`

automatically satisfies the requirements of a `Semigroup`

making `Semigroup`

a strict superset of `Monoid`

. The `Monoid`

typeclass however does not enforce it's instances to already be instances of `Semigroup`

`Semigroup`

is a particularly forgiving typeclass in it's requirements, and datatypes may have many instances of `Semigroup`

as long as they have functions which satisfy the requirements.### Semigroup Laws

In addition to the class requirements above, potential instances of `Semigroup`

must obey a single law in order to become instances:

- The binary operation
`<>`

must be associative (a <> b) <> c == a <> (b <> c)

- As long as you do not change the order of the arguments, you can insert parenthesis anywhere, and the result will be the same.

For example, addition (`a + (b + c) == (a + b) + c`

), and multiplication (`a * (b * c) == (a * b) * c`

) satisfy this requirement. Therefore `<>`

could be defined as `+`

or `*`

for instances of class `Num a`

. Division (`div`

) however, would not be a candidate as it is not associative: `8 `div` (4 `div` 2) == 8 `div` 2 == 4`

is not equal to `(8 `div` 4) `div` 2 == 2 `div` 2 == 1`

.

`<>`

function could do anything, as long as it doesn't matter where you put parenthesis.### Rules for Monoids

Instances of `Monoid`

have to obey an additional rule:

(<>) == mappend

This is to ensure that the instance of `Monoid`

is equivalent to a more strict instance of `Semigroup`

.

## Methods

(<>) :: a -> a -> a

- An associative binary operation.

sconcat :: [[Data.List.Nonempty|Nonempty]] a -> a

- Take a nonempty list of type
`a`

and apply the`<>`

operation to all of them to get a single result.

stimes :: Integral b => b -> a -> a

- Given a number
`x`

and a value of type`a`

, apply`<>`

to the value`x`

times.

## Examples

### Sum numbers using `<>`

:

Sum 3 <> Sum 4 -- with the type "Sum a", "<>" becomes "+" -- returns: Sum {getSum = 7} because (3 <> 4) == (3 + 4) == 7

### Exponents using `stimes`

:

stimes 4 (Product 3) -- with the type "Product a" "<>" becomes "*" -- returns: Product {getProduct = 81} -- This is because (3 <> 3 <> 3 <> 3) == (3 * 3 * 3 * 3) == 81 -- i.e. 3 multiplied by itself 4 times.

### Test for any elements which are True in a non-empty list using `sconcat`

:

sconcat (Any True :| [Any False, Any True, Any False]) -- sconcat will apply "<>" to all of the members in a list -- returns: Any {getAny = True} -- If any elements in the list are True than the whole expression is True -- The type "Any" converts "<>" to "||" -- (True <> False <> True <> False) == (True || False || True || False) == True

### Test if all elements are True in a non-empty list using `sconcat`

:

sconcat (All True :| [All False, All True, All False]) -- returns: All {getAll = False} -- If all elements in the list are True than the whole expression is True -- The type "All" converts "<>" to "&&" -- (True <> False <> True <> False) == (True && False && True && False) == True

## See Also

- Data.Monoid: a special case of
`Semigroup`

with an identity element`mempty`