# Difference between revisions of "Data.Semigroup"

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-- returns: Product {getProduct = 81} | -- returns: Product {getProduct = 81} | ||

-- This is because (3 <> 3 <> 3 <> 3) == (3 * 3 * 3 * 3) == 81 | -- This is because (3 <> 3 <> 3 <> 3) == (3 * 3 * 3 * 3) == 81 | ||

− | -- i.e. 3 | + | -- i.e. 4 copies of 3 combined via multiplication. |

</pre> | </pre> | ||

## Revision as of 08:36, 15 January 2020

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`

, combine`x`

numbers of the value`a`

by repeatedly applying`<>`

.

## 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. 4 copies of 3 combined via multiplication.

### 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`