## Concept:

# Monoid

## Headline

A type with an associative operation and a neutral element

## Illustration

The notion of monoid is precisely defined in group theory, but we focus here on its illustration in a programming setting. Specifically, in functional programming, a monoid is essentially a type with an associative operation and a neutral element. For instance, lists form a monoid with the empty list as neutral element and list append as the associative operation. Monoids are useful, for example, in aggregating results.

In Language:Haskell, monoids are modeled through the type class *Monoid* with first two members needed for a minimal complete definition:

```
-- The type class Monoid
class Monoid a
where
mempty :: a -- neutral element
mappend :: a -> a -> a -- associative operation
mconcat :: [a] -> a -- fold
mconcat = foldr mappend mempty
```

Algebraically, the following properties are required for any monoid (given in Haskell notation):

```
mempty `mappend` x = x -- left unit
x `mappend` mempty = x -- right unit
x `mappend` (y `mappend` z) = (x `mappend` y) `mappend` z -- associativity
```

See the following monoids for continued illustration:

## Metadata

- Data type
- Vocabulary:Functional programming
- Vocabulary:Mathematics
- http://en.wikipedia.org/wiki/Monoid
- http://mathworld.wolfram.com/Monoid.html
- http://en.wikibooks.org/wiki/Haskell/Monoids
- http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Monoid.html

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