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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:

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