# # Monoid

## # An instance of Monoid for lists

```
instance Monoid [a] where
mempty = []
mappend = (++)
```

Checking the `Monoid`

laws for this instance:

```
mempty `mappend` x = x <-> [] ++ xs = xs -- prepending an empty list is a no-op
x `mappend` mempty = x <-> xs ++ [] = xs -- appending an empty list is a no-op
x `mappend` (y `mappend` z) = (x `mappend` y) `mappend` z
<->
xs ++ (ys ++ zs) = (xs ++ ys) ++ zs -- appending lists is associative
```

## # Collapsing a list of Monoids into a single value

`mconcat :: [a] -> a`

is another method of the `Monoid`

typeclass:

```
ghci> mconcat [Sum 1, Sum 2, Sum 3]
Sum {getSum = 6}
ghci> mconcat ["concat", "enate"]
"concatenate"
```

Its default definition is `mconcat = foldr mappend mempty`

.

## # Numeric Monoids

Numbers are monoidal in two ways: **addition** with 0 as the unit, and **multiplication** with 1 as the unit. Both are equally valid and useful in different circumstances. So rather than choose a preferred instance for numbers, there are two `newtypes`

, `Sum`

and `Product`

to tag them for the different functionality.

```
newtype Sum n = Sum { getSum :: n }
instance Num n => Monoid (Sum n) where
mempty = Sum 0
Sum x `mappend` Sum y = Sum (x + y)
newtype Product n = Product { getProduct :: n }
instance Num n => Monoid (Product n) where
mempty = Product 1
Product x `mappend` Product y = Product (x * y)
```

This effectively allows for the developer to choose which functionality to use by wrapping the value in the appropriate `newtype`

.

```
Sum 3 <> Sum 5 == Sum 8
Product 3 <> Product 5 == Product 15
```

## # An instance of Monoid for ()

`()`

is a `Monoid`

. Since there is only one value of type `()`

, there's only one thing `mempty`

and `mappend`

could do:

```
instance Monoid () where
mempty = ()
() `mappend` () = ()
```