## Concept:

# Sum monoid

## Headline

A monoid leveraging addition for the associative operation

## Illustration

Number types may be completed into monoids in different ways. The sum monoid favors addition for the associative operation of the monoid. We illustrate the sum monoid in Language:Haskell on the grounds of the type class *Monoid* with the 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
```

The sum monoid relies on a designated type which essentially wraps a number type:

```
-- The type of the sum monoid
newtype Sum a = Sum { getSum :: a }
```

Here is the type-class instance for the sum monoid:

```
-- The Monoid instance for numbers under addition
instance Num a => Monoid (Sum a)
where
mempty = Sum 0
x `mappend` y = Sum (getSum x + getSum y)
```

For further illustration, we can reconstruct the standard *sum* function in a monoidal way. To this end, we first review the normal definition in terms of foldr:

```
-- A foldr-based definition of sum
sum' :: Num a => [a] -> a
sum' = foldr (+) 0
```

```
-- A monoidal definition of sum
sum'' :: Num a => [a] -> a
sum'' = getSum . mconcat . map Sum
```

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