difference-monoid

This package provides the Difference Monoid, which adds subtraction to arbitrary monoids.

This has a number of uses:

• Diff (Product a) will give you a type similar to Data.Ratio. Here, the "subtraction" operation is division. For example:

  >>> (1 :-: 2) <> (3 :-: 4) :: Diff (Product Int)
  Product {getProduct = 3} :-: Product {getProduct = 8}
  

• In a similar vein, Diff (Sum a) will add subtraction to a numeric type:

  >>> runDiff (-) (diff 2 <> diff 3 <> invert (diff 4)) :: Sum Natural
  Sum {getSum = 1}
  

  This will let you work with nonnegative types, where you need some form of subtraction (for, e.g., differences, hence the name), and you only want to check for underflow once.

• Using the above example, in particular, we get a monoid for averages:

  >>> import Data.Function (on)
  >>> let avg = runDiff ((%) `on` getProduct.getSum) . foldMap (fmap Sum . diff . Product)
  >>> avg [1,4,3,2,5]
  3 % 1
  

The Monoid and Semigroup laws hold in a pretty straightforward way, provided the underlying type also follows those laws.

For the Group laws, the underlying type must be a cancellative semigroup.

A cancellative semigroup is one where

• a <> b = a <> c implies b = c
• b <> a = c <> a implies b = c

If this does not hold, than the equivalence only holds modulo the the addition of some constant

Most common semigroups are cancellative, however notable exceptions include the cross product of vectors, matrix multiplication, and sets:

fromList [1] <> fromList [1,2] = fromList [1] <> fromList [2]`

This type is known formally as the Grothendieck group.

The package also provides the Parity monad and comonad, which is left-adjunct to the difference monoid.