PartialInverse

The PartialInverse type class describes an associative binary operator for a type A that has an identity element and an inverse binary operator. Combining any value with itself with the inverse operator must return the identity element. There may be an exception, where the result is not defined. For example, for multiplication of reals, one is an identity element and division is an inverse operation, because dividing any value by itself always returns one.

Because PartialInverse defines a binary rather than a unary operator it can be used to describe inverse operations for types that do not have inverse values. For example, the natural numbers do not have inverses because the set of natural numbers does not include negative numbers. But we can still define a subtraction operation that is the inverse of addition for the natural numbers, since subtracting a number from itself always returns zero.