Your solution would imply[1] that all Rational are multiplicatively invertible -- which they are not. The Rationals are not a multiplicative group -- although the _positive_ Rationals are. You can't express this in Haskell's type system AFAIK. Your basic point is correct: if you are willing to use a tag (like Multiply and Add), then you can indeed have a domain be seen as matching an interface in 2 different ways. Obviously, this can be extended to n different ways with appropriate interfaces. Jacques [1] imply in the sense of intensional semantics, since we all know that Haskell's type system is not powerful enough to enforce axioms. PS: if you stick to 2 Monoidal structures, you'll be on safer grounds. Brian Hulley wrote:
If the above is equivalent to saying "Monoid is a *superclass* of SemiRing in two different ways", then can someone explain why this approach would not work (posted earlier):
data Multiply = Multiply data Add = Add
class Group c e where group :: c -> e -> e -> e identity :: c -> e inverse :: c -> e -> e
instance Group Multiply Rational where group Multiply x y = ... identity Multiply = 1 inverse Multiply x = ...
instance Group Add Rational where group Add x y = ... identity Add = 0 inverse Add x = ...
(+) :: Group Add a => a -> a -> a (+) = group Add
(*) = group Multiply
class (Group Multiply a, Group Add a) => Field a where ...
If the objection is just that you can't make something a subclass in two different ways, the above is surely a counterexample. Of course I made the above example more fixed than it should be ie:
class (Group mult a, Group add a) => Field mult add a where ...
and only considered the relationship between groups and fields - obviously other classes would be needed before and in-between, but perhaps the problem is that even with extra parameters (to represent *all* the parameters in the corresponding tuples used in maths), there is no way to get a hierarchy?
Thanks, Brian.