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.
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