[Haskell-beginners] Monad for Pair

[Marcin Mrotek]

Hello,

I'm pretty sure a Pair (like any other fixed-length type, besides the
corner case of a single field like in Writer, Identity, etc) can't be
a monad. Perhaps instead of struggling with >>=, consider join. It has
a type:

join :: Monad m => m (m a) -> m a

for Pairs that would be

join :: Pair (Pair a) -> Pair a
join (Pair (Pair a1 a2) (Pair b1 b2)) = Pair _ _

How do you want to fit four values into two boxes? You cannot place
any constraints on the type inside the pair, so it can't be a monoid
or anything that would let you combine the values somehow. You could
only choose two of the values and drop the other two on the floor.

Getting back to >>=, it's assumed to follow these laws:

1) return a >>= k  =  k a
2) m >>= return    =  m
3) m >>= (\x -> k x >>= h)  =  (m >>= k) >>= h

As for the firs, return a = Pair a a. Then the first two laws become

1) Pair a a >>= k = k a
2) Pair a b >>= (\a -> Pair a a) = Pair a b

The first law could work if >>= just chose one of the values
arbitrarily. But the second law is a hopeless case. You would need to
pick one element of a pair, plug it into a function that repeats the
argument, and somehow get back the other element that you've already
dropped.

Concluding, either I'm sorely mistaken or there indeed isn't a Monad
instance for Pair.

Best regards,
Marcin Mrotek