how to achieve "product of two monads" effect?

Question

Suppose we have two monads, m and m'. Now, suppose we have variables,

-- in real problems, the restriction is some subclass MyMonad, so don't worry
-- if it's the case here that mx and f must essentially be pure.
mx :: Monad m'' => m'' a
f :: Monad m'' => a -> m'' b

Is there a way to create anything similar to the product m x m'? I know this is possible with Arrows, but it seems more complicated (impossible?) for monads, especially when trying to write what mx >>= f should do.

To see this, define

data ProdM a = ProdM (m a) (m' a)
instance Monad ProdM where
    return x = ProdM (return x) (return x)

but now, when we define mx >>= f, it's not clear which value from mx to pass to f,

    (ProdM mx mx') >>= f
        {- result 1 -} = mx >>= f
        {- result 2 -} = mx' >>= f

I want (mx >>= f) :: ProdM to be isomorphic to ((mx >>= f) :: m) x ((mx >>= f) :: m').

Answer 1

Yes, this type is a monad. The key is simply to pass both results to f, and only keep the matching field from the result. That is, we keep the first element from the result of passing mx's result, and the second element from the result of passing mx''s result. The instance looks like this:

instance (Monad m, Monad m') => Monad (ProdM m m') where
  return a = ProdM (return a) (return a)
  ProdM mx mx' >>= f = ProdM (mx >>= fstProd . f) (mx' >>= sndProd . f)
    where fstProd (ProdM my _) = my
          sndProd (ProdM _ my') = my'

ProdM is available in the monad-products package under the name Product.