Why can AccValidation not have a Monad instance?

Question

From the documentation of the validation package:

The AccValidation data type is isomorphic to Either, but has an instance of <

Answer 1

The (<*>) = ap exigence can be stated explicitly in terms of (>>=):

u <*> v = u >>= \f -> fmap f v -- [1]

Now, given the Functor and Applicative instances for AccValidation, we have:

fmap _ (AccFailure e) = AccFailure e -- [2]

AccFailure e1 <*> AccFailure e2 = AccFailure (e1 <> e2) -- [3]

If we make u = AccFailure e1 and v = AccFailure e2 in [1], we get:

AccFailure e1 <*> AccFailure e2 = AccFailure e1 >>= \f -> fmap f (AccFailure e2)

Substituting [2] and [3] into that leads us to:

AccFailure (e1 <> e2) = AccFailure e1 >>= \_ -> AccFailure e2 -- [4]

The problem is that it is impossible to write a (>>=) such that [4] holds. The left-hand side depends on an e2 value which must originate, on the right-hand side, from applying \_ -> AccFailure e2 :: Semigroup e => a -> AccValidation e b. However, there is nothing to which it can be applied -- in particular, e1 has the wrong type. (See the final two paragraphs of Cactus' answer for further discussion of this point.) Therefore, there is no way of giving AccValidation a Monad instance which is consistent with its Applicative one.