I'm still slightly confused. The author suggests that a monad is composed of 3 functions. It seems to me you've shown 2 of these functions.
a -> m b
>>=
What is the third function? Did I misunderstand you?
a -> m b
>>=
What is the third function? Did I misunderstand you?
Also, the type of return is `a -> m a`. The type you cite `a -> m b` is a parameter to >>= (which has type `m a -> (a -> m b) -> m b.
The author might be refering to a quirk of the Haskell monad interface, which has to additional functions:
">>" which can be easily defined in terms of >>=. I assume this is part of the interface so users can override it with a more efficient implementation, but every implementation I have seen just uses the default one.
"fail" which is an error handler. This is being depreciated as a function of Monads, and moved into its own MonadFail typeclass. This also has a default implementation that just throws an exception.
Probably because monads are frequently described as triples -- two functions and a functor. The author seems to have reduced that to three functions (presumably fmap) without realizing that it isn't sufficient.