Degrading Lists

We discuss the relationship between monads and their known generalisation, graded monads, which are especially useful for modelling computational effects equipped with a form of sequential composition. Specifically, we ask if a graded monad can be extended to a monad, and when such a degrading is in some sense canonical. Our particular examples are the graded monads of lists and non-empty lists indexed by their lengths, which gives us a pretext to study the space of all (non-graded) monad structures on the list and non-empty list endofunctors. We show that, in both cases, there exist infinitely many monad structures. However, while there are at least two ways to complete the graded monad structure on length-indexed lists to a monad structure on the list endofunctor, such a completion for non-empty lists is unique.