A Counterexample to Tensorability of Effects

Monads are widely used in programming semantics and in functional programming to encapsulate notions of side-effect, such as state, exceptions, input/output, or continuations. One of their advantages is that they allow for a modular treatment of effects, using composition operators such as sum and tensor. Here, the sum represents the non-interacting combination of effects, while the tensor imposes a high degree of interaction in the shape of a commutation law. Although many important effects are ranked, i.e. presented by algebraic operations of bounded arity, there is also a range of relevant unranked effects, with prominent examples including continuations and unbounded non-determinism. While the sum and tensor of ranked effects always exist, this is not so clear already when one of the components is unranked, in which case size problems come into play. In contrast to the case of sums where a counterexample can be constructed rather trivially, the general existence of tensors has, so far, been an open issue — as the tensor identifies more terms than the sum, it does exist in many cases where the sum fails to exist. As a possible counterexample, tensors of continuations with unranked effects have been discussed; however, we have disproved that possibility in recent work. In the present work, we nevertheless settle the question in the negative by presenting a well-order monad whose tensor with a simple ranked monad fails to exist; as a consequence, we show also that there is an unranked monad whose tensor with the finite list monad fails to exist.