Abstract and concrete categories
Abstract and concrete categories
Notions of computation and monads
Information and Computation
Handbook of logic in computer science (vol. 3)
ACM Computing Surveys (CSUR)
Composing monads using coproducts
Proceedings of the seventh ACM SIGPLAN international conference on Functional programming
Revised Report on the Algorithmic Language Scheme
Higher-Order and Symbolic Computation
Notions of Computation Determine Monads
FoSSaCS '02 Proceedings of the 5th International Conference on Foundations of Software Science and Computation Structures
Combining effects: sum and tensor
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
Electronic Notes in Theoretical Computer Science (ENTCS)
Combining algebraic effects with continuations
Theoretical Computer Science
Handbook of Weighted Automata
From Comodels to Coalgebras: State and Arrays
Electronic Notes in Theoretical Computer Science (ENTCS)
Kleene monads: handling iteration in a framework of generic effects
CALCO'09 Proceedings of the 3rd international conference on Algebra and coalgebra in computer science
Powermonads and Tensors of Unranked Effects
LICS '11 Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer Science
A Semantics For Evaluation Logic
Fundamenta Informaticae
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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.