Complete Elgot Monads and Coalgebraic Resumptions

Monads are used to abstractly model a wide range of computational effects such as nondeterminism, statefulness, and exceptions. Complete Elgot monads are monads that are equipped with a (uniform) iteration operator satisfying a set of natural axioms, which allows to model iterative computations just as abstractly. It has been shown recently that extending complete Elgot monads with free effects (e.g. operations of sending/receiving messages over channels) canonically leads to generalized coalgebraic resumption monads, which were previously used as semantic domains for non-wellfounded guarded processes. In this paper, we continue the study of complete Elgot monads and their relationship with generalized coalgebraic resumption monads. We give a characterization of the Eilenberg-Moore algebras of the latter. In fact, we work more generally with Uustalu's parametrized monads; we introduce complete Elgot algebras for a parametrized monad and we prove that they form an Eilenberg-Moore category. This is further used for establishing a characterization of complete Elgot monads as those monads whose algebras are coherently equipped with the structure of complete Elgot algebras for the parametrized monads obtained from generalized coalgebraic resumption monads.