  Monads are extensively used nowadays to abstractly model a wide range of
computational effects such as nondeterminism, statefulness, and exceptions. It
turns out that equipping a monad with a (uniform) iteration operator satisfying
a set of natural axioms allows for modelling iterative computations just as
abstractly. The emerging monads are called complete Elgot monads. 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, previously used as semantic domains
for non-wellfounded guarded processes. In this paper, we continue the study of
the relationship between abstract complete Elgot monads and those that capture
coalgebraic resumptions, by comparing the corresponding categories of
(Eilenberg-Moore) algebras. To this end we first provide a characterization of
the latter category; even more generally, we formulate this characterization in
terms of Uustalu's parametrized monads. This is further used for establishing a
characterization of complete Elgot monads as precisely those monads whose
algebras are coherently equipped with the structure of algebras of coalgebraic
resumption monads.
