| Summary: | Models of iterated computation, such as (completely) iterative monads, often
depend on a notion of guardedness, which guarantees unique solvability of
recursive equations and requires roughly that recursive calls happen only under
certain guarding operations. On the other hand, many models of iteration do
admit unguarded iteration. Solutions are then no longer unique, and in general
not even determined as least or greatest fixpoints, being instead governed by
quasi-equational axioms. Monads that support unguarded iteration in this sense
are called (complete) Elgot monads. Here, we propose to equip (Kleisli
categories of) monads with an abstract notion of guardedness and then require
solvability of abstractly guarded recursive equations; examples of such
abstractly guarded pre-iterative monads include both iterative monads and Elgot
monads, the latter by deeming any recursive definition to be abstractly
guarded. Our main result is then that Elgot monads are precisely the
iteration-congruent retracts of abstractly guarded iterative monads, the latter
being defined as admitting unique solutions of abstractly guarded recursive
equations; in other words, models of unguarded iteration come about by
quotienting models of guarded iteration.
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