Iterative reflections of monads

Abstract

Iterative monads were introduced by Calvin Elgot in the 1970's and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.