Iterative reflections of monads

Iterative monads, introduced by Calvin Elgot in the 1970’s, are those ideal monads in which every guarded system of recursive equations has a unique solution. For every ideal monad M we prove that an iterative reflection, i.e., an embedding M ↪−→ M̂ into an iterative monad with the expected universal property, exists. We also introduce the concept of iterativity for algebras for the monad M, following the footsteps of Evelyn Nelson and Jerzy Tiuryn, and we prove that M is iterative if and only if all free algebras for M are iterative algebras. In mathematics you don’t understand things. You just get used to them. John von Neumann (1903–1957)