How iterative reflections of monads are constructed

Abstract

Every ideal monad M on the category of sets is known to have a reflection Mˆ in the category of all iterative monads of Elgot. Here we describe the iterative reflection Mˆ as the monad of free iterative Eilenberg–Moore algebras for M. This yields numerous concrete examples: if M is the free-semigroup monad, then Mˆ is obtained by adding a single absorbing element; if M is the monad of finite trees then Mˆ is the monad of rational trees, etc.