Free Iterative Theories: A Coalgebraic View

Iterative algebraic theories were introduced by Calvin C. Elgot in Elgot (1975) as a concept serving the study of computation (on, say, Turing machines) at a level abstracting from the nature of external memory. The main example presented by Elgot is the theory of rational trees, that is, infinite trees that are solutions of systems of finitary iterative equations. Or, equivalently, that possess only finitely many subtrees. He and his coauthors later proved that this theory is a free iterative theory on a given (finitary) signature (Elgot et al. 1978). The purpose of the present paper is to generalise Elgot’s result from signatures (in other words, polynomial endofunctors of the category of sets) to finitary endofunctors of Set and some ‘set-like’ categories, for example, the category of posets. Using a very general Solution Theorem, developed in previous work, which shows by coalgebraic methods how iterative equations can be solved in categories, we prove that finitary endofunctors generate free iterative theories (in other words, finitary monads), called rational monads. We construct the rational monad in two steps: