From Iterative Algebras to Iterative Theories

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with “iterative algebras”, i. e., algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. And our result is, nevertheless, much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A. This allows us, e. g., to consider iterative algebras over any equationally specified class A of finitary algebras.