On Algebras with Iteration

Several concepts of algebras with solutions of recursive equation systems are compared: CPO-enrichable algebras are proved to be iteration algebras of Z. Esik, and iteration algebras are a special case of the recently introduced Elgot algebras (which are the monadic algebras for the free iterative monad). Another special case of iteration algebras are the iterative algebras of E. Nelson and J. Tiuryn, which are algebras with unique solutions of all guarded systems. For each of the above classes of algebras an example is provided showing that the inclusion in a wider class is proper.