The seminal work of Calvin Elgot on iteration theories, done in the mid 1970s, never received the full recognition it deserved, probably because of its high degree of abstraction and complexity. In this paper, the author studies completely iterative algebras, namely, those algebras in which every system of recursive equations has a unique solution. Rather than restricting himself to the context of universal algebras, as Elgot did, or to other more concrete categories, as researchers such as Bloom and Esik [1,2,3] do, the author prefers to work in much more general categories. This has the disadvantage of distancing the reader from definite applications, in such areas as automata and flowchart schemes, but does make the proofs conceptually clearer, and mathematically neater.

Such a rarified mathematical atmosphere is not, needless to say, for the casual reader. However, for those who do have the mathematical tools to be able to cope with this level of abstraction, this paper is definitely worth reading and, more importantly, applying.