On generalized methods of successive approximations

There are several generalized methods of monotone successive approximations in ordered sets (A,<)(A,<) which generalize the usual sequences of the iterative, of a point aa of AA under a selfmapping ff on AA, to some well-ordered subsets of AA. These are: smallest admissible set containing aa by Büber and Kirk, the set W(a)W(a) by Abian and Brown, well-ordered chain of ff-iterations of aa by Heikkilä, and smallest complete ff-chain containing aa by Fuchssteiner. The well-ordered chains of ff-iterations were compared by Heikkilä in 1999 with the admissible sets. We compare here all of the above four methods. Our study is based on a notion of a generalized orbit which is a generalization of the one which was introduced by the author in 1988. As an answer to a problem posed by Banaschewski in 1992, we use in our proofs a minimum of tools. In particular, we assume that the relation << is only asymmetric (we do not use the transitivity postulated for orderings) and use only extensionality, pairing, union and separation axioms of Zermelo’s set theory.