proceedings of the american mathematical society Volume 116, Number 4, December 1992 THE AXIOM OF CHOICE, FIXED POINT THEOREMS, AND INDUCTIVE ORDERED SETS MILAN R. TASKOVlC (Communicated by Andrew M. Odlyzko) Abstract. This paper continues the study of the inductiveness of posets in terms of fixed apexes and points. The author proves some new equivalents of the Axiom of Choice, i.e., Zorn's lemma. These statements are of fixed apex type and fixed point type theorems. The paper includes comments about these theorems and presents new characterizations of inductiveness and quasi- inductiveness of posets in terms of fixed apexes and fixed points. 1. Introduction We shall first discuss an assumption that appears to be independent of, and yet consistent with (see [G]), the usual logical assumptions regarding classes and correspondences, but whose absolute validity has been seriously questioned by many authors. This is the so-called Axiom of Choice, which has excited more controversy than any other axiom of set theory since its formulation by Zermelo in 1908. In this sense, many results are known in the set theory (see references). The fixpoint problem for a given mapping f\P is very easy to formulate: the question is whether some Ç £ P satisfies /(£) = £ . Many problems are re- ducible to the existence of fixpoints of certain mappings. The question remains whether some statement (of the axiom of choice type) could be equivalently expressed in the fixpoint language as well. The answer is affirmative. In this paper we prove some equivalents of the Axiom of Choice. These statements are of fixed point type theorems and fixed apex type theorems. Also, this paper presents new characterizations of inductiveness of posets in terms of fixed apexes and fixed point. Call a poset (partially ordered set) P inductive (chain complete) when every nonempty chain in P has an upper bound (least upper bound, i.e., supremum) in P. Also, call a poset P quasi-inductive (quasi-chain complete) when every nonempty well ordered chain has an upper bound (supremum) in P. In [Ta2] we consider the concept of fixed apexes for the mapping / of a Received by the editors July 7, 1988 and, in revised form, April 12, 1991. Presented at the Department of Mathematics, Indiana University, Bloomington, Indiana (Mathematics Seminars and Colloquia), October, 1987; and at the University of Missouri-Rolla, Rolla, December, 1987. 1991 Mathematics Subject Classification. Primary 05A15, 04A25, 47H10, 54H25. Key words and phrases. Inductive partially ordered sets, Axiom of Choice, Zorn's lemma, fixed apexes, fixed points. ©1992 American Mathematical Society 0002-9939/92 $1.00+ $.25 per page 897 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use