PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 2, June 1975 LOS' THEOREMAND THE BOOLEAN PRIME IDEAL THEOREM IMPLY THE AXIOM OF CHOICE PAUL E. HOWARD ABSTRACT. In this paper it is shown that t-os' theorem and the Boolean prime ideal theorem imply the axiom of choice. The possibility of eliminat- ing the use of the Boolean prime ideal theorem from the proof is also dis- cussed. 1. The results of this paper are proved in Zermelo-Fraenkel set theory without the axiom of choice. We denote this theory by ZF. Let AC and BPI denote the following statements in the theory ZF: AC (The axiom of choice). For every collection X of nonempty sets, there is a function j with domain X such that (Vy £ X)(/(y) e y). (j is called a choice function for X.) BPI (The Boolean prime ideal theorem). Every Boolean algebra has a prime ideal. Now suppose that M = (A, R.).£. is a relational system, i.e. A is a set and for each j £ J, R is a finitary relation on A. Let X be any set and let J be a filter in the Boolean algebra of all subsets of X. We denote by Ax the set of all functions from X to A, and by A /j we mean the set of equivalence classes of A under the relation S defined by /, Otf2 iff \t ex\fy(t) =f2it)\ e3\ If / e A , let [/] denote the equivalence class of /. For each j £ J we de- fine the relation R. as follows: Suppose R. is an 72-ary relation on A, then R . is the zz-ary relation on A /'S given by KM' . . . , [f^) iff \t £ X\R]ifyit), ..., fnit))\£S. One can easily show that R is well defined. Finally we define the reduced ultra power Received by the editors March 28, 1974. AMS(MOS) subject classifications (1970). Primary 02K20, 04A25. Key words and phrases. Axiom of choice, Boolean prime ideal theorem, tos' theorem. Copyright © 1975. American Mathematical Society 426 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use