HOMOMORPHIC IMAGES OF a-COMPLETE BOOLEAN ALGEBRAS

It is a well-known theorem of R. S. Pierce that, for every infinite cardinal a, a 0 = a if and only if there is a complete Boolean algebra B s.t. card B = a (see [3, Theorem 25.4]). Recently, Comfort and Hager proved [l] that, for every infinite a-complete Boolean algebra B, (card B) 0 _ card B. We extend this result to the class of homomorphic images of cr-complete algebras, following closely Comfort's and Hager's proof. As a corollary, an improvement of Shelah's theorem on the cardinality of ultraproducts of finite sets [2] is derived (Theorem 2).1 We denote the finite operations on a Boolean algebra A by +, -, and —, the infinite operations by S and II. If a £ A and a > 0, A\a is the algebra {x £ A\x < a]. We write card a tot card A\a (if a = 0, card a = 1). A sequence (a \n £ a>) in A is disjointed if a 4 0 and a -a = 0 for n, m £ a>, n 4 ^ n' ' ' 77 77 777 ' t??. For every sequence (A \n £ a>) of Boolean algebras, ÍI A is the product algebra. Theorem 1. Let B be a cr-complete Boolean algebra, p an epimorphism /rorrz B onto A and a = card A > NQ. Then a 0 = a. Proof. We first prove three lemmas. (a) Suppose x, y £ B s.t. x < y, put a = p(x) and b p(y). If c £ A s.t. a < c < b, there is some z £ B s.t. x < z < y and p(z) = c. If d £ A s.t. a • d = 0, there is some t £ B s.t. x • t = 0 and p(t) = d. Proof. Choose z', t' £ B s.t. p(z') = c, p(t') = d; put z = z' • y + x, t = t' • -x. (b) Suppose I is an ideal of A s.t. card / = card A = a > 2 °; every countable subset of I has an upper bound in I and for every a £ I, (card a) K X equals 2 0 or card a. Then a 0 = a. Received by the editors May 7, 1974. AMS iMOS) subject classifications (1970). Primary 02H13, 06A40. 1 This answers a question of M. Ziegler whether Theorem 1 allows to generalize Shelah's result to arbitrary reduced products of finite sets; the authot's original proof only established Shelah's theorem. Copyright © 1975, American Mathematical Society 171 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 172 SABINE KOPPELBERG