On some small cardinals for Boolean algebras

Assume that all algebras are atomless. (1) Spind(A x B) = Spind(A) U Spind(B). (2) Spind(H1-I~ Ai) = {co } U U,Ej Spind(Ai). Now suppose that K and 2 are infinite cardinals, with r uncountable and regular and with r. < 2. (3) There is an atomless Boolean algebra A such that u(A) = te and i(A) = A. (4) If 2 is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = K and a(A) = 2. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [oo]. Introduction. First we define the cardinal functions considered in this paper. In these definitions, assume that A is an atomless Boolean algebra. A subset X of A splits A provided that for every nonzero a E A there is a b E X such that a b 5 0 a -b. A partition of unity of A is a collection X of nonzero pairwise disjoint elements of A with supremum 1. A tower is a subset X of A\{ 1 } well-ordered by the Boolean ordering, with supremum 1. i(A) = min{ X : X is a maximal independent subset of A}; u(A) = min{ X : X generates an ultrafilter of A}; s(A) = min{ lX : X splits A}; a(A) = min{ X : X is an infinite partition of unity in A}; t(A) = min{ XI : X is a tower in A}; Spind(A) = {IX : X is a maximal independent subset of A}. The main results are then as indicated in the abstract. As a corollary of (1) we have i(A x B) = min{i(A), i(B)}. (1) answers questions raised in Monk [01] and Monk [oo]. With A = 9(ow)/fin, models M, N of ZFC in which u(A) < i(A) and s(A) < a(A) respectively hold have been known for a long time; see Blass, Shelah [87] and Balcar, Simon [89]. Thus a contribution here is a construction of such Boolean algebras in ZFC. The problems about obtaining such examples in ZFC were also raised in Monk [01]. In fact, the only inequality holding in all atomless Boolean algebras between the cardinal functions i, u, s, a, and t is t < s; these two results, together with examples in Monk [01], establish this. All of these cardinal functions for Boolean algebras generalize known ones for 9(w)/fin, and they are extensively discussed in Monk [01] and Monk [oo]. This article is self-contained, however. Received December 11, 2002; revised January 30, 2004. c 2004, Association for Symbolic Logic 0022-4812/04/6903-0004/$1.90