6 L.S. GRINBLAT Some remarks and notation. Unless otherwise stated, all algebras, measures and ul- trafilters are defined on the same set X, which will be regarded as a topological space with the discrete topology. As usual, f3X will denote the Cech compactification of X. The points of /3X are ultrafilters over X. Ultrafilters will therefore be denoted by lower case letters. If M C fiX (in particular, if M C X), we let M denote the closure M in 0X. The symbol \M\ will denote the cardinality of a set M. The memoir consists essentially of two parts. The first part comprises Sections 4-9, the second part Sections 10-12. The main theorems will be stated in Section 2 and numbered, to set them apart from other theorems, by Roman numerals. Theorems I-IV and Theorem II* will be proved in the first part, Theorems V-XII in the second. It should already be clear from the formulations of these theorems in what sense the memoir splits into two parts. On the other hand, these two parts of our memoir combine quite naturally to form a unified whole - this will become particularly clear in the second part of Section 7. The basic idea of our method is outlined in Section 3. Section 13 may be regarded as an appendix to the memoir. It concerns what we shall call semi-lattices of subsets, and lattices of subsets, given, of course, on X. In Section 14 we list some unsolved problems which arose during our work on this memoir. Acknowledgment. It is a pleasure to acknowledge the friendly support extended to me by M. Gitik throughout the lengthy research of which this memoir is the outcome, and his many useful comments. I am deeply grateful to D. Fremlin and the referee for their remarks. 2. MAI N RESULTS Theore m I. (1) Consider a finite sequence of algebras Ai,... ,An such that for every k 7^ 2, 1 k n, there exist more than |(fc — 1) pairwise disjoint sets not in Ak (if A2 is taken into consideration, A2 ^ V(X)). Then there exists Q £ Ak, 1 k n.7 (2) The hound I(k — 1) is best possible in the following sense: For every natural number n 1 one 7 In order to give the reader an idea of the character of the material, we have stated two claims in the Introduction. The only difference between Claim 1.1 and the first part of Theorem I is that Claim 1.1 demands the existence of at least two pairwise disjoint sets not belonging to A2 (if Ai is taken into consideration). If X £ A2, then the statements of Claim 1.1 and the first part of Theorem I are identical.

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