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.