10 L.S. GRINBLAT
exists a matrix of pairwise disjoint sets
•u\ ... u*mi\
ir?"'•'••"vi'„ I
where rrik §(fc 1), Uf £ Ak- Then there exist pairwise disjoint sets V, U\,..., 17jt,...
such that i£Uk CQ,Vf)Q = 0, then Q £ Ak-
Theore m XI . Consider a finite sequence of algebras A\,..., An- Suppose there exists a
matrix of pairwise disjoint sets
(°L::.5.Y
where rrtk 3(k—1), Uf £ Ak- Then there exist pairwise disjoint sets U\,..., Un, V\,..., Vn
such that ifUk C Q, Vk H Q = 0, then Q £ Ak, 1 k n.
Theore m XII . Consider a countable sequence of a-algebras A\,..., Ak-,... Suppose
there exists a matrix of pairwise disjoint sets
•u\ ... u^\
u? ... u*
mk
where rrik 3(A; 1), Uf ^ Ak- Then there exist pairwise disjoint sets U\,..., Uk, - - -,
V i , . . . , Vi,.. . such that if Uk C 0 , Vk O Q = 0, then Q $ Ak.
We do not claim that the bound |(fc 1) in Theorem IX and the bound 3(k 1) in
Theorem XI are best possible, though they correspond to such bounds in Theorems 10.5
and 10.6. Theorems IX and X are apparently true if rrik k 1. Then, as is shown in
Section 14 (Proposition 14.1), this bound is in a certain sense best possible. Theorems XI
and XII are apparently true if mi 0 and rrik 2k 1 (k 1). Then, as is shown in
Section 14 (Proposition 14.2), this bound is in a certain sense best possible.
Remark 2.2. Obviously, the statement about the existence of sets V, {7i,...,£/
n
(in the
first part of Theorem V, and in Theorem IX) is equivalent to the following statement: there
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