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