8 L.S. GRINBLAT
Theore m IV. (1) Consider an at most countable sequence of algebras A\,..., Ak,....
Suppose there exists a matrix
of pairwise disjoint sets such that Uf (£ Ak', ni = ri2 = I; rik 1 for all k 2; if k — » oo,
then nk —• oo. Then there exists a set U £ Ak for all k. (2) If k — oo but limn* oo,
then the corresponding set U may not exist.
A particular case of Theorem IV is the following
Corollary 2.1. Consider a finite sequence of algebras A\,... ,*4n such that there exists
a matrix of pairwise disjoint sets
(each row from the third row on contains two sets) for which Uf £ Ak- Then there exists
a set U
Ak for all k n.
Remark 2.1. Corollary 2.1 follows quite obviously from the arguments presented in the
proof of Theorem III (see Remark 8.2).
Theore m V. (1) Consider a finite sequence of algebras A\,... , An such that for every
k y£ 2, I k n, there exist more than |(fc — 1) pairwise disjoint sets not in Ak (if A2 is
taken into consideration, A2 7^ V(X)), and such that if n 1, there exist three pairwise
disjoint sets each of which is not a member of either A\, A2- Then there exist pairwise
disjoint sets V, U\,..., Un such that if Uk C Q, V f) Q = 0, then Q £ Ak, 1 k n.
(2) The bound |(fc — 1) is best possible in the following sense: For every natural number
n one can construct a sequence of algebras A\,..., An, A„+i such that for any k (k ^ 2,
1 k n) there exist more than |(fc — 1) pairwise disjoint sets not in Ak', there exists
[^p] pairwise disjoint sets not in A%+1, and such that if n 1, A2 7^ V(X), and there