8 L.S. GRINBLAT

Theore m IV. (1) Consider an at most countable sequence of algebras A\,..., Ak,....

Suppose there exists a matrix

I

l

V..

V

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

ui ui

(each row from the third row on contains two sets) for which Uf £ Ak- Then there exists

a set U

$L

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