Families of sets not belonging to algebras and combinatorics of finite sets of ultrafilters

This article is a part of the theory developed by the author in which the following problem is solved under natural assumptions: to find necessary and sufficient conditions under which the union of at most countable family of algebras on a certain set X is equal to (Formula presented.). Here the following new result is proved. Let (Formula presented.) be a finite collection of algebras of sets given on a set X with (Formula presented.), and for each λ there exist at least (Formula presented.) pairwise disjoint sets belonging to (Formula presented.). Then there exists a family (Formula presented.) of pairwise disjoint subsets of (Formula presented.) except the case (Formula presented.); and for each λ the following holds: if (Formula presented.) and Q contains one of the two sets (Formula presented.), and its intersection with the other set is empty, then (Formula presented.).