Finite and countable families of algebras of sets

Let {Aλ} λ ε ∧ be a family of algebras of sets defined on a set X, 0 < ≠(∧) ≥ א0, and Aλ ≠ P(X) for each λ ε ∧. We assume that Aλ are σ-algebras if ≠( ∧) = א0. We obtained the necessary and sufficient conditions for which ∪λ ε ∧ Aλ = P(X). In the formulation of these conditions we use ω-saturated algebras and finite sequences of ultrafilters on X.