Finite and countable families of algebras of sets
L. Grinblat · Mathematical Research Letters · 2010
Let {A λ } λ∈Λ be a family of algebras of sets defined on a set X, 0 < #(Λ) ≤ ℵ 0 , and A λ = P(X) for each λ ∈ Λ.We assume that A λ are σ-algebras if #(Λ) = ℵ 0 .We obtained the necessary and sufficient conditions for which λ∈Λ A λ = P(X).In the formulation of these conditions we use ω-saturated algebras and finite sequences of ultrafilters on X. 1.The formulation of results 1.1.The object of our present investigation is algebras of sets.The present article is a further development of the theory formulated in [Gr1],[Gr2],[Gr3], [Gr4], [Gr5].The results of other authors from [E],[S],[G-S],[W] bear a relation to the subject of our research.Definition.By an algebra on a set X we mean a non-empty system of subsets X with the following properties: (1) if M ∈ A, then X M ∈ A; (2) if M 1 , M 2 ∈ A, then M 1 ∪ M 2 ∈ A. Some notations and terms.All algebras and measures are considered on some abstract set X.As usual, P(M ) denotes the set of all subsets of the set M .When it is clear from the context, we will not state explicitly that a set belongs to P(X).The symbol #(M ) denotes the cardinality of the set M .The set M is called countable if #(M ) = ℵ 0 .We assume that #(X) ≥ ℵ 0 .We denote the set of natural numbers by N + .If n ∈ N + , then we defineAs usual, an algebra A is called a σ-algebra, if for any countable sequence M 1 , . . ., M k , . . .∈ A, we have that A ∞ k=1 M k .We will consider ultrafilters on X.Each ultrafilter is a point βX and vice-versa -each point βX is an ultrafilter on X. (Here, as usual, βX is the Stone-Čech compactification of X with discrete topology). Example.There exists a finite sequence of algebras A 1 , ..., A ν , where ν is an odd number ≥ 3, and A i = P(X) for each i ∈ N ν , such that ν i=1 A i = P(X).