Finite and countable families of algebras of sets

L. S. Grinblat
2010 Mathematical Research Letters  
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
more » ... 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: 1.2. 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 define N n = {k ∈ N + | k ≤ n}. As 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). 2000 Mathematics Subject Classification. Primary 03E05, Secondary 54D35.
doi:10.4310/mrl.2010.v17.n4.a3 fatcat:spues2soirc7piyopxegy6xn7y