Special Subsets of Cf(µ) Μ, Boolean Algebras and Maharam Measure Algebras Sh620

The original theme of the paper is the existence proof of “there is η̄ = 〈ηα : α < λ〉 which is a (λ, J)-sequence for Ī = 〈Ii : i < δ〉, a sequence of ideals”. This can be thought of as a generalization to Luzin sets and to Sierpinski sets, but for the product ∏ i<δ Dom(Ii), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B have a free caliber λ (i.e. if X ⊆ B and |X| = λ, then for some Y ⊆ X with |Y | = λ and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and κ-cc Boolean algebras. A central case is λ = (iω) or more generally, λ = μ for μ strong limit singular of “small” cofinality. Second case is μ = μ < λ < 2; the main case is λ regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) etc.