Special subsets of cf(μ)μ, Boolean algebras and Maharam measure algebras

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 Sierpinski sets, but for the product ∏i<δdom(Ii), the existence proofs are related to pcf. cond 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=(ℶω)+, or more generally, λ=μ+ for μ strong limit singular of “small” cofinality. A second one is μ=μ<κ<λ<2μ; 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.