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More constructions for Boolean algebras. (English) Zbl 1023.03043

Summary: We construct Boolean algebras with prescribed behaviour concerning depth for the free product of two Boolean algebras over a third, in ZFC using pcf; assuming squares we get results on ultraproducts. We also deal with the family of cardinalities and topological density of homomorphic images of Boolean algebras (you can translate it to topology – on the cardinalities of closed subspaces); and lastly we deal with inequalities between cardinal invariants, mainly \(d(B)^\kappa < |B|\Rightarrow \text{ind}(B) > \kappa \vee \text{Depth}(B) \geq {\log}(|B|)\).

MSC:

03E04 Ordered sets and their cofinalities; pcf theory
06E05 Structure theory of Boolean algebras
03E35 Consistency and independence results
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