More constructions for Boolean algebras

by Shelah. [Sh:652]
Archive for Math Logic, 2002
We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than kappa . We give a condition (for lambda, mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda, mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth (B_1 oplus_{A_theta} B_1) >= lambda . We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary conditions, involving consistency results. Further we prove that e.g. if B is a BA of cardinality lambda, lambda >= mu and lambda, mu are strong limit singular of the same cofinality, then B has a homomorphic image of cardinality mu (and with mu ultrafilters). Next we show that for a BA B, if d(B)^kappa <|B| then ind (B)> kappa or Depth (B) >= log (|B|) . Finally we prove that if square_lambda holds and lambda = lambda^{aleph_0} then for some BAs B_n, Depth (B_n) <= lambda but for any uniform ultrafilter D on omega, prod_{n< omega} B_n/D has depth >= lambda^+ .

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