### 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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