### Special Subsets of ${}^{{\rm cf}(\mu)}\mu$, Boolean Algebras and Maharam measure Algebras

by Shelah. [Sh:620]

Topology and its Applications, 1999

The original theme of the paper is the existence proof of ``there
is bar {eta}=< eta_alpha : alpha < lambda > which is a
(lambda,J)-sequence for bar {I}=< I_i:i< delta >, a sequence
of ideals. This can be thought of as in a generalization to
Luzin sets and Sierpinski sets, but for the product
prod_{i< delta} Dom(I_i), the existence proofs are related to
pcf
. The second theme is when does a Boolean algebra B has free
caliber
lambda (i.e. if X subseteq B and |X|= lambda, then for some
Y subseteq X with |Y|= lambda and Y is independent). We consider
it for B being a Maharam measure algebra, or B a (small) product
of
free Boolean algebras, and kappa-cc Boolean algebras. A central
case
lambda = (beth_omega)^+ or more generally, lambda = mu^+ for
mu strong limit singular of ``small'' cofinality. A second one
is
mu = mu^{< kappa}< lambda < 2^mu ; the main case is lambda
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(-) Length, etc.

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