### On ultraproducts of Boolean Algebras and irr

by Shelah. [Sh:703]

Archive for Math Logic, 2003

We prove the consistency of irr (prod limits_{i< kappa}
B_i/D)< prod limits_{i< kappa} irr (B_i)/D, where D is an
ultrafilter on kappa and each B_i is a Boolean Algebra. This
solves the last problem of this form from the Monk's list of
problems, that is number 35. The solution applies to many other
properties, e.g., Souslinity.
Next, we get similar results with kappa = aleph_1 (easily we
cannot have it for kappa = aleph_0) and Boolean Algebras B_i
(i< kappa) of cardinality < beth_{omega_1} .

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