  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}.
