On Monk's questions

by Shelah. [Sh:479]
Fundamenta Math, 1996
Monk asks (problems 13, 15 in his list; pi is the algebraic density): ``For a Boolean algebra B, aleph_0 <= theta <= pi (B), does B have a subalgebra B' with pi (B')= theta ?'' If theta is regular the answer is easily positive, we show that in general it may be negative, but for quite many singular cardinals - it is positive; the theorems are quite complementary. Next we deal with pi chi and we show that the pi chi of an ultraproduct of Boolean algebras is not necessarily the ultraproduct of the pi chi 's. We also prove that for infinite Boolean algebras A_i (i< kappa) and a non-principal ultrafilter D on kappa : if n_i< aleph_0 for i< kappa and mu = prod_{i< kappa} n_i/D is regular, then pi chi (A) >= mu . Here A= prod_{i< kappa}A_i/D . By a theorem of Peterson the regularity of mu is needed.


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