On Monk's questions

Abstract

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.