### 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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