# Sh:479

- Shelah, S. (1996).
*On Monk’s questions*. Fund. Math.,**151**(1), 1–19. arXiv: math/9601218 MR: 1405517 -
Abstract:

Monk asks (problems 13, 15 in his list; \pi is the algebraic density): "For a Boolean algebra B, \aleph_0\le\theta\le\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)\ge \mu. Here A=\prod_{i<\kappa}A_i/D. By a theorem of Peterson the regularity of \mu is needed. - Version 1996-01-08_10 (22p) published version (19p)

Bib entry

@article{Sh:479, author = {Shelah, Saharon}, title = {{On Monk's questions}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {151}, number = {1}, year = {1996}, pages = {1--19}, issn = {0016-2736}, mrnumber = {1405517}, mrclass = {03E05 (03E35 03G05 04A20 06E05)}, note = {\href{https://arxiv.org/abs/math/9601218}{arXiv: math/9601218}}, arxiv_number = {math/9601218} }