# Sh:703

- Shelah, S. (2003).
*On ultraproducts of Boolean algebras and irr*. Arch. Math. Logic,**42**(6), 569–581. arXiv: math/0012171 DOI: 10.1007/s00153-002-0167-6 MR: 2001060 -
Abstract:

We prove the consistency of {\rm irr}(\prod\limits_{i< \kappa} B_i/D)<\prod\limits_{i<\kappa}{\rm 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}. - published version (13p)

Bib entry

@article{Sh:703, author = {Shelah, Saharon}, title = {{On ultraproducts of Boolean algebras and irr}}, journal = {Arch. Math. Logic}, fjournal = {Archive for Mathematical Logic}, volume = {42}, number = {6}, year = {2003}, pages = {569--581}, issn = {0933-5846}, doi = {10.1007/s00153-002-0167-6}, mrclass = {03E35 (03E05 03E55 03G05)}, mrnumber = {2001060}, mrreviewer = {Andrzej Ros{\l}anowski}, doi = {10.1007/s00153-002-0167-6}, note = {\href{https://arxiv.org/abs/math/0012171}{arXiv: math/0012171}}, arxiv_number = {math/0012171} }