# Sh:712

- Fuchino, S., Geschke, S., Shelah, S., & Soukup, L. (2001).
*On the weak Freese-Nation property of complete Boolean algebras*. Ann. Pure Appl. Logic,**110**(1-3), 89–105. arXiv: math/9911230 DOI: 10.1016/S0168-0072(01)00023-9 MR: 1846760 -
Abstract:

The following results are proved: (a) In a Cohen model, there is always a ccc complete Boolean algebras without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a ccc complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of \neg0^\#, the weak Freese-Nation property of ({\mathcal P}(\omega),{\subseteq}) is equivalent to the weak Freese-Nation property of any of {\mathbb C}(\kappa) or {\mathbb R}(\kappa) for uncountable \kappa. (d) Modulo consistency of (\aleph_{\omega+1},\aleph_\omega) \longrightarrow(\aleph_1,\aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding \aleph_\omega Cohen reals destroys the weak Freese-Nation property of ({\mathcal P}(\omega),{\subseteq}). - Version 2001-01-29_10 (19p) published version (17p)

Bib entry

@article{Sh:712, author = {Fuchino, Saka{\'e} and Geschke, Stefan and Shelah, Saharon and Soukup, Lajos}, title = {{On the weak Freese-Nation property of complete Boolean algebras}}, journal = {Ann. Pure Appl. Logic}, fjournal = {Annals of Pure and Applied Logic}, volume = {110}, number = {1-3}, year = {2001}, pages = {89--105}, issn = {0168-0072}, mrnumber = {1846760}, mrclass = {03E35 (03E55 06E10)}, doi = {10.1016/S0168-0072(01)00023-9}, note = {\href{https://arxiv.org/abs/math/9911230}{arXiv: math/9911230}}, arxiv_number = {math/9911230} }