# 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.
• 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}).
• 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},
doi = {10.1016/S0168-0072(01)00023-9},
mrclass = {03E35 (03E55 06E10)},
mrnumber = {1846760},
mrreviewer = {Judith Roitman},
doi = {10.1016/S0168-0072(01)00023-9},
note = {\href{https://arxiv.org/abs/math/9911230}{arXiv: math/9911230}},
arxiv_number = {math/9911230}
}