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)

@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}
}