TY - JOUR
T1 - On the weak Freese-Nation property of complete Boolean algebras
AU - Fuchino, Sakaé
AU - Geschke, Stefan
AU - Shelah, Saharon
AU - Soukup, Lajos
PY - 2001/6/20
Y1 - 2001/6/20
N2 - The following results are proved: (a) In a model obtained by adding א2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form □μ and cof([μ]א0, ⊆) = μ+ hold for each μ > cf(μ) = ω, then the weak Freese-Nation property of 〈℘(ω), ⊆〉 is equivalent to the weak Freese-Nation property of any of ℂ(κ) or ℝ(κ) for uncountable κ. (d) Modulo the consistency of (אω+1,אω) ↠ (א1,א0), it is consistent with GCH that ℂ(אω) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding אω Cohen reals destroys the weak Freese-Nation property of 〈℘(ω), ⊆〉. These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159-176, and some other problems posed by Geschke.
AB - The following results are proved: (a) In a model obtained by adding א2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form □μ and cof([μ]א0, ⊆) = μ+ hold for each μ > cf(μ) = ω, then the weak Freese-Nation property of 〈℘(ω), ⊆〉 is equivalent to the weak Freese-Nation property of any of ℂ(κ) or ℝ(κ) for uncountable κ. (d) Modulo the consistency of (אω+1,אω) ↠ (א1,א0), it is consistent with GCH that ℂ(אω) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding אω Cohen reals destroys the weak Freese-Nation property of 〈℘(ω), ⊆〉. These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159-176, and some other problems posed by Geschke.
KW - Chang's conjecture
KW - Cohen algebra
KW - Cohen model
KW - Complete Boolean algebras
KW - Random algebra
KW - Weak Freese-Nation property
UR - http://www.scopus.com/inward/record.url?scp=0347749537&partnerID=8YFLogxK
U2 - 10.1016/S0168-0072(01)00023-9
DO - 10.1016/S0168-0072(01)00023-9
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AN - SCOPUS:0347749537
SN - 0168-0072
VL - 110
SP - 89
EP - 105
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1-3
ER -