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- W2003672364 abstract "The following results are proved: (a) In a model obtained by adding ℵ 2 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 of □ μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ >cf( μ )= ω , then the weak Freese-Nation property of 〈 P (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 (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold , and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (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." @default.
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- W2003672364 date "2001-06-01" @default.
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- W2003672364 title "On the weak Freese-Nation property of complete Boolean algebras" @default.
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- W2003672364 doi "https://doi.org/10.1016/s0168-0072(01)00023-9" @default.
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