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On the weak Freese-Nation property of complete Boolean algebras |
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| Authors: | Saka Fuchino, Stefan Geschke, Saharon Shelah,Lajos Soukup |
| Affiliation: | a Kitami Institute of Technology, Kitami, Japan b II. Mathematisches Institut, Freie Universität Berlin, Germany c Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel d Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA e Institute of Mathematics, Hungarian Academy of Science, Budapest, Hungary |
| Abstract: | The following results are proved: (a) In a model obtained by adding 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]0,)=μ+ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (ω+1,ω)(1,0), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ω Cohen reals destroys the weak Freese-Nation property of . 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. |
| Keywords: | Weak Freese-Nation property Complete Boolean algebras Cohen model Chang's conjecture Cohen algebra Random algebra |
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