On the weak Freese-Nation property of complete Boolean algebras

Sakaé Fuchino*, Stefan Geschke, Saharon Shelah, Lajos Soukup

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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+1ω) ↠ (א10), 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.

Original languageEnglish
Pages (from-to)89-105
Number of pages17
JournalAnnals of Pure and Applied Logic
Volume110
Issue number1-3
DOIs
StatePublished - 20 Jun 2001

Keywords

  • Chang's conjecture
  • Cohen algebra
  • Cohen model
  • Complete Boolean algebras
  • Random algebra
  • Weak Freese-Nation property

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