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On the weak Freese-Nation property of complete Boolean algebras

✍ Scribed by Sakaé Fuchino; Stefan Geschke; Saharon Shelah; Lajos Soukup

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
164 KB
Volume
110
Category
Article
ISSN
0168-0072

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✦ Synopsis

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 of and cof ([ ] ℵ 0 ; ⊆) = + hold for each ¿ cf ( ) = !, then the weak Freese-Nation property of P(!); ⊆ is equivalent to the weak Freese-Nation property of any of C(Ä) or R(Ä) for uncountable Ä.

(d) Modulo the consistency of (ℵ!+1; ℵ!) (ℵ1; ℵ0), it is consistent with GCH that C(ℵ!) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ! Cohen reals destroys the weak Freese-Nation property of P(!); ⊆ .

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A familiar construction for a Boolean algebra A is its normal completion NA, given by its normal ideals or, equivalently, the intersections of its principal ideals, together with the embedding A → NA taking each element of A to its principal ideal. In the classical setting of Zermelo-Fraenkel set th