TY - JOUR
T1 - Partial orderings with the weak Freese-Nation property
AU - Fuchino, Sakaé
AU - Koppelberg, Sabine
AU - Shelah, Saharon
PY - 1996/7/15
Y1 - 1996/7/15
N2 - A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f : P → [P] ≤א0 such that, for any a, b ∈ P, if a ≤ b then there exists c ∈ f(a)∩f(b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, every ccc complete Boolean algebra has this property while, under b ≥ א2, there exists no complete Boolean algebra with the WFN (Theorem 6.2).
AB - A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f : P → [P] ≤א0 such that, for any a, b ∈ P, if a ≤ b then there exists c ∈ f(a)∩f(b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, every ccc complete Boolean algebra has this property while, under b ≥ א2, there exists no complete Boolean algebra with the WFN (Theorem 6.2).
UR - http://www.scopus.com/inward/record.url?scp=0030585894&partnerID=8YFLogxK
U2 - 10.1016/0168-0072(95)00047-X
DO - 10.1016/0168-0072(95)00047-X
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AN - SCOPUS:0030585894
SN - 0168-0072
VL - 80
SP - 35
EP - 54
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1
ER -