### Partial orderings with the weak Freese-Nation property

by Fuchino and Koppelberg and Shelah. [FKSh:549]

Annals Pure and Applied Logic, 1996

A partial ordering P is said to have the weak
Freese-Nation property (WFN) if there is a mapping
f:P ---> [P]^{<= aleph_0} such that, for any a, b in P, if a
<= b then there exists c in f(a) cap 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 BAs
with the WFN seem to be quite sensitive to additional axioms of
set theory: e.g., under CH, every ccc cBA has this property
while, under b >= aleph_2, there exists no cBA with the
WFN.

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