Partial orderings with the weak Freese-Nation property

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).