Partial orderings with the weak Freese-Nation property Original Research Article

A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping View the MathML source such that, for any a, bϵP, if a ⩽ b then there exists cϵtf(a)∩tf(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 View the MathML source, there exists no complete Boolean algebra with the WFN (Theorem 6.2).