3 M ay 1 99 6 More set - theory around the weak Freese - Nation property

For a regular κ, a partial ordering P is said to have the κ Freese-Nation property (the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P ] such that for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q. Freese and Nation [5] used the א0-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness. L. Heindorf gave a negative answer to the question showing that the Boolean algebras with the א0-FN are exactly those which are openly generated. It is known that the class of openly generated Boolean algebras contains projective Boolean algebras as a proper subclass (see [8] — openly generated Boolean algebras are called ‘rcfiltered’ there). Heindorf and Shapiro [8] then studied the א1-FN which they called the weak Freese-Nation property and proved some elementary properties of the Boolean algebras with this property. Partial orderings with the κ-FN for arbitrary regular κ were further studied in Fuchino, Koppelberg and Shelah [6]. Koppelberg [10] gives some nice applications of the א1-FN. In the following we shall quote some elementary facts from [6] which we need later. First of all, it can be readily seen that every small partial ordering has the κ-FN: