1 5 A ug 1 99 5 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 ]0 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). 0) Institut für Mathematik II, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany. 1) Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel. 2) Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, U.S.A. This paper was written while the first author was at the Hebrew University of Jerusalem. He would like to thank The Israel Academy of Science and Humanities for enabling his stay there. The second author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. Ko 490/7–1. The third author would like to thank “Basic Research Foundation” of The Israel Academy of Science and Humanities and the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Germany) for their support. The present paper is the third author’s Publication No. 549. The authors would like to thank the referee for some variable suggestions.