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](less than or equal to aleph 0 ) such that, for any a, b is an element of P, if a less than or equal to b then there exists c is an element of f(a)boolean AND f(b) such th at a less than or equal to c less than or equal to b. In this note, we study the WFN and some of its generalizations. Some features of the c lass of Boolean algebras with the WFN seem to be quite sensitive to ad ditional axioms of set theory: e.g. under CH, every ccc complete Boole an algebra has this property while, under b greater than or equal to a leph(2), there exists no complete Boolean algebra with the WFN (Theore m 6.2).