A partial ordering P is said to have the weak Freese-Nation property if there is a mapping tf : P → [P]0 such that, for any a, b ε P, if a b then there exists c ε tf∩tf 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 2, there exists no complete Boolean algebra with the WFN. (shrink)

We consider a class of Lindström extensions of first-order logic which are susceptible to a natural Skolemization procedure. In these logics Ehrenfeucht Mostowski (EM) functors for theories with arbitrarily large models can be obtained under suitable restrictions. Characteristic dependencies between algebraic properties of the quantifiers and the maximal domains of EM functors are investigated.Results are applied to Magidor Malitz logic,L(Q <ω), showing e.g. its Hanf number to be equal to ℶω(ℵ1) in the countably compact case. Using results of Baumgartner, the (...) maximal number of isomorphism types of linearly ordered models of regular cardinality is shown to be achieved for theories that admit an EM functor on a typically restricted domain. (shrink)

We show how to build various models of first-order theories, which also have properties like: tree with only definable branches, atomic Boolean algebras or ordered fields with only definable automorphisms. For this we use a set-theoretic assertion, which may be interesting by itself on the existence of quite generic subsets of suitable partial orders of power λ + , which follows from ♦ λ and even weaker hypotheses . For a related assertion, which is equivalent to the morass see Shelah (...) and Stanley [16]. The various specific constructions serve also as examples of how to use this set-theoretic lemma. We apply the method to construct rigid ordered fields, rigid atomic Boolean algebras, trees with only definable branches; all in successors of regular cardinals under appropriate set- theoretic assumptions. So we are able to answer the following algebraic question. Saltzman's Question . Is there a rigid real closed field, which is not a subfield of the reals? (shrink)