Maximal Chains in Positive Subfamilies of P(ω)

摘要

A family P ⊂ [ω]~ω is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/ Fin. For example, if I ⊂ P(ω) is an ideal containing the ideal Fin of finite subsets of ω, then P(ω) I is a positive family and the set Dense (Q) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(Q). We prove that, for a positive family V, the order types of maximal chains in the complete lattice (P ∪ {O}, ⊂) are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg [0, 1)_R and the poset E(Q), where E(Q) is the set of elementary submodels of the rational line.