Maximal Chains in Positive Subfamilies of P(ω)

Abstract

A family \({\mathcal P} \subset [\omega]^\omega\) is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if \({\mathcal I} \subset P(\omega)\) is an ideal containing the ideal Fin of finite subsets of ω, then \(P(\omega) \setminus {\mathcal I}\) is a positive family and the set \(\mbox{Dense}({\mathbb Q})\) of dense subsets of the rational line is a positive family which is not the complement of some ideal on \(P({\mathbb Q})\). We prove that, for a positive family \({\mathcal P}\), the order types of maximal chains in the complete lattice \(\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle\) 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 \(\mbox{Intalg}[0,1)_{{\mathbb R}}\) and the poset \(E({\mathbb Q})\), where \(E({\mathbb Q})\) is the set of elementary submodels of the rational line.