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.