Maximal chains in positive subfamilies of \(P(\omega )\)
A family \(\mathcal{P} \subseteq P(\omega)\) is positive if (i) \(\emptyset \notin \mathcal{P} \neq \emptyset\); (ii) if \(A \in \mathcal{P}\) and \(A \subseteq^* B \subseteq \omega\), then \(B \in \mathcal{P}\); and (iii) there is some \(A \in \mathcal{P}\) such that \(\omega \setminus A\) is infinite. The author investigates the order types of maximal chains in positive sets. He shows 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 \(\mathbb{R}\) having the minimum nonisolated. He compares this result with the corresponding results concerning maximal chains in the Boolean algebra \(P(\omega)\) and Intalg\([0,1]_{\mathbb{R}}\) and also for the poset \(E(\mathbb{Q})\) of elementary submodels of the rational line.
scientific article; zbMATH DE number 6018073