Maximal chains in positive subfamilies of \(P(\omega )\)
type: Item
instance of: scholarly article
title: Maximal chains in positive subfamilies of \(P(\omega )\)
zbMATH Open document ID: 1257.06001
author: Miloš S. Kurilić
published in: Order
publication date: 2012-03-23T00:00:00Z
review text: 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.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 6018073
zbMATH Keywords: positive family
zbMATH Keywords: maximal chain
zbMATH Keywords: complete lattice
zbMATH Keywords: Boolean algebra
zbMATH Keywords: Boolean linear order
zbMATH Keywords: interval algebra
MaRDI profile type: Publication
full work available at URL: s11083-011-9201-9
OpenAlex ID: W2076890888
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DOI: 10.1007/S11083-011-9201-9
description: scientific article; zbMATH DE number 6018073