Countably complementable linear orderings
type: Item
instance of: scholarly article
title: Countably complementable linear orderings
zbMATH Open document ID: 1121.06001
author: Antonio Montalbán
published in: Order
publication date: 2007-04-26T00:00:00Z
review text: Let \({\mathbf L}\) denote the set of countable linear orderings, and \(\preccurlyeq\) the embeddability relation. Given \({\mathcal L}\in{\mathbf L}\), let \({\mathbf L}({\mathcal L})= \{{\mathcal B}\in{\mathbf L}:{\mathcal L}\npreceq{\mathcal B}\}\). A countable linear ordering \({\mathcal L}\) is said to be countably complementable if there exists a linear ordering \(\overline{\mathcal L}\), possibly uncountable, such that for every countable linear ordering \({\mathcal B}\) there holds \({\mathcal B}\in{\mathbf L}({\mathcal L})\Leftrightarrow{\mathcal B}\preceq\overline{\mathcal L}\). For \({\mathcal L}\in{\mathbf L}\) the following is defined: (1) \({\mathcal L}\) has the finitely faithful extension property if for all \({\mathcal A},{\mathcal B}\in{\mathbf L}({\mathcal L})\) there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \({\mathcal A}\preceq{\mathcal C}\) and \({\mathcal B}\preceq{\mathcal C}\). (2) \({\mathcal L}\) has the completely faithful extension property if for every set \(\{{\mathcal A}_i: i\in\omega\}\subseteq{\mathbf L}({\mathcal L})\) there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \((\forall i\in\omega)\) \({\mathcal A}_i\preceq{\mathcal C}\). (3) \({\mathcal L}\) has the totally faithful extension property if for every set \(\{{\mathcal A}_i: i\in\omega\}\subseteq{\mathbf L}({\mathcal L})\) that is totally ordered by embeddability there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \((\forall i\in\omega)\) \({\mathcal A}_i\preceq{\mathcal C}\). The main theorem of this paper gives four characterization of countable linear orderings that are countably complementable. In particular the following three properties are equivalent: \({\mathcal L}\) is countably complementable, \({\mathcal L}\) has the completely faithful extension property, \({\mathcal L}\) has the finitely faithful extension property. \textit{R. Fraïssé} [C. R. Acad. Sci., Paris 226, 1330--1331 (1948; Zbl 0034.17604)] had menti…
reviewed by: Egbert Harzheim
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 5146176
zbMATH Keywords: countable linear ordering
zbMATH Keywords: extension property
zbMATH Keywords: embeddability
MaRDI profile type: Publication
full work available at URL: s11083-006-9049-6
OpenAlex ID: W1997631835
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DOI: 10.1007/S11083-006-9049-6
description: scientific article; zbMATH DE number 5146176