Complete embeddings of linear orderings and embeddings of lattice-ordered groups
type: Item
instance of: scholarly article
title: Complete embeddings of linear orderings and embeddings of lattice-ordered groups
zbMATH Open document ID: 0619.06001
DOI: 10.1007/BF02782939
author: Manfred Droste
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: For any linearly ordered set \((S,\leq)\), let \((\bar S,\leq)\) denote its Dedekind completion. The main theorem proved is: If \((C,\leq)\) is any linearly ordered set, then no non-trivial closed interval of C is dense if and only if there is a doubly homogeneous linearly ordered set \((S,\leq)\) and a complete embedding \(\Phi\) of \((C,\leq)\) into \((\bar S,\leq)\) such that \(\Phi(\bar C)\subseteq \bar S\setminus S\). Moreover, if this holds and \(k<| C|\), then (S,\(\leq)\) can be chosen of cardinality k, each \(s\in S\) has countable coterminality and whenever \(a<b\in C\) with \((a,b)=\emptyset\), \((\Phi(a),\Phi(b))\subseteq \bar S\) has countable coterminality. The proof follows the construction of Theorem 2.11 of \textit{M. Droste} and \textit{S. Shelah} [Isr. J. Math. 51, 223-261 (1985; Zbl 0608.06001)]. It is further shown that for each regular cardinal k and all regular cardinals \(\mu,\nu\leq k\), there exist up to isomorphism precisely \(2^ k\) doubly homogeneous linearly ordered sets \((S,\leq)\) of cardinality k with pairwise non-isomorphic and non- antiisomorphic Dedekind completions such that each \(s\in S\) has cofinality \(\mu\) and coinitiality \(\nu\). Moreover Aut\((C,\leq)\) can be embedded in Aut\((S;\leq)\) so as to preserve finite maxima and minima of functions. The proofs use the results of Droste and Shelah (op. cit), clever but elementary analysis, and \textit{R. M. Solovay}'s work [in Axiomatic Set Theory, Proc. Symp. Pure Math. 13 I, 397-428 (1971; Zbl 0222.02078)].
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 20B07
zbMATH DE Number: 4004232
zbMATH Keywords: automorphism group
zbMATH Keywords: divisible lattice-ordered groups
zbMATH Keywords: Dedekind completion
zbMATH Keywords: complete embedding
zbMATH Keywords: regular cardinals
zbMATH Keywords: doubly homogeneous linearly ordered sets
reviewed by: A. M. W. Glass
MaRDI profile type: Publication
cites work: Normal Subgroups of Doubly Transitive Automorphism Groups of Chains
cites work: Structure of partially ordered sets with transitive automorphism groups
cites work: The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets
cites work: Completeness properties of certain normal subgroup lattices
cites work: Partially Ordered Sets with Transitive Automorphism Groups
cites work: A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets
cites work: Q3927300
cites work: Q5830676
cites work: The lattice-ordered group of automorphisms of an ordered set
cites work: Transitive lattice-ordered permutation groups
cites work: Bi-Interpretable Groups and Lattices
cites work: The closed prime subgroups of certain ordered permutation groups
cites work: The lattice-ordered group of automorphisms of an \(\alpha\)-set
cites work: Groups of homeomorphisms with manageable automorphism groups
cites work: Q4133662
cites work: Q3949052
cites work: Q5626665
cites work: The logic of time. A model-theoretic investigation into the varieties of temporal ontology and temporal discourse
cites work: Embedding in a Divisible Lattice-Ordered Group
Recommended article: Q3326426
Recommended article: Dedekind Completeness and the Algebraic Complexity of o-Minimal Structures
Recommended article: Q3216704
Recommended article: \(\mathbf{L}(\mathbb{R})\) with determinacy satisfies the Suslin hypothesis
Recommended article: Cardinal Representations for Closures and Preclosures
Recommended article: Sur une propriété des constituantes des ensembles analytiques.
Recommended article: On completeness
Recommended article: Galvin's question on non-\( \sigma \)-well ordered linear orders
Recommended article: Interrelations between the length, the structure and the cardinality of a chain
Recommended article: Dedekind Order Completion ofC(X) by Hausdorff Continuous Functions
description: scientific article; zbMATH DE number 4004232