A characterization of the Dedekind completion of a totally ordered group of infinite rank
type: Item
instance of: scholarly article
title: A characterization of the Dedekind completion of a totally ordered group of infinite rank
zbMATH Open document ID: 1184.06008
DOI: 10.1016/S0019-3577(08)80025-0
author: E. Olivos
author: Herme Soto
author: Angélica Mansilla
published in: Indagationes Mathematicae. New Series
publication date: 2009-10-22T00:00:00Z
review text: Let \(G\) be a directed partially ordered abelian group and \(X\) a nonempty subset of \(G\), throughout. If \(X\) is bounded from above (below), let \(U(X)\) (respectively, \(L(X)\)) denote the set of all the upper (lower) bounds of \(X\). To each \(X\) bounded from above we can associate \(X^{\#}=L(U(X))\). The sets \( X^{\#}\) serve as a generalization of Dedekind cuts to p.o. sets. The set \( G^{\#}\) of all \(X^{\#}\) for suitable subsets \(X\) of \(G\) can be shown to be a conditionally complete lattice under inclusion and a semigroup under a compatible multiplication defined by \((XY)^{\#}=(X^{\#}Y)^{\#}\) [\textit{L. Fuchs}, Partially ordered algebraic systems. Oxford-London-New York-Paris: Pergamon Press (1963; Zbl 0137.02001)]. Also \(G\) can be embedded in \(G^{\#}\) via the mapping \(a\mapsto (a)^{\#}=L(a)\). \(G^{\#}\) is called the Dedekind completion of \(G\). If \(G\) is a lattice-ordered group it can be regarded as a sublattice of \(G^{\#}\). Finally, \(G^{\#}\) is a group if and only if \(G\) is completely integrally closed. In search of norms of vectors and linear operators in non-Archimedian functional analysis the authors of the paper under review study the Dedekind completion \(\Gamma _{\alpha }^{\#}\) of the totally ordered group \(\Gamma _{\alpha }\) where \(\alpha \) is an ordinal and \(\Gamma _{\alpha }\) is a sort of Hahn product of totally ordered groups \(\{G_{\beta }\}_{\beta <\alpha }\) of rank one. The rank of each \(\Gamma _{\alpha }\) is the order type of \( \alpha\). The first author has discussed \(\Gamma _{\alpha }\) in detail in [\textit{E. Olivos}, Ann. Math. Blaise Pascal 12, No. 1, 79--90 (2005; Zbl 1085.06010)]. For this review, we note that with \(\alpha \) and \(G_{\beta }\) as above \(\Gamma _{\alpha }=\{f:\alpha \rightarrow \bigcup G_{\beta }:f(\beta )\in G_{\beta }\) and supp(\( f \)) is finite\}. The multiplication in \(\Gamma _{\alpha }\) is componentwise and the ordering is antilexicographic. The aim of this study is…
reviewed by: Muhammad Zafrullah
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 5619298
zbMATH Keywords: totally ordered group
zbMATH Keywords: Dedekind completion
zbMATH Keywords: rank of a group
MaRDI profile type: Publication
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full work available at URL: s0019-3577(08)80025-0
OpenAlex ID: W2091031375
description: scientific article; zbMATH DE number 5619298