On finite and totally finite elements in vector lattices
type: Item
instance of: scholarly article
title: On finite and totally finite elements in vector lattices
zbMATH Open document ID: 0881.46007
DOI: 10.1007/BF01911127
published in: Analysis Mathematica
publication date: 1998-02-19T00:00:00Z
review text: Let \(X\) be an Archimedean vector lattice. An element \(\phi\) in \(X\) is called finite if there exists a \(z\in X\) such that for every \(x\in X\) there exists a number \(C_x\) such that \(|x|\wedge n|\phi|\leq C_xz\) for all \(n\in\mathbb{N}\). The element \(z\) is then called a majorant for \(\phi\). The set of all finite elements is an ideal in \(X\) and is denoted by \(\Phi_1(X)\). An element of \(X\) which has a majorant which itself belongs to \(\Phi_1(X)\), is called totally finite. The set of all totally finite elements is an ideal in \(X\) and is denoted by \(\Phi_2(X)\). It is clear that \(\{0\}\subset\Phi_1(X)\subset \Phi_2(X)\subset X\). If, in the latter chain of inclusions, one replaces all of the inclusion signs by either equality or strict inclusion signs, the question arises whether each of the eight ensuing possibilities is realized by some vector lattice \(X\). That question is fully answered in the present paper. For instance, the situation \(\{0\}= \Phi_1(X)\neq \Phi_2(X)\neq X\) turns out to be impossible. Furthermore, a certain topological condition in the set of all maximal ideals of \(X\) equipped with the hull-kernel topology is introduced, under which \(\Phi_1(X)= \Phi_2(X)\).
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 817242
zbMATH Keywords: Archimedean vector lattice
zbMATH Keywords: totally finite
zbMATH Keywords: maximal ideals
zbMATH Keywords: hull-kernel topology
author: Martin R. Weber
reviewed by: Gerard Buskes
MaRDI profile type: Publication
cites work: Lattices of continuous functions
cites work: Lattices of Continuous Functions II
cites work: Einige Untersuchungen des Raumes der maximalen Ideale eines Vektorverbandes mit Hilfe finiter Elemente II. (Einige Eigenschaften des Raumes der (ml)-Ideale eines Vektorverbandes vom Typ (σ))
cites work: Über die Realisierung von Vektorverbänden III
cites work: Q5682782
cites work: [Russian Text Ignored.]
full work available at URL: https://doi.org/10.1007/bf01911127
OpenAlex ID: W1784122517
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description: scientific article; zbMATH DE number 817242