Invariant homomorphisms of nonstandard extensions of Boolean algebras and vector lattices
type: Item
instance of: scholarly article
title: Invariant homomorphisms of nonstandard extensions of Boolean algebras and vector lattices
zbMATH Open document ID: 0868.46003
DOI: 10.1007/BF02674623
published in: Siberian Mathematical Journal
publication date: 1997-07-22T00:00:00Z
review text: Let \(B\) be a Boolean algebra and let \({}^{\ast}B\) be its nonstandard extension in a polysaturated Robinson superstructure. A mapping \(h:{}^{\ast}B\rightarrow B\) is called a \(\ast\)-invariant Boolean homomorphism (briefly, \(\ast\)-IBH) if \(h\) is a Boolean homomorphism and \(h(b)=b\) for all \(b\in B\). The main theorem asserts that a \(\ast\)-IBH exists if and only if \(B\) is complete; a \(\ast\)-IBH is uniquely defined if and only if \(B\) is atomic. The method of Boolean-valued models is used in the proof. Further, this result is extended to \(\ast\)-invariant linear homomorphisms of vector lattices. Finally, the author considers a Boolean algebra \(B\) with positive \(\sigma\)-additive measure \(\mu : B\rightarrow \mathbf{R}_+\). He proves that if \((B, \mu)\) is a nonatomic complete normed Boolean algebra and \(h: {}^{\ast}B\rightarrow B\) is a \(\ast\)-IBH then, for every standard \(\varepsilon >0\), there exists a \(\chi_{\varepsilon}\in {}^{\ast}B\) such that \({}^{\ast}\mu (\chi_{\varepsilon})<\varepsilon\) and \(h(b\land\chi_{\varepsilon})=h(b)\) for all \(b\in {}^{\ast}B\). This implies that the measure \(\mu\circ h\) is singular with respect to the Loeb measure \(\text{st}\circ\mu\).
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46S20
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 1037968
zbMATH Keywords: Boolean algebra
zbMATH Keywords: nonstandard extension
zbMATH Keywords: Boolean-valued model
zbMATH Keywords: polysaturated nonstandard extension
zbMATH Keywords: Boolean homomorphism
zbMATH Keywords: nonatomic Boolean algebra
author: Eduard Yu. Emel'yanov
MaRDI profile type: Publication
cites work: Q4192051
cites work: Q4317904
cites work: The order and regular hulls of Riesz spaces
cites work: Q3659775
full work available at URL: https://doi.org/10.1007/bf02674623
OpenAlex ID: W2328312792
Recommended article: Q4219235
Recommended article: Q3484892
Recommended article: Q3197876
Recommended article: Extension of Vector Lattice Homomorphisms
Recommended article: Homogeneous Boolean algebras with very nonsymmetric subalgebras
Recommended article: Q3717602
Recommended article: Q3697277
Recommended article: Q4654563
Recommended article: Invariants, Boolean algebras and ACA₀⁺
description: scientific article; zbMATH DE number 1037968