Injectivity of Archimedean \(\ell \)-groups with order unit
type: Item
instance of: scholarly article
title: Injectivity of Archimedean \(\ell \)-groups with order unit
zbMATH Open document ID: 1192.06013
published in: Algebra Universalis
publication date: 2010-06-16T00:00:00Z
review text: Let \(\mathbf W\) be the category of Archimedean \(\ell\)-groups with specified weak order unit and unit-preserving \(\ell\)-group homomorphisms. In unpublished notes (circa 1985), Hager proved by a lengthy argument that \(\mathbf W\) has only trivial injectives. The main contribution of the present paper consists in presenting two different proofs of the mentioned result, which are shorter and simpler than Hager's previous argument. Further, the authors deal with two subcategories of \(\mathbf W\), which are denoted as \(\mathbf{SW}\) and \(\mathbf W^*\). The subcategory \(\mathbf{SW}\) is given by the \(A\in \mathbf W\) for which the order unit is singular; the subcategory \(\mathbf W^*\) is given by the objects whose order unit is strong. It is proved that \(\mathbf{SW}\) has only trivial injectives and that \(\mathbf W^*\) has injective hulls.
reviewed by: Ján Jakubík
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 5721490
zbMATH Keywords: Archimedean \(\ell\)-group
zbMATH Keywords: weak order unit
zbMATH Keywords: injectivity
author: Bernhard Banaschewski
author: Anthony W. Hager
MaRDI profile type: Publication
Wikidata QID: Q122961137
full work available at URL: s00012-010-0042-y
OpenAlex ID: W2150169326
cites work: Q3830990
cites work: Q4396195
cites work: Q4440515
cites work: On the function ring functor in pointfree topology
cites work: Stone-Čech compactification of locales. II
cites work: The essential closure of an Archimedean lattice-ordered group
cites work: Projective topological spaces
cites work: Lindelöf locales and realcompactness
cites work: Nontrivial 𝔪-injective Boolean algebras do not exist
cites work: Some Homological Pathology in Vector Lattices
DOI: 10.1007/S00012-010-0042-Y
description: scientific article; zbMATH DE number 5721490