More on the laterally \(\sigma\)-complete reflection of an Archimedean lattice-ordered group
type: Item
instance of: scholarly article
title: More on the laterally \(\sigma\)-complete reflection of an Archimedean lattice-ordered group
zbMATH Open document ID: 0936.06016
DOI: 10.1023/A:1006274508999
author: Anthony W. Hager
author: Jorge Martinez
published in: Order
publication date: 2000-05-14T00:00:00Z
review text: This paper can be considered as a continuation of a previous article of the authors [``The laterally \(\sigma\)-complete reflection of an Archimedean lattice-ordered group'', in: W. C. Holland and J. Martinez (eds.), Ordered algebraic structures. Proc. Curacao Conf. 1995 (Kluwer, Dordrecht) (1997; Zbl 0874.06013)]. Three categories of \(\ell\)-groups are considered: (i) \textbf{Arch} -- the category of all Archimedean \(\ell\)-groups and all \(\ell\)-homomorphisms between them; (ii) \textbf{W} -- the category of all Archimedean \(\ell\)-groups with a distinguished order unit, whose morphisms preserve the unit; (iii) \textbf{W}\(_s\) -- the full subcategory of \textbf{W} whose objects are the \(\ell\)-groups such that the distinguished order unit is singular. Further, let \(\Sigma^A\) (resp. \(\Sigma^W\)) be the class of all laterally \(\sigma\)-complete \(\ell\)-groups in \textbf{Arch} (resp. in \textbf{W}); in the paper quoted above it has been proved that \(\Sigma^A\) is a monoreflective subcategory in \textbf{Arch}, and \(\Sigma^W\) in \textbf{W}. The results of the present paper are too technical to be reproduced here; the main result is characterized in the authors' abstract as follows: ``The laterally \(\sigma\)-complete reflection is given a new, one-step construction, which, in some notable cases affords an explicit realization. For example, if \(X\) is a compact Hausdorff topological space, the laterally \(\sigma\)-complete reflection of \(C(X)\) consists of all real-valued functions on \(X\) which are countably Baire-piecewise continuous''.
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18A20
zbMATH DE Number: 1340446
zbMATH Keywords: Archimedean \(\ell\)-group
zbMATH Keywords: strong unit
zbMATH Keywords: singular element
zbMATH Keywords: laterally \(\sigma\)-complete reflection
reviewed by: Ján Jakubík
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1340446