Special-valued \(\ell\)-groups
type: Item
instance of: scholarly article
title: Special-valued \(\ell\)-groups
zbMATH Open document ID: 0582.06017
DOI: 10.1007/BF01224024
author: J. Patrick Bixler
author: Michael R. Darnel
published in: Algebra Universalis
publication date: 1986-01-01T00:00:00Z
review text: Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call such \(\ell\)- groups special-valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof that S, the class of special-valued \(\ell\)-groups, is closed with respect to joins of convex \(\ell\)-subgroups, incidentally giving a direct proof that S is a quasitorsion class. This proof is then used to show that the special- valued and finite-valued kernels of \(\ell\)-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valued \(\ell\)-group is special-valued and is an \(a^*\)- extension of the original \(\ell\)-group. Our most important result is that the lateral completion of a completely distributive normal-valued \(\ell\)-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valued \(\ell\)-group can be v-embedded into a special-valued \(\ell\)-group.
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 3931085
zbMATH Keywords: special values
zbMATH Keywords: lattice-ordered groups
zbMATH Keywords: totally-ordered groups
zbMATH Keywords: lexicographic extensions
zbMATH Keywords: regular subgroups
zbMATH Keywords: polars
zbMATH Keywords: minimal prime subgroups
zbMATH Keywords: special elements
zbMATH Keywords: special-valued \(\ell \)-groups
zbMATH Keywords: joins of convex \(\ell \)-subgroups
zbMATH Keywords: quasitorsion class
zbMATH Keywords: lateral completion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3931085