The structure of free Abelian \(\ell\)-groups and free vector lattices
type: Item
instance of: scholarly article
title: The structure of free Abelian \(\ell\)-groups and free vector lattices
zbMATH Open document ID: 0617.06014
DOI: 10.1007/BF01237715
author: Michael R. Darnel
published in: Algebra Universalis
publication date: 1986-01-01T00:00:00Z
review text: The paper gives the prime and regular subgroup structures for free abelian \(\ell\)-groups (lattice-ordered groups). An \(\ell\)-subgroup H (both a subgroup and a sublattice) of an \(\ell\)-group G is convex if \(0\leq g\leq h\in H\) forces \(g\in H\). A convex \(\ell\)-subgroup P is prime if \(a\wedge b=0\) forces either \(a\in P\) or \(b\in P\), and regular, if it is maximal with respect to not containing an element of G. If \(\mu\) is a cardinal, let \(A_{\mu}\) denote the free abelian \(\ell\)-group of rank \(\mu\). For an \(\ell\)-group G, the depth of G is defined to be the maximal cardinality of any chain of primes of G. \(\Gamma\) (G) denotes the set of the regular subgroups of G, partially ordered by inclusion. \(\Gamma\) (G) is then a root system: for any \(M\in \Gamma (G)\) the set \(\{\) \(N\in \Gamma (G)|\) \(M\subset N\}\) is totally ordered. Theorem 2.8. For a finite n, \(depth(A_ n)=n\). Thus every proper prime is regular. - Theorem 2.15. For finite \(n\geq 2\), \(\Gamma (A_ n)\) consists of countably infinitely many copies of \(\Gamma (A_{n- 1}{\overset \leftarrow \times}{\mathbb{Z}})\) and \(2^{\omega}\) many copies of \(\Gamma (A_{n-1})\). With minor adaptations, analogues for free vector lattices are proved. Some results are obtained in the infinite rank case.
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 4000001
zbMATH Keywords: free abelian \(\ell \)-groups
zbMATH Keywords: lattice-ordered groups
zbMATH Keywords: convex \(\ell \)- subgroup
zbMATH Keywords: regular subgroups
zbMATH Keywords: root system
zbMATH Keywords: free vector lattices
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
cites work: Groupes et anneaux reticules
cites work: Q5672682
cites work: Free abelian \(\ell\)-groups and vector lattices
cites work: An Even Better Representation for Free Lattice-Ordered Groups
cites work: Free lattice-ordered Abelian groups
full work available at URL: https://doi.org/10.1007/bf01237715
OpenAlex ID: W1989033511
description: scientific article; zbMATH DE number 4000001