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Forum Mathematicum

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Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 20, Issue 3 (May 2008)


Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free lattice-ordered Abelian groups

Vincenzo Marra
  • Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano, via Comelico, 39-41, I-20135 Milan, Italy.
  • Email:
Published Online: 2008-05-23 | DOI: https://doi.org/10.1515/FORUM.2008.025


We investigate the structure of lattice-preserving homomorphisms of free lattice-ordered Abelian groups to the ordered group of integers. For any lattice-ordered group, a choice of generators induces on such homomorphisms a partial commutative monoid canonically embedded into a direct product of the group of integers. Free lattice-ordered Abelian groups can be characterised in terms of this dual object and its embedding. For finite sets of generators, we obtain the stronger result: a lattice-ordered Abelian group is free on a finite generating set if and only if the generators make ℤ-valued homomorphisms a free Abelian group of finite rank. One of the main points of the paper is that all results are proved in an entirely elementary and self-contained manner. To achieve this end, we give a short new proof of the standard result of Weinberg that free lattice-ordered Abelian groups have enough ℤ-valued homomorphisms. The argument uses the ultrasimplicial property of ordered Abelian groups, first established by Elliott in a different connection. The paper is made self-contained by a new proof of Elliott's result.

2000 Mathematics Subject Classification: 06F20.

About the article

Received: 2006-09-10

Revised: 2006-11-16

Published Online: 2008-05-23

Published in Print: 2008-05-01

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/FORUM.2008.025. Export Citation

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