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Forum Mathematicum

Managing Editor: Brüdern, Jörg

Editorial Board Member: Bruinier, Jan Hendrik / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Gallavotti, Giovanni / Garnier, Josselin / Neeb, Karl-Hermann / Noguchi, Junjiro / Ranicki, Andrew / Segal, Dan / Shahidi, Freydoon / Shigekawa, Ichiro / Sogge, Christopher D. / Strambach, Karl

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

Citation Information: Forum Mathematicum. Volume 20, Issue 3, Pages 505–513, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/FORUM.2008.025, May 2008

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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.

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