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

Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

Marra, Vincenzo
Forum Mathematicum , Volume 20 (3) – May 1, 2008
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References (13)

Publisher
de Gruyter
Copyright
© Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2008.025
Publisher site
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Abstract

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.

Journal

Forum Mathematicum – de Gruyter

Published: May 1, 2008

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