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.
Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free lattice-ordered Abelian groups / V. Marra. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 20:3(2008 May), pp. 505-513.
Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free lattice-ordered Abelian groups
V. MarraPrimo
2008
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.Pubblicazioni consigliate
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