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From free abelian groups to free abelian \(\ell \)-groups. (English) Zbl 1265.06055
Using the Baer-Specker theorem, A. Glass and V. Marra proved that the underlying group of any finitely generated abelian lattice-ordered group is free. The authors of the present paper give an elementary geometric proof of this theorem without using the Baer-Specker theorem. Moreover, they prove the converse statement: for each natural number \(n = 2, 3,\dots \), the free abelian group \(F\) on countably many free generators can be equipped with a distinguished monoid \(P_n\) making \((F,P_n)\) a free \(n\)-generated abelian lattice-ordered group.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20K99 Abelian groups
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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