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Finitely presented, coherent, and ultrasimplicial ordered abelian groups

Jean-François Caillot·Friedrich Wehrung
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Abstract

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, the positive cone G^+ is well-founded as a partially ordered set, and the set of minimal elements of (G^+)-{0} is finite. (ii) Torsion-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Z^n, with a finitely generated submonoid of (Z+)^n as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.

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Jean-François Caillot, Friedrich Wehrung. 2005-01-25. Finitely presented, coherent, and ultrasimplicial ordered abelian groups. https://arxiv.org/abs/math/0501432

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