2005/01/25 by Jean-François Caillot, Friedrich Wehrung
Mathematics · #math.GM #msc:06F20 #msc:06F25 #msc:15A39 #msc:12J15
published as Semigroup Forum 61 (2000) 116--137
arxiv created 2005/01/25 · arxiv updated 2009/12/01
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 Zn, 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.