Finitely presented, coherent, and ultrasimplicial ordered abelian groups
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.