Finitely presented, coherent, and ultrasimplicial ordered abelian groups
  We study notions such as &#64257;nite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is &#64257;nitely presented if and only if G is &#64257;nitely 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 &#64257;nite. (ii) Torsion-free, &#64257;nitely presented partially ordered abelian groups can be represented as subgroups of some Z^n, with a &#64257;nitely generated submonoid of (Z+)^n as positive cone. (iii) Every unperforated, &#64257;nitely 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 &#64257;nite subset of G is contained into a &#64257;nitely presented ordered subgroup. 
Jean-Fran\c{c}ois Caillot (LMNO), Friedrich Wehrung (LMNO)