Lattice-ordered groups with a prescribed minimum for given elements (Q1185234)

Let \(G\) be a divisible torsion-free Abelian group. As is well known, \(G\) admits in general many lattice orders transforming it into a lattice- ordered group. Let \(S\) be a finite subset of \(G\). The authors show that \(G\) admits a lattice order in which \(O=\inf S\) if and only if either (a) \(S\cup\{O\}\) is colinear and \(O\) is an extreme point of \(S\), or (b) \(S\cup\{O\}\) is not colinear and \(O\) is not contained in the interior of the linear convex hull of \(S\) (defined in the natural way, considering \(G\) as a vector space over \(\mathbb{Q}\)). It is also shown, by example, that for free Abelian groups the situation is much more complex.