We know that the necessary condition for any partially ordered group to be a group of divisibility is that the group must be a directed group. What is the sufficient condtion for partially ordered group to be a group of divisibility?

Let me begin by saying I am far from an expert on this area; however, I attended a seminar in which M. Zafrullah gave a few talks on this subject. It appears this question you posted has not been answered in quite some time, so I don't feel too terribly about at least posting this in hopes that it might help. At worst, hopefully it is better than nothing. I read a couple of articles back when he was talking just to get some background in this subject area for his talks. One I found particularly insightful and seemed to answer your question at least partly was one by Joe L. Mott entitled "Convex directed subgroups of a group of divisibility." In it he mentions $D$ is a UFD if and only if $G(D)$ is a cardinal sum of copies of $\mathbb{Z}$. $D$ is a valuation ring if and only if $G(D)$ is totally ordered. $D$ is a GCD domain if and only if $G(D)$ is lattice ordered. But I think this is not exactly what you are asking. You want to know, given any partially ordered group, can I tell if it arises as the group of divisibility of some ring $R$. It sounds like this is not an easy question to answer, and I think from your comment it appears you know of Krull's theorem that answers this question in the affirmative for totally ordered groups. And in more generality, it is true for abelian lattice ordered groups by the theorem of Jaffard, Kaplansky and Ohm theorem, that they occur as groups of divisibility of Bezout domains that can be constructed. There is a nice paper, that I came across by Yi Chuan Yang in 2008 called "Some remarks on almost lgroups" where they define an almost lgroup, and show that almost GCD domains have group of divisibility which is an almost lgroup, and seek to answer the converse. They answer in the negative and construct a counterexample which shows that that converse is not true. There is an almost GCD domain which is not an almost lgroup. He also provides references to several other possibly ways of generalizing this converse question. I hope this may at least point you in a positive direction! 

