I suggest to rename the tag divisors in a way which distinguishes it more clearly from divisors-multiples. The tag-info for divisors (revision history) clearly says: For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question...

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?

It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable semigroup (is cancellative and torsion-free, and) embeds into a linearly orderable monoid (see here ...