Before revision: We establish the lattice theorem,  rural hospitals theorem,  and a group-incentive-compatibility result for terminal buyers (sellers) with unit demand, in a general bilateral trading network without making the assumption of quasi-linear utility in transfers.
After revision: We establish several structural results for the set of competitive equilibria in bilateral trading networks with transfers in the case of imperfectly transferable utility and frictions: The lattice theorem, rural hospitals theorem, and a group-incentive-compatibility result for terminal buyers (sellers) with unit demand, in a general bilateral trading network without making the assumption of quasi-linear utility in transfers.