Abstract: In recent years, the familiar Bose-Hubbard system, of interacting bosons in a lattice potential, has been generalised to a lattice system of coupled Jaynes-Cummings atom cavities. These Jaynes-Cummings-Hubbard systems have been predicted to exhibit a broadly similar phase diagram for the superfluid-to-Mott insulator quantum phase transition as for the Bose-Hubbard model. In this talk, I shall discuss how the replacement of the familiar two-level atomic system in each cavity, by a three-level atomic system, can induce the coexistence of two-body and three-body interactions inside each cavity. The strength and sign of the two-body interaction, relative to the three-body repulsion, is found to be dependent on the relative ratio of the two relevant transition matrix elements between the three atomic levels. Subsequently, I will discuss the implications of the possibility of a local two-body attraction, balanced by a three-body repulsion, for the mean-field superfluid-Mott insulator phase diagram: the coexistence of first- and second-order phase transitions, as well as a possible pair superfluid phase.”