  We analyze the hard-core Bose-Hubbard model with both the three-body and
nearest neighbor repulsions on the triangular lattice. The phase diagram is
achieved by means of the semi-classical approximation and the quantum Monte
Carlo simulation. For a system with only the three-body interactions, both the
supersolid phase and one third solid disappear while the two thirds solid
stably exists. As the thermal behavior of the bosons with nearest neighbor
repulsion, the solid and the superfluid undergo the 3-state Potts and the
Kosterlitz-Thouless type phase transitions, respectively. In a system with both
the frustrated nearest neighbor two-body and three-body interactions, the
supersolid and one third solid revive. By tuning the strength of the three-body
interactions, the phase diagram is distorted, because the one-third solid and
the supersolid are suppressed.
