  We investigate two-species bosons in a two-dimensional square lattice by
quantum Monte Carlo method. We show that the inter-species attraction and
nearest-neighbor intra-species repulsion results in the pair-supersolid phase,
where a diagonal solid order coexists with an off-diagonal pair-superfluid
order. The quantum and thermal transitions out of the pair-supersolid phase are
characterized. It is found that there is a direct first order transition from
the pair-supersolid phase to the double-superfluid phase without an
intermediate region. Furthermore, the melting of the pair-supersolid occurs in
two steps. Upon heating, first the pair-superfluid is destroyed via a KT
transition then the solid order melts via an Ising transition
