  We investigate the nature of the time-reversal breaking pairing state in the
hole-doped monolayer MoS$_{2}$ on the basis of the realistic three-orbital
attractive Hubbard-like model with the atomic spin-orbit coupling. Due to the
multi-band features arising from the Mo $d$ orbitals in the noncentrosymmetric
crystal structure, the Lifshitz transition takes place upon hole doping. Across
the Lifshitz transition point, the sign of the relative phase between the
Cooper-pair components drastically changes, leading to the emergence of the
time-reversal breaking phase with complex gap functions. It is shown that this
intriguing pairing state is characterized by the finite momentum-space
distributions of the orbital and spin angular momentum with three-fold
rotational symmetry on the Fermi-surface pockets around K and K$'$ points. The
present mechanism for the time-reversal breaking superconductivity can
ubiquitously be applied to spin-orbit-coupled metals in noncentrosymmetric
crystal structures.
