  In this paper, we generalize the original majority-vote (MV) model with noise
from two states to arbitrary $q$ states, where $q$ is an integer no less than
two. The main emphasis is paid to the comparison on the nature of phase
transitions between the two-state MV (MV2) model and the three-state MV (MV3)
model. By extensive Monte Carlo simulation and mean-field analysis, we find
that the MV3 model undergoes a discontinuous order-disorder phase transition,
in contrast to a continuous phase transition in the MV2 model. A central
feature of such a discontinuous transition is a strong hysteresis behavior as
noise intensity goes forward and backward. Within the hysteresis region, the
disordered phase and ordered phase are coexisting.
