  Using Monte Carlo calculations we analyze the order and the universality
class of phase transitions into a low density (honeycomb) phase of a triangular
antiferromagnetic three-state Bell-Lavis model. The results are obtained in a
whole interval of chemical potential $\mu$ corresponding to the honeycomb
phase. Our results demonstrate that the phase transitions might be attributed
to the three-state Potts universality class for all $\mu$ values except for the
edges of the honeycomb phase existence. At the honeycomb phase and the low
density gas phase boundary the transitions become of the first order. At
another, honeycomb-to-frustrated phase boundary, we observe the approach to the
crossover from the three-state Potts to the Ising model universality class. We
also obtain the Schottky anomaly in the specific heat close to this edge. We
show that the intermediate planar phase, found in a very similar
antiferromagnetic triangular Blume-Capel model, does not occur in the
Bell-Lavis model.
