  The existence of juxtaposed regions of distinct cultures in spite of the fact
that people's beliefs have a tendency to become more similar to each other's as
the individuals interact repeatedly is a puzzling phenomenon in the social
sciences. Here we study an extreme version of the frequency-dependent bias
model of social influence in which an individual adopts the opinion shared by
the majority of the members of its extended neighborhood, which includes the
individual itself. This is a variant of the majority-vote model in which the
individual retains its opinion in case there is a tie among the neighbors'
opinions. We assume that the individuals are fixed in the sites of a square
lattice of linear size $L$ and that they interact with their nearest neighbors
only.
  Within a mean-field framework, we derive the equations of motion for the
density of individuals adopting a particular opinion in the single-site and
pair approximations. Although the single-site approximation predicts a single
opinion domain that takes over the entire lattice, the pair approximation
yields a qualitatively correct picture with the coexistence of different
opinion domains and a strong dependence on the initial conditions. Extensive
Monte Carlo simulations indicate the existence of a rich distribution of
opinion domains or clusters, the number of which grows with $L^2$ whereas the
size of the largest cluster grows with $\ln L^2$. The analysis of the sizes of
the opinion domains shows that they obey a power-law distribution for not too
large sizes but that they are exponentially distributed in the limit of very
large clusters. In addition, similarly to other well-known social influence
model -- Axelrod's model -- we found that these opinion domains are unstable to
the effect of a thermal-like noise.
