Iterative behaviour of generalized majority functions

Let us consider a society of n persons {P₁,...,P_n} with interaction coefficient a_ij between P_i and P_j. Let {θ₁,...,θ_p} be the set of opinions which may be assumed by any person. We study the dynamical behaviour of such a society under local majority rules. The results presented in this paper generalize many partial results previously obtained by several authors (Tchuente, 1977; Goles and Olivos, 1980; Poljak and Sura, 1982). Furthermore, by introducing algebraic invariants, analogous to those exhibited in Goles (1980) for the study of Boolean threshold functions, we provide a tool for a general approach of such dynamical models.