  We analyze the generalized mean-field q-state Potts model which is obtained
by replacing the usual quadratic interaction function in the mean-field
Hamiltonian by a higher power z. We first prove a generalization of the known
limit result for the empirical magnetization vector of Ellis and Wang which
shows that in the right parameter regime, the first-order phase-transition
persists. Next we turn to the corresponding generalized fuzzy Potts model which
is obtained by decomposing the set of the q possible spin-values into 1 < s < q
classes and identifying the spins within these classes. In extension of earlier
work which treats the quadratic model we prove the following: The fuzzy Potts
model with interaction exponent bigger than four (respectively bigger than two
and smaller or equal four) is non-Gibbs if and only if its inverse temperature
is larger or equal than the critical inverse temperature of the corresponding
Potts model with number of spin-labels equal to the size of the smallest class
which is greater or equal than two (respectively greater or equal than three.)
We also provide a dynamical interpretation considering sequences of fuzzy Potts
models which are obtained by increasingly collapsing classes at finitely many
times and discuss the possibility of a multiple in- and out of Gibbsianness,
depending on the collapsing scheme.
