  We give sufficient conditions ensuring the strong ergodic property of unique
mixing for $C^*$-dynamical systems arising from Yang-Baxter-Hecke quantisation.
We discuss whether they can be applied to some important cases including
monotone, Boson, Fermion and Boolean $C^*$-algebras in a unified version. The
monotone and the Boolean cases are treated in full generality, the Bose/Fermi
cases being already widely investigated. In fact, on one hand we show that the
set of stationary stochastic processes are isomorphic to a segment in both the
situations, on the other hand the Boolean processes enjoy the very strong
property of unique mixing with respect to the fixed point subalgebra and the
monotone ones do not
