  We develop a general framework for studying ergodicity of order-preserving
Markov semigroups. We establish natural and in a certain sense optimal
conditions for existence and uniqueness of the invariant measure and
exponential convergence of transition probabilities of an order-preserving
Markov process. As an application, we show exponential ergodicity and
exponentially fast synchronization-by-noise of the stochastic
reaction-diffusion equation in the hypoelliptic setting. This refines and
complements corresponding results of Hairer, Mattingly (2011).
