Increasing coupling of probabilistic cellular automata

We give a necessary and sufficient condition for the existence of an increasing coupling of N (N⩾2) synchronous dynamics on SZd(PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.