Markov matrix, *i.e* non-negativity and sum to one property of the entries, follows from the fact that the entries of the local operator $U$ are non-negative and sum to one on each column (thanks to the Markovian property of the R-matrix). Note that the local operators $U$ involved in the definition of ${{\mathbb U}}^{o}$ all commute one with each other and thus the order in the product is not important. The same is true for ${{\mathbb U}}^{e}$. From the expression it is clear that the Markov matrix $M$ can be interpreted as describing a Floquet dynamics composed of two steps: the first one embodied in ${{\mathbb U}}^{e}$ and the other one in ${{\mathbb U}}^{o}$. We stress that ${{\mathbb U}}^{e}$ and ${{\mathbb U}}^{o}$ do not commute. The operator ${{\mathbb U}}^{e}$ updates every pair of consecutive sites with the first site located at an even position. The local stochastic update is encoded in the matrix $U$ acting on $\cV\otimes\cV$. We will give precise examples of such matrices in subsections \[subsec:examples\] and \[subsec:fusion\]. The operator ${{\mathbb U}}^{o}$ updates every pair of consecutive sites with the first site located at an odd position. A pictorial representation of the discrete time dynamics is given in 