of operators generated by $t(z)$ $$[M,t(z)] = 0$$ which is direct consequence of the commutation relation and of the definition .

#### Open boundary conditions.

To define our process on a lattice with open boundaries, we need to consider a lattice with an odd number of sites $L$. The idea is essentially the same as for the periodic boundary conditions case. We fix $z_1=z_3=\dots=z_L=\kappa$ and $z_2=z_4=\dots=z_{L-1}=\frac{1}{\kappa}$. The inhomogeneous transfer matrix then takes the following staggered form $$\label{eq:transfer_matrix_open}
  t(z)=tr_0 \left(\widetilde{K}_0(z) R_{0L}\left(\frac{z}{\kappa}\right)\dots R_{02}\left(z\kappa\right) R_{01}\left(\frac{z}{\kappa}\right)
  K_0(z) R_{10}(z\kappa)R_{20}\left(\frac{z}{\kappa}\right) \dots R_{L0}(z\kappa) \right).$$ A direct computation yields $$t(\kappa) = B_1 U_{23}U_{45}\dots U_{L-1,L}\, U_{12}U_{34} \dots U_{L-2,L-1} \overline{B}_L,$$ where $B=K(\kappa)$ and $\overline{B}=\overline{K}(1/\kappa)$. Once again we can define a Markov matrix as $$\label{eq:Markov_matrix_open}
 M = t(\kappa) = {{\mathbb U}}^{e} {{\mathbb U}}^{o}$$ with operators ${{\mathbb U}}^{e}$ and ${{\mathbb U}}^{o}$ having similar expression as in the periodic case but with additional boundary terms $${{\mathbb U}}^{o} = \prod_{k=1}^{\frac{L-1}{2}} U_{2k-1,2k} \, \overline{B}_L \quad \mbox{and} \quad {{\mathbb U}}^{e} =  B_1 \prod_{k=1}^{\frac{L-1}{2}} U_{2k,2k+1}.$$ The local operators involved in ${{\mathbb U}}^{o}$ (respectively in ${{\mathbb U}}^{e}$) commute one with each other because they are acting on different sites of the lattice. ${{\mathbb U}}^{o}$ realizes a stochastic update on every pair of consecutive sites with the first site 