of the entries of each column is equal to one. In our construction the $R$-matrix will indeed play the role of a local Markovian operator, see for instance . We will also require that $R$-matrices satisfy the regularity property $$\label{eq:regularity}
 R(1)=P$$ where $P$ is the permutation operator in $\cV\otimes \cV$, *i.e* we have $P {{|\tau\rangle}} \otimes {{|\tau'\rangle}} = {{|\tau'\rangle}} \otimes {{|\tau\rangle}}$ for all $\tau,\tau'=0,\dots,s$. In the continuous time setting, it is known that this property ensures that one obtains a continuous time Markov matrix (or a quantum Hamiltonian in the context of spin chains) acting locally on the lattice from the transfer matrix constructed below. In the discrete time setting, we will show that this property allows us to simplify the expression of the transfer matrix to end up with a Markov matrix which has an easy interpretation in the parallel update picture. We also require the $R$-matrix to satisfy the unitarity property $$\label{eq:unitarity}
 R_{12}(z) R_{21}\left(\frac{1}{z}\right) = 1$$ where $R_{21}(z)=PR_{12}(z)P$. This property is used to show the commutation of the transfer matrix below[^4] and it appears as a consistency relation when constructing the stationary state in a matrix product form, see subsection \[subsec:stationary\_state\].

Up to now, we wrote all the 