rest of the paper. We will provide explicit examples of the construction of discrete time models and of the associated stationary state.

Examples: the SSEP and ASEP case. \[subsec:examples\]
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In this subsection we apply the generic construction introduced above in subsection \[subsec:parallel\_update\] to the $R$ and $K$ matrices associated with the SSEP and the ASEP. We define the discrete time stochastic dynamics and provide explicitly the transition probabilities between the configurations. The stationary states are then computed following the general procedure presented in \[subsec:stationary\_state\]. The matrix ansatz algebra is found to be closely related to the one used to solve the continuous time models.

#### The SSEP case.

We are first interested in the SSEP, which is a stochastic model where particles can jump to the left or to the right with equal probability (*i.e* there is no driving force in the bulk) and with an exclusion constraint. We introduce an $R$-matrix which reads in the basis ${{|0\rangle}}\otimes{{|0\rangle}}$, ${{|0\rangle}}\otimes{{|1\rangle}}$, ${{|1\rangle}}\otimes{{|0\rangle}}$ and ${{|1\rangle}}\otimes{{|1\rangle}}$ (ordered this way) $$\label{eq:SSEP_R}
  R(z) = \begin{pmatrix}
          1 & 0 & 0 & 0 \\ 
          0 & \frac{z}{z+1} & \frac{1}{z+1} & 0 \\
          0 & \frac{1}{z+1} & \frac{z}{z+1} & 0 \\
          0 & 0 & 0 & 