nearest neighbor interactions). We are going to see in the next subsection \[subsec:parallel\_update\] that the inhomogeneity parameters can be efficiently used to define discrete time processes from the transfer matrix. We stress that the construction is quite general in the sense that it does not rely on a particular choice for the $R$-matrix and the $K$-matrices (they only have to fulfill the properties mentioned above).

In the spectral parameter picture, the inhomogeneous transfer matrix reads $$t(z|\mathbf{z})=tr_0 \left(\widetilde{K}_0(z) R_{0,L}(z-z_L) \dots R_{0,1}(z-z_1)
  K_0(z) R_{1,0}(z+z_1) \dots R_{L,0}(z+z_L) \right)$$ and the matrices $\overline{K}$ and $\widetilde{K}$ are related for instance through the equation $$\overline{K}_1(z)= tr_0\left( \widetilde{K}_0(-z)R_{01}(-2z)P_{01} \right).$$

Definition of the process. \[subsec:parallel\_update\]
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The goal of this subsection is to use the transfer matrix to define a discrete time Markov matrix. The general idea is to specify particular values of the spectral parameter $z$ and of the inhomogeneity parameters $z_i$ in the transfer matrix to obtain Markovian dynamics on the lattice with simple stochastic rules. Several different approaches have already been proposed in the literature.

Among the recent developments we can mention the work [@KunibaMMO] where a large class of integrable discrete time exclusion processes were introduced using particular values for the inhomogeneity parameters (this 