This question arises from pages 14 and 15 of this review paper on quantum stochastic processes (in a section on classical stochastic processes).
Suppose we have a stochastic process, with statistical states $\mathbb P(X_t)$ at each $t \in \mathcal T$ (say in a $d$-dimensional system, i.e. an observation at each time has $d$ possible outcomes $\{x_1, \dots, x_d\}$, e.g. rolling a die repeatedly would be 6-dimensional). So $\mathbb P(X_t)$ is a $d$-dimensional vector, perhaps of the form $(\mathbb P(X_t = x_1), \dots , \mathbb P(X_t = x_d))^T$.
Then a stochastic matrix $\Gamma_{t':t}$ would map the state $\mathbb P(X_t)$ to a later state $\mathbb P(X_{t'})$. The paper states in Eq. 38 that a divisible process is one where the following condition is satisfied: $$ \Gamma_{t:r} = \Gamma_{t:s} \Gamma_{s:r} \quad \forall t > s > r. $$
Question 1. What is the form of $\Gamma_{t':t}$? Presumably they must be $d \times d$ matrices, possibly of the form $$ \Gamma_{t':t} = \begin{pmatrix} \mathbb P(X_{t'} = x_1 | X_t = x_1) & P(X_{t'} = x_1 | X_t = x_2) & \cdots & P(X_{t'} = x_1 | X_t = x_d) \\ P(X_{t'} = x_2 | X_t = x_1) & P(X_{t'} = x_2 | X_t = x_2) & \cdots & P(X_{t'} = x_2 | X_t = x_d) \\ \vdots & \vdots & \ddots & \vdots \\ P(X_{t'} = x_d | X_t = x_1) & P(X_{t'} = x_d | X_t = x_2) & \cdots & P(X_{t'} = x_d | X_t = x_d) \end{pmatrix}. $$ Is this correct? Are there any other possible forms for this stochastic matrix?
Question 2. What is an example of an indivisible process? If both $\Gamma_{t:r}$ and $\Gamma_{t:s}\Gamma_{s:r}$ correctly map every $\mathbb P(X_r)$ to $\mathbb P(X_t)$, how can they be different?