I have a question about what classifies as a Markov chain and what does not. Consider a system with state space $\left\{ 1,\ldots,n \right\}$ and a trajectory for the system defined by the following transition matrix
$$P_{ij} = 1 \mathrm{ \ if \ } j = i+1$$
$$P_{n1} = 1$$
This is just a cycle over the n states of the system. Given the initial state of the system, it is clearly a deterministic process. Can this be called a first order time-homogenous Markov chain? If not, is there something in the definition of a Markov chain (besides the fact that it is a random process) based on which it can be shown that this process is not Markov?
Maybe I'm missing out something obvious, but I can't seem to find an answer to my query. Any thoughts on this would be appreciated!