Partitions on bigrams and Markov property of block ciphers

A model of iterated block ciphers with alphabet $X$, independent uniform round keys and a key addition group $(X,\otimes)$ is considered. We find conditions ensuring the preservation of Markov property under lumping of Markov chain with state space $X^2$ corresponding to bigrams of intermediate ciphertexts. We describe properties of Markov ciphers considered and lumping transforms.