On the general transformation of Markov chains (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2022)
In this work, we consider a class of homogeneous irreducible discrete states Markov process  ,  taking values in a partition of some topological space  . Using  a notion of fractional state,  we introduce a general transformation   such that      remains a Markov process. This general transformation includes also lumpability. Infinite kinds of this expanded Markov chain    can be constructed, however lumped  Markov chains are finite or may not exist .Using general transformation   and fractional states, we give a necessary and sufficient condition under which the transformed process   remains still a Markov chain