In this paper, the problem of aggregation Markov chains has been considered. The necessary and sufficient conditions of the transition probability matrix of the original Markov chain to be a doubly stochastic matrix when the transition probability matrix of the lumped chain was doubly stochastic have been given for lumping Markov chain for all (certain) partitions within isomorphic with respect to the cases (k, k+1), (k, k+2),and (2,k). also, special cases have been introduced for each case, for each partition.
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