A finite characterization of weak lumpable Markov processes. Part I: The discrete time case

We consider an irreducible and homogeneous Markov chain (discrete time) with finite state space. Given a partition of the state space, it is of interest to know if the aggregated process constructed from the first one with respect to the partition is also Markov homogeneous. We give a characterization of this situation by means of a finite algorithm. This algorithm computes the set of all initial probability distributions of the starting homogeneous Markov chain leading to an aggregated homogeneous Markov chain.