Markov chain lumpability on fuzzy partitions

We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition and for the resulting aggregated process we study two aspects emerging from the classical theory on hard partitions. The first aspect is lumpability, a technique for recovering from the large state space of a stochastic system. We provide necessary and sufficient conditions for strong lumpability on the transition probabilities of the original chain for the lumped process to have the Markov property. The second aspect is the asymptotic behavior of the lumped chain. The results are compared with those existing in the classical theory of hard partitions.