Lumpability of absorbing Markov chains and replacement chains on fuzzy partitions

We consider a finite, aperiodic, time homogeneous, absorbing Markov chain on a fuzzy partition, for which each time an absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The resulting process is a Markov replacement chain on a fuzzy partition. We study certain aspects of the aggregated process 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 chains for the lumped processes to have the Markov property. The second aspect is the asymptotic characteristics of the lumped Markov chains. The results are compared with those existing in the classical theory of hard partitions.