  We construct different classes of lumpings for a family of Markov chain
products which reflect the structure of a given finite poset. We use
essentially combinatorial methods. We prove that, for such a product, every
lumping can be obtained from the action of a suitable subgroup of the
generalized wreath product of symmetric groups, acting on the underlying poset
block structure, if and only if the poset defining the Markov process is
totally ordered, and one takes the uniform Markov operator in each factor state
space. Finally we show that, when the state space is a homogeneous space
associated with a Gelfand pair, the spectral analysis of the corresponding
lumped Markov chain is completely determined by the decomposition of the group
action into irreducible submodules.
