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We construct different classes of lumpings for a family of Markov chainproducts which reflect the structure of a given finite poset. We useessentially combinatorial methods. We prove that, for such a product, everylumping can be obtained from the action of a suitable subgroup of thegeneralized wreath product of symmetric groups, acting on the underlying posetblock structure, if and only if the poset defining the Markov process istotally ordered, and one takes the uniform Markov operator in each factor statespace. Finally we show that, when the state space is a homogeneous spaceassociated with a Gelfand pair, the spectral analysis of the correspondinglumped Markov chain is completely determined by the decomposition of the groupaction into irreducible submodules.

The lumpability property for a family of Markov chains on poset block structures