On Moebius duality and Coarse-Graining

We study duality relations for zeta and M\"{o}bius matrices and monotone conditions on the kernels. We focus on the cases of family of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive M\"{o}bius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an $h-$transform is needed to preserve stochasticity. We give conditions in order that zeta and M\"{o}bius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.