Exactly solvable models through the generalized empty interval method: multi-species and more-than-two-site interactions

Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E_n^{\mathbf{a}}(t)$'s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$.