Exactly solvable models through the generalized empty interval method, for multi-species interactions

Multi-species reaction-diffusion systems, with nearest-neighbor 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 $E^{\mathbf a}_n(t)$'s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$. The constraints are solved for the single-species left-right-symmetric systems. Also, examples of multi-species system for which the evolution equations of $E^{\mathbf a}_n(t)$'s are closed, are given.