Ranks of functions specified by minimal reaction systems and induced by images of singletons

This paper studies mathematical properties of reaction systems, which is a formal model introduced by Ehrenfeucht and Rozenberg and inspired by biochemical reactions that occur in living cells. Numerous studies have focused on reaction system ranks of functions specified by minimal reaction systems, where the rank refers to the smallest size among the specifying reaction systems. We particularly study the reaction system ranks for a class of union-additive functions specified by minimal reaction systems introduced by Salomaa, which is closed under taking composition. More precisely, when the signature size is two, we obtain a general formula for the reaction system ranks that shows the reaction system rank of each function from this subclass depends on the characteristic of the function. Then, we study the reaction system ranks of such functions with signature size three, as well as establishing a general upper bound for reaction system ranks of such functions with one-to-one signatures for any background set.