Reaction systems, formal models motivated by biochemical reactions inside the living cells, were introduced nearly two decades ago by Ehrenfeucht and Rozenberg. This study investigates the mathematical properties of \({{\,\mathrm{\textit{rs}}\,}}\) functions, namely, subset functions specified by reaction systems. A number of studies have investigated ranks of \({{\,\mathrm{\textit{rs}}\,}}\) functions, where the rank denotes the smallest achievable number of reactions in a reaction system that specifies the function. This work focuses on the class \({\mathcal {F}}_U(S)\) introduced by Salomaa, where each function is union-additive and can be associated to a minimal reaction system. Building on our earlier works on ranks of functions belonging to \({\mathcal {F}}_U(S)\) , in this paper, we propose the variations of minimal reaction system rank and almost minimal reaction system rank, along with a novel hierarchy of \({{\,\mathrm{\textit{rs}}\,}}\) functions according to rank witnessing reaction systems. Naturally, we obtain the minimal reaction system rank for every function from this class \({\mathcal {F}}_U(S)\) . Additionally, for those within the class that have a size three or one-to-one signature, we determine their almost minimal reaction system ranks. Furthermore, we acquire preliminary results about classifying functions from the class \({\mathcal {F}}_U(S)\) according to our rank-dependent hierarchy.
