Chemical Systems as Ternary $Γ$-Semirings:Theory, Case Studies, and Operational Tests
Chandrasekhar Gokavarapu, Venkata Rao Kaviti, Srinivasa Rao Thirunagari, D. Madhusudhana Rao
TL;DR
This work proposes a novel algebraic framework in which chemical systems are modeled as ternary Γ-semirings (TGS), with chemical states forming an additive semigroup, mediators encoding context, and mediated transformations encoded by a ternary operation $\mathsf{s}_a\gamma\mathsf{s}_b$. The authors interpret the axioms of distributivity and associativity as physical laws governing ideal parallel reactions and path-independent equilibria, respectively, and demonstrate how classical processes like Michaelis–Menten kinetics and thermodynamic equilibrium map onto associative or non-associative regimes. They develop operational tests—the distributivity index $D$ and an associativity test—to empirically classify any chemical system within the TGS framework and illustrate how non-distributive, non-associative regions capture information-processing motifs such as allostery and chemical computation. The framework unifies equilibrium, kinetics, regulation, and computation within a single algebraic language and offers axiom-based design principles for smart materials, including self-limiting polymerization, linking algebraic constraints to concrete molecular targets and material functionalities.
Abstract
Chemical systems are traditionally described by lists of species, reactions, and externally imposed kinetic laws, a framework that lacks an intrinsic algebraic structure governing how transformations compose. We propose an axiomatic reformulation in which a chemical system is modelled as a ternary Gamma semiring (TGS), where chemical states form an additive semigroup, mediators encode catalytic or environmental context, and mediated transformations are represented by a ternary operation. We show that the TGS axioms admit direct physical interpretations: distributivity corresponds to ideal, non-interfering parallel reactions, while associativity characterizes thermodynamic path-independence. Classical systems including Michaelis-Menten kinetics, global equilibrium, and allosteric regulation are recovered as different algebraic regimes, and we develop operational tests that quantify departures from the axioms through experimentally measurable indices. The resulting framework unifies equilibrium, kinetics, regulation, and chemical computation within a single algebraic language, offering new principles for the analysis and design of responsive or self-regulating materials.