Axiomatic Foundations of Chemical Systems as Ternary Γ-Semirings

Chemical transformations depend not only on the identities of the reacting species but also on the catalytic, environmental, and intermediate conditions under which they occur. Classical binary reaction formalisms usually treat such conditions as external annotations, which obscures the genuinely multi-state and multi-parameter character of real chemical processes. In this paper we introduce an axiomatic framework in which a chemical system is modeled by a ternary Γ-semiring. The elements of the state set represent chemical states, while the parameter set encodes catalytic and environmental conditions. A Γ- dependent ternary operation is used to describe mediated transformations, treating reactants, intermediates, and mediators as intrinsic arguments of the transformation law. We develop the algebraic axioms governing these mediated interactions and interpret their associativity, distributivity, and Γ- linearity in terms of multi-step pathways, parallel processes, and controlled environmental dependence. We introduce chemical ideals and Γ-ideals as algebraic structures modeling reaction-closed subsystems and pathway-stable domains, and study their prime and semiprime forms. Homomorphisms between TGS-chemical systems are shown to preserve reaction pathways and describe consistent changes of chemical environment. Abstract examples from catalysis, thermodynamic phase control, and field-induced quantum transitions illustrate how familiar chemical phenomena fit within this framework. The resulting theory provides a unified algebraic foundation for multi-parameter chemical behavior and establishes the structural basis for subsequent developments involving kinetics, geometric methods, and computational or AI-assisted models.