Zariski Topology and Cohomology for Commutative Ternary Gamma Semirings

Abstract: This paper develops the algebraic foundation required to build a Zariski-type geometry for \emph{commutative ternary $Γ$-semirings}, where multiplication is an inherently triadic, multi-parametric interaction $(a,b,c,γ)\mapsto{abc}γ$. Rather than treating triadic multiplication as an optional variation of binary algebra, we adopt it as an \emph{algebraic necessity} for modeling systems whose elementary interactions are intrinsically three-body and whose operational modes are indexed by parameters $Γ$. We construct the prime spectrum $\SpecΓ(T)$ and its Zariski topology, prove functoriality, and build the structure sheaf $\Ocal_{\Spec_Γ(T)}$ via local fraction descriptions that must simultaneously respect triadic associativity and the sheaf gluing axioms. A key technical point is ensuring that local representations by ternary-parametric fractions glue uniquely, despite the absence of a binary product and despite the parameter dependence of the multiplication law. We then define sheaves of $Γ$-modules, quasi-coherent sheaves associated to algebraic modules, and the cohomology groups $H^i(X,\Fcal)$ as derived functors of global sections. Finally, we give a concrete finite \emph{structural example} (a ternary $Γ$-version of $\mathbb{Z}/n\mathbb{Z}$) and compute its $Γ$-spectrum explicitly, thereby exhibiting nontrivial spectral behavior in a fully finite setting.