Zariski Topology and Cohomology for Commutative Ternary Gamma Semirings
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arXiv
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| Format: | Preprint |
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2025
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| author | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| contents | 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 $\Spec_Γ(T)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20784 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zariski Topology and Cohomology for Commutative Ternary Gamma Semirings Gokavarapu, Chandrasekhar Rao, D. Madhusudhana Rings and Algebras 16Y60, 14A15, 18G10, 18E3 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 $\Spec_Γ(T)$ 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. |
| title | Zariski Topology and Cohomology for Commutative Ternary Gamma Semirings |
| topic | Rings and Algebras 16Y60, 14A15, 18G10, 18E3 |
| url | https://arxiv.org/abs/2512.20784 |