Homological and Categorical Foundations of Ternary $Γ$-Modules and Their Spectra
Chandrasekhar Gokavarapu, Madhusudhana Rao Dasari
TL;DR
The paper addresses the absence of a unified homological and geometric foundation for commutative ternary Γ-semirings by introducing a rigorous theory of left ternary $\Gamma$-modules and establishing foundational isomorphism theorems, annihilator–primitive correspondences, and a Schur-density framework. It builds a robust homological scaffold with additive/exact structure, projectives/injectives, and well-defined Ext and Tor functors, then augments this with a monoidal closed categorical structure and a tensor–Hom adjunction. The work further develops spectral dualities via the spectrum $\operatorname{Spec}_\Gamma(T)$ and endomorphism sheaves, tying local module theory to global geometric pictures and enabling localization, structure sheaves, and Grothendieck-style spectral sequences. It culminates with analytic, fuzzy, and computational enrichments that integrate $\Gamma$-spectra into analytic and data-driven contexts, setting the stage for fuzzy and computational extensions (Paper D) and broad applications across algebra, geometry, and computation. This framework provides a comprehensive algebraic–geometric toolkit for derived and spectral analysis of ternary semirings, with explicit pathways to finite-algebra computation and practical representations in analytic and fuzzy settings.
Abstract
Purpose: To develop a unified homological categorical foundation for commutative ternary Gamma semirings by formulating a general theory of ternary Gamma modules that integrates algebraic, geometric, and computational layers, extending the ideal theoretic and algorithmic bases of Papers A [Rao2025A] and B [Rao2025B1]. Methods: We axiomatize ternary Gamma modules and establish the fundamental isomorphism theorems, construct annihilator primitive correspondences, and prove Schur density embeddings. Categorical analysis shows that T Gamma Mod is additive, exact, and monoidal closed, enabling the definition of derived functors Ext and Tor via projective injective resolutions and yielding a tensor Hom adjunction. We develop geometric dualities between module objects and the spectrum Spec Gamma (T) and extend them to analytic, fuzzy, and computational settings. Results: The category T Gamma Mod admits kernels, cokernels, coequalizers, and balanced exactness; monoidal closure ensures internal Homs and coherent tensor Hom adjunctions. Derived functors Ext and Tor are well defined and functorial, with long exact sequences and base change compatibility. Schur density yields faithful embedding criteria, while annihilator primitive correspondences control primitivity and support theory. Geometric dualities provide contravariant equivalences linking submodule spectra with closed sets in Spec Gamma (T), persisting under analytic, fuzzy, and computational enrichments. Conclusion: These results complete the algebraic homological geometric synthesis for commutative ternary Gamma semirings, furnish robust tools for derived and spectral analysis, and prepare the framework for fuzzy and computational extensions developed in Paper D [Rao2025D], extending the algebraic framework first established in [Rao2025]