Unified Geometric, Fuzzy, and Computational Framework for Ternary Gamma Semirings

TL;DR

This paper develops a unified geometric framework for ternary gamma semirings, integrating ideal theory, fuzzy, and computational geometries, with new invariants, dualities, and algorithms for finite models.

Contribution

It introduces a comprehensive geometric and computational framework for ternary gamma semirings, unifying multiple theoretical layers and providing new invariants and algorithms.

Findings

Constructed structure sheaves and Grothendieck topologies for ternary G-products.

Proved dualities linking spectra, embeddings, and derived functors.

Developed criteria and algorithms for finite models.

Abstract

Aim. This paper (Paper D) unifies the ideal-theoretic, computational, and homological layers developed in Papers A (Rao 2025), B (Rao 2025), and C (Rao 2025) into a geometric framework that includes fuzzy and computational geometries on the spectrum Spec_G(T) and derived invariants in TGMod. Scope. We construct structure sheaves and Grothendieck topologies adapted to ternary G-products, develop fuzzy and weighted sites, and prove dualities bridging primitive spectra, Schur-density embeddings, and derived functors Ext and Tor. Outcomes. We obtain comparison theorems between radical/primitive strata and cohomological supports, and supply computable criteria and algorithms for finite models.