arXiv subjects

Chandrasekhar Gokavarapu

Publications and source records attributed to Chandrasekhar Gokavarapu.

43 records · Page 3Linked to original sources

arXiv2025

Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings

This paper develops the homological backbone of the theory of non-commutative $n$-ary $Γ$-semirings. Starting from an $n$-ary $Γ$-semiring $(T,+,\tildeμ)$ and its $Γ$-ideals, we work in the slot-sensitive categories of left, right, and bi-$Γ$-modules, and endow the bi-module category with a Quillen exact structure compatible with the $n$-ary multiplication. Within this exact framework we construct bar-type projective resolutions and cofree-based injective resolutions under natural $Γ$-Noetherian and $Γ$-regular hypotheses on $T$, and we obtain finite projective resolutions for finitely presented bi-modules under $Γ$-Noetherian conditions. On this basis we define the derived functors $\ExtG$ and $\TorG$ for bi-$Γ$-modules, prove their balance with respect to projective and injective resolutions, establish long exact sequences and a Yoneda interpretation via iterated extensions, and construct Künneth-type spectral sequences and base-change isomorphisms. Interpreting bi-$Γ$-modules as quasi-coherent sheaves on the non-commutative $Γ$-spectrum $\SpecGnC{T}$, these homological invariants provide the appropriate derived language for a non-commutative $Γ$-geometry and prepare the ground for the spectral and geometric analysis carried out in the third part of this series.

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arXiv2025

Exact Categories and Homological Foundations of Non-Commutative n-ary Gamma. Semirings

This paper establishes the homological and geometric foundations of non-commutative n-ary Gamma-semirings, unifying two previously distinct directions in Gamma-algebra: the derived Gamma-geometry developed for the commutative ternary case and the structural and spectral theory for general non-commutative n-ary systems. We introduce categories of left, right, and bi-Gamma-modules that respect positional asymmetry and prove that they form additive and exact categories in Quillen's sense. Within this setting, we construct projective and injective resolutions, define the derived functors Ext^Gamma and Tor_Gamma, and establish long exact sequences and spectral balance theorems in the n-ary regime. By extending sheaf-theoretic and homological tools to the non-commutative Gamma-spectrum Spec_Gamma^nc(T), we obtain a coherent framework of non-commutative derived Gamma-geometry that parallels the classical paradigms of Grothendieck and Kontsevich in homological algebra and non-commutative geometry. The framework developed here establishes the foundational exact-categorical and homological structures that enable Morita-type analyses and spectral interpretations in the subsequent parts of this series.

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arXiv2025

Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings

This paper introduces a new class of error-correcting codes constructed from the ideal lattices of finite commutative ternary Gamma-semirings (TGS). Unlike classical linear or ring-linear codes, which rely on binary operations, TGS codes arise from the intrinsic ternary operation $[x,y,z]$ and the op-plus order that governs coordinatewise absorption. The fundamental parameters of a TGS code are determined by the $k$-ideal structure of the underlying semiring: the dimension is given by the index $|T/I|$, while the minimum distance depends on the minimal nonzero elements of the distributive ideal lattice $L(T)$. This leads to parameter sets that are not achievable over finite fields, group algebras, or standard semiring frameworks. A quotient-based decoding method is developed in which the ternary syndrome $S(c) = Phi(c) + I$ lies in the quotient TGS $T/I$ and partitions the ambient space into cosets determined by ideal absorption. Minimal nonzero lattice elements yield canonical error representatives, producing a decoding procedure that resembles classical syndrome decoding but comes from higher-arity interactions. A concrete finite example illustrates the computation of parameters, the structure of syndrome classes, and the performance of the decoding method. These results show that ternary Gamma-semirings provide a new algebraic foundation for nonlinear, nonbinary, and higher-arity coding theory. Their ideal-lattice structure and ternary quotient behavior generate new decoding mechanisms and error profiles, expanding algebraic coding theory beyond the limitations of classical linear systems.

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arXiv2025

A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization Problems

Classical shortest-path methods rely on binary tropical semirings $(\min,+)$, whose dyadic structure limits them to pairwise cost interactions. However, many real-world systems, including logistics, supply chains, communication networks, and reliability-aware infrastructures, exhibit inherently ternary dependencies among cost, time, and risk that cannot be decomposed into pairwise components. This paper introduces the \emph{Ternary Tropical Gamma Semiring} (TTGS), a $Γ$-indexed algebraic structure that generalizes tropical semirings by replacing binary additive composition with a non-separable ternary operator. We establish the axioms of TTGS, prove associativity, distributivity, and monotonicity, and show that TTGS forms a well-structured foundation for multi-parameter optimization. Building on this framework, we develop TTGS-Pathfinder, a ternary analogue of the Bellman--Ford algorithm. We derive its dynamic-programming recurrence, prove correctness through an invariant-based argument, analyze convergence under the TTGS order, and obtain an $O(n^2 m)$ complexity bound. Applications demonstrate that TTGS naturally models systems whose behaviour depends on triadic cost interactions, offering a principled alternative to binary tropical, vector, or scalarized multi-objective methods.

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arXiv2025

Ternary Gamma Semirings as a Novel Algebraic Framework for Learnable Symbolic Reasoning

Binary semirings such as the tropical, log, and probability semirings form a core algebraic tool in classical and modern neural inference systems, supporting tasks like Viterbi decoding, dynamic programming, and probabilistic reasoning. However, these structures rely on a binary multiplication operator and therefore model only pairwise interactions. Many symbolic AI tasks are inherently triadic, including subject-predicate-object relations in knowledge graphs, logical rules involving two premises and one conclusion, and multi-entity dependencies in structured decision processes. Existing neural architectures usually approximate these interactions by flattening or factorizing them into binary components, which weakens inductive structure, distorts relational meaning, and reduces interpretability. This paper introduces the Neural Ternary Semiring (NTS), a learnable and differentiable algebraic framework grounded in the theory of ternary Gamma-semirings. The central idea is to replace the usual binary product with a native ternary operator implemented by neural networks and guided by algebraic regularizers enforcing approximate associativity and distributivity. This construction allows triadic relationships to be represented directly rather than reconstructed from binary interactions. We establish a soundness result showing that, when algebraic violations vanish during training, the learned operator converges to a valid ternary Gamma-semiring. We also outline an evaluation strategy for triadic reasoning tasks such as knowledge-graph completion and rule-based inference. These insights demonstrate that ternary Gamma-semirings provide a mathematically principled and practically effective foundation for learnable symbolic reasoning.

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arXiv2025

Derived $Γ$-Geometry, Sheaf Cohomology, and Homological Functors on the Spectrum of Commutative Ternary $Γ$-Semirings

This paper develops a comprehensive geometric and homological framework for derived Gamma-geometry, extending the theory of commutative ternary Gamma-semirings established in our earlier works. Building upon the ideal-theoretic, computational, and categorical foundations of Papers A to D (Rao 2025A, Rao 2025B1, Rao 2025B2, Rao 2025C, Rao 2025D), the present study constructs the algebraic and geometric infrastructure necessary to place Gamma-semirings within the modern language of derived and categorical geometry. We define the affine Gamma-spectrum Spec_Gamma(T) together with its structure sheaf O_{Spec_Gamma(T)}, establishing a Zariski-type topology adapted to ternary Gamma-operations. In this setting, the category of Gamma-modules is shown to be additive, exact, and monoidal-closed, supporting derived functors Ext_Gamma and Tor^Gamma, whose existence is ensured through explicit projective and injective resolutions. The derived category D(T-Gamma-Mod) is then constructed to host homological dualities and Serre-type vanishing theorems, culminating in a categorical analogue of the Serre-Swan correspondence (Swan 1962, Gelfand 1960) for Gamma-modules. Geometric and categorical unification is achieved through fibered and derived Gamma-stacks, which provide a natural environment for studying morphisms of affine Gamma-schemes and cohomological descent. The theory connects with noncommutative geometry and higher-arity (n-ary) generalizations, showing that derived Gamma-geometry forms a coherent homological universe capable of expressing algebraic, geometric, and physical dualities within one categorical law. Computational methods for finite Gamma-semirings and applications to mathematical physics, where ternary operations model triadic couplings, are also discussed.

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arXiv2025

Structure and Spectral Theory of Non-Commutative and $n$-ary $Γ$-Semirings

This paper develops the structural and spectral foundations of noncommutative and n-ary Gamma semirings, extending the commutative ternary framework established in earlier studies. We introduce left, right, and two-sided ideals in the noncommutative setting, derive quotient characterizations of prime and semiprime ideals, and construct corresponding Gamma-Jacobson radicals. For general n-ary operations, we define (n,m)-type ideals and establish diagonal criteria for n-ary primeness and semiprimeness. A unified radical theory and a Zariski-type spectral topology are then formulated, connecting primitive ideals with simple module representations. The results culminate in a noncommutative Wedderburn-Artin-type decomposition, revealing a triadic spectral geometry that unifies commutative, noncommutative, and higher-arity cases

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