arXiv subjects

D. Madhusudhana Rao

Publications and source records attributed to D. Madhusudhana Rao.

8 recordsLinked to original sources

arXiv2026

Operational Regimes of Ternary $Γ$-Semiring Chemical Systems

Ternary $Γ$-semiring chemical systems provide an algebraic language in which mediators such as catalysts, solvents, inhibitors, thermodynamic conditions, and external fields enter the transformation law as internal arguments. Building on the published MATCH foundation article, this paper studies operational regimes of the compressed TGS notation rather than proposing a second axiomatization. Distributivity is interpreted as observable non-interference of mediated channels, while associativity is interpreted as observable path-independence of bracketed mediated protocols. Since the state space is only semigroup- or semiring-level, subtraction is not used at the state level; instead, observable defect and ratio indices are defined after applying a non-negative observable. We prove that distributivity implies observable non-interference under observable additivity, and that associativity implies observable path-independence. Nonzero defects detect observable distinguishability. To strengthen the diagnostic layer, we prove recovery theorems showing that separating families of observables can recover algebraic distributivity and associativity from vanishing observable defects. We also introduce approximate operational regimes and prove ratio bounds controlled by uniform defect estimates. Under strict convexity, uniqueness, and common global-constraint assumptions, a potential-generated equilibrium operation is shown to be associative. A symbolic intermediate-dependent Michaelis--Menten type construction shows how non-associativity may arise in kinetic regimes, and regulatory/allosteric-type non-distributivity is discussed cautiously through observable coupling.

physics.chem-ph↗

arXiv2026

Axiomatic Foundations of Chemical Systems as Ternary $Γ$-Semirings

Chemical transformations depend not only on the identities of the reacting species but also on the catalytic, environmental, and intermediate conditions under which they occur. Classical binary reaction formalisms usually treat such conditions as external annotations, which obscures the genuinely multi-state and multi-parameter character of real chemical processes. In this paper we introduce an axiomatic framework in which a chemical system is modelled by a ternary $Γ$-semiring. The elements of the state set represent chemical states, while the parameter set encodes catalytic and environmental conditions. A $Γ$-dependent ternary operation is used to describe mediated transformations, treating reactants, intermediates, and mediators as intrinsic arguments of the transformation law. We develop the algebraic axioms governing these mediated interactions and interpret their associativity, distributivity, and $Γ$-linearity in terms of multi-step pathways, parallel processes, and controlled environmental dependence. We introduce chemical ideals and $Γ$-ideals as algebraic structures modelling reaction-closed sub-systems and pathway-stable domains, and study their prime and semiprime forms. Homomorphisms between TGS-chemical systems are shown to preserve reaction pathways and describe consistent changes of chemical environment. Abstract examples from catalysis, thermodynamic phase control, and field-induced quantum transitions illustrate how familiar chemical phenomena fit within this framework. The resulting theory provides a unified algebraic foundation for multi-parameter chemical behaviour and establishes the structural basis for subsequent developments involving kinetics, geometric methods, and computational or AI-assisted models.

math.RA↗

arXiv2026

Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras

Binary idempotent semirings govern classical path algebras. Their multiplicative structure is dyadic. We examine whether this restriction is structural or accidental. We define ternary idempotent $Γ$-semirings as higher-arity ordered algebraic systems admitting associative ternary composition compatible with idempotent addition. We prove that such structures strictly extend classical semiring path algebras. In particular, we construct a ternary associative operation which cannot be represented as an iterated associative binary operation. This establishes non-reducibility. We formulate a higher-order path problem in directed graphs with weights in a ternary idempotent $Γ$-semiring. The associated relaxation operator is shown to be monotone on a complete lattice and to admit a least fixed point. Convergence follows under a finite acyclicity condition. The combinatorial growth of interaction windows yields a distinct complexity class relative to binary path schemes. These results indicate that dyadic semiring frameworks do not exhaust algebraic path formalisms. Higher-arity composition introduces structural phenomena absent in binary systems.

math.RA↗

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.

math.RA↗

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.

math.OC↗

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.

math.RA↗

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.

math.RA↗

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

math.RA↗