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

Chandrasekhar Gokavarapu, D. Madhusudhana Rao

Abstract

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.

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

Abstract

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 -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 -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.
Paper Structure (67 sections, 20 theorems, 69 equations)

This paper contains 67 sections, 20 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Idempotent Semilattices and Canonical Order
  4. Idempotent Semirings
  5. Monotone Operators and Fixed Points
  6. Associative Triple Systems
  7. Semiring Path Frameworks
  8. Structural Limitation of Dyadic Composition
  9. Ternary Idempotent $\Gamma$-Semirings
  10. Axiomatic Definition
  11. Order Compatibility
  12. Window Independence
  13. Degeneration to Binary Semirings
  14. Non-Degenerate Systems
  15. Dimensional Extension Principle
  16. ...and 52 more sections

Key Result

Proposition 2.3

$(T,\le)$ is a partially ordered set.

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Knaster--Tarski
  • Definition 2.7
  • Definition 2.8
  • Definition 3.1
  • ...and 39 more