Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras
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| Format: | Preprint |
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2026
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| author | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_21310 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras Gokavarapu, Chandrasekhar Rao, D. Madhusudhana Rings and Algebras Logic 16Y60, 68R10, 06B35, 68Q25 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. |
| title | Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras |
| topic | Rings and Algebras Logic 16Y60, 68R10, 06B35, 68Q25 |
| url | https://arxiv.org/abs/2602.21310 |