Ternary Idempotent $\Gamma$-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 $\Gamma$-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 $\Gamma$-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.