Relational Atomic Representations, Binary Reducibility, and Witness Profiles for Complete Ternary Gamma-Semirings

Abstract

Let the carrier and index reducts of a ternary $\Gamma$-semiring be complete atomic Boolean algebras and let its five-variable product preserve arbitrary joins. Atomic products need not be atoms: they may vanish or contain several output atoms. We encode this behaviour by a six-place, two-sorted relation and formulate three relational-composition identities corresponding exactly to the type-correct ternary $\Gamma$-associativity laws. We prove that complex algebras of these relational alternating quinary frames are precisely the completely additive complete atomic ternary $\Gamma$-semirings. With strong frame morphisms and atom-preserving complete-join morphisms, the construction is an equivalence of categories. Deterministic atom-total cores occur as the functional subcategory. For diagonal functional cores, binary reducibility is equivalent to an atom-preserving, completely union-distributive associative binary collapse of the complex algebra. For each finite relational frame, minimal witness antichains give canonical profiles for carrier-sort polynomial functions with fixed index coefficients and yield direct equality and universal-inclusion tests. Finally, a four-point irreducible symmetric five-ary band gives an index-sensitive $16$-element example without binary collapse; one polynomial inequation recovers its parity quotient on atoms.