Symmetric Dual Arithmetic (SDA) Revisited: Fully Distributive Products on MV-Chains
Ihor Ivliev, LLMs GenAI · Zenodo (CERN European Organization for Nuclear Research) · 2025
This work studies fully distributive arithmetic on MV-chain intervals with truncated addition and exact complement. It proves that a commutative semiring multiplication exists exactly when the interval has a least positive element, which is necessarily the multiplicative unit. Every infinite model contains a canonical endpoint subalgebra identified with repaired Symmetric Dual Arithmetic, and the full arithmetic–complement structures are shown to be simple. On the three-level lexicographic MV-chain, all two-sided distributive unital products are completely classified, together with their homomorphisms, isomorphism classes, endomorphisms, and subalgebras. The nonmaximal products admit polynomial-cap representations and are classified up to isomorphism by a quadratic discriminant. The work also gives finite-realization and bounded-information results while situating the theory within established MV-algebra, MVW-rig, and capped-semiring mathematics. Historical priority for the new structural classifications is not assumed.