Topological Consistency of q-Rung Orthopair Fuzzy Normed Rings Under Strict Archimedean Operators

Abstract

The integration of q-rung orthopair fuzzy sets (q-ROFSs) with algebraic structures relies on idempotent minimum and maximum operators. To formalize the topological behavior of fuzzy sets in unbounded normed rings, we define the norm-compatibility condition. We show that applying classical idempotent operators violates this condition, which intuitively requires the membership grades of divergent sequences to vanish at infinity, and restricts the fuzzy support to the unit ball, creating a metric limitation. To resolve this metric incompatibility, this paper introduces the axiom of Archimedean Dual Norm Compatibility. By substituting idempotent operators with Archimedean t-norms and t-conorms, we construct topologically consistent q-ROF (TCq-ROF) normed rings that preserve the unbounded metric geometry. Under this framework, we formulate the TCq-ROF metric kernel and prove the First Isomorphism Theorem for TCq-ROF normed rings. Furthermore, we establish TCq-ROF direct sums, maximal and prime ideals, zero divisors over product rings, and TCq-ROF radical extensions. Our proposed Archimedean framework provides a metric-preserving mathematical foundation for fuzzy abstract algebra over infinite-dimensional commutative spaces.