A Unified Framework for Constructing Two-Branched Fuzzy Implications and Copulas via Monotone and Convex Function Composition

This paper presents a unified framework for constructing two-branched fuzzy implications and families of copulas based on the same composition principles involving monotone and convex functions. The proposed methodology yields operators with a genuine dual structure, where each branch satisfies distinct boundary and monotonicity conditions while remaining consistent with the general axioms of copulas. By systematically combining monotone generators with convex transformations, new families of fuzzy implications and copulas are obtained, both exhibiting enhanced analytical properties such as strengthened two-increasing behavior, adjustable dependence strength, and flexible convexity with continuous transitions. Convexity ensures the two-increasing property, while continuity guarantees the completeness and mathematical soundness of the constructions. Remarkably, certain copulas produced under this framework display Archimedean-like features—symmetry and associativity—thus providing new theoretical instruments for the advancement of fuzzy logic and dependence modeling.