Second Order Fuzzy Relational Equations with Sup and Inf Compositions: A Theoretical and Computational Framework

Abstract

               Second-order fuzzy relations (SOFRs), where membership grades are themselves fuzzy numbers, can model multiple levels of uncertainty. Max-min, max-product, and sum-product have already been established within the efficient epsilon-delta ( ) formalism, however the exploration of more general fuzzy relational connectives remains limited. This paper introduces two non-standard compositions, the Supremum- norm (Sup ) and the Infimum- conorm (Inf ) compositions, within a comprehensive  framework for formulating and solving Second Order Fuzzy Relational Equations (SOFREs). We extend these compositions founded by Klir & Yuan [1] for Type-1 relations to the second-order context. We prove key algebraic properties, including a necessary condition for solvability, and demonstrate the application of these compositions through a comparative numerical example in a medical diagnostic setting. Our results demonstrate that the Sup  composition offers a tunable, pessimistic chaining of relations, as shown with the product norm. In contrast, the Inf  composition provides an optimistic, evidence-aggregating behavior when implemented with probabilistic sum conorm; it yields strong diagnostic discrimination but at the cost of significant uncertainty amplification. Together, these compositions significantly expand the modeling flexibility of the second-order fuzzy relational calculus.