Interval-Valued Fuzzy WI-Ideals in Wajsberg Semi-Algebras

This paper introduces interval-valued fuzzy WI-ideals in Wajsberg semi-algebras as a natural extension of fuzzy WI-ideals by incorporating interval-based membership grades to model uncertainty. A foundational decomposition principle is established, demonstrating that an interval-valued fuzzy set qualifies as an WI-ideal if and only if its lower and upper component fuzzy sets are fuzzy WI-ideals of the underlying algebra. Building on this structural equivalence, the study systematically investigates the fundamental algebraic properties of these structures, establishing that they exhibit the max-at-zero property, are order-reversing, and remain invariant under arbitrary intersections. Additionally, it analyses their behaviour under algebraic transformations, showing that interval-valued fuzzy WI-ideals are preserved under inverse homomorphisms and surjective homomorphic images. Finally, it explores the Cartesian product of interval-valued fuzzy WI-ideals in direct product Wajsberg semi-algebras, establishing necessary and sufficient conditions for product structures along with their precise level-set characterisations. These findings provide a rigorous algebraic foundation for further developments in fuzzy logic algebras and uncertainty-based structural analysis.