Interval-Valued Fuzzy WI-Ideals in Wajsberg Semi-Algebras
K. Perarasan *
PG and Research Department of Mathematics, Annai Vailankanni Arts and Science College (Affiliated to Bharathidasan University), Thanjavur -07, Tamil Nadu, India.
C. Shajitha Begum
Department of Science and Humanities, CMS College of Engineering and Technology (Affiliated to Anna University), Coimbatore, Tamil Nadu, India.
B. Mohamed Harif
Department of Mathematics, Government Arts College (Affiliated to Bharathidasan University), Tiruchirappalli -02, Tamil Nadu, India.
M. Kaliraja
PG and Research Department of Mathematics, Rajah Serfoji Government College (A), (Affiliated to Bharathidasan University), Thanjavur, Tamil Nadu, India.
D. Sargunasundari
PG and Research Department of Mathematics, Annai Vailankanni Arts and Science College (Affiliated to Bharathidasan University), Thanjavur -07, Tamil Nadu, India.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Wajsberg semi-algebra, Interval-valued fuzzy set, Fuzzy WI-ideal, Cartesian product, Level set