Analysis of Translations and Algebraic Properties of Intuitionistic Fuzzy Ideals across Diverse Algebraic Systems

Abstract

In this paper, a detailed investigation of the rich class of Intuitionistic Fuzzy Ideals (IFIs) is carried out in different algebraic structures with their different translations with respect to the algebraic properties. Intuitionistic fuzzy sets generalize the classical fuzzy set theory from degrees of membership to degrees of both membership and non-membership, and hence provide a powerful model for uncertainty representation. Their extension and application to abstract algebra results in the introduction of intuitionistic fuzzy ideals, generalizations of the classical ideals which are used to reason under conditions of uncertainty. The study is based on the analysis of the behaviour of IFIs under the convolutional fuzzy translations within BCK and BCI-algebras, and on the examination of anti-intuitionistic fuzzy ideals through level-set-based translation mechanisms. Furthermore, Cartesian product operations of opposite IF a-translation on division HA algebras are defined and give new insights into structure relations between algebraic systems. The paper combines sophisticated decision making models and distance measure in fuzzy set theory to justify the theoretical constructs and bring out their practical implications. This work is expected to promote further progress of mathematics, decision science, engineering and computer application by integrating intuitionistic fuzzy logic and algebraic theory for better understanding of uncertainty modelling and its algebraic foundation.