Optimistic multi-granulation roughness of intuitionistic fuzzy ideals based on soft relations over dual universes

<h2>Abstract</h2> The major point of proposed work is to study optimistic multi-granulation roughness of intuitionistic fuzzy ideals by using soft relations that is free from all the complications that challenged by many researchers. The combination of soft binary relations, optimistic multi-granulation rough sets, and intuitionistic fuzzy sets is a useful technique to deal with uncertainty. In modern theories, this combination provides a powerful tool to manage daily life critical situations. Furthermore, the lower and upper approximations of intuitionistic fuzzy subsemigroups, intuitionistic fuzzy left (right) ideals, intuitionistic fuzzy interior ideals and intuitionistic fuzzy bi-ideals of semigroups are studied by using soft relations. Compatible relation is used for upper approximation and complete relation is used for lower approximation. This approach has potential impacts on fields like coding theory, automata theory, and cryptography, providing a flexible and robust method for handling imprecise data, where the use of optimization, and analysis in the industrial domain identify input data with uncertainty and vagueness and reachable multi-source information. At the end, a comparison study and conclusions of the presented work are given that proves how our technique is effective in contrast to fuzzy set.