Roughness of Bipolar Fuzzy Ideals in Semigroups Using Soft Relations Under Decision-Making Applications

Abstract

Examining the interactions between multiple theories, including rough sets, fuzzy sets, and soft sets, is essential for understanding their relationships and applications. This article presents a combination of rough sets, bipolar fuzzy sets, and semigroups equipped with soft relations. Thus, the roughness of bipolar fuzzy substructures in semigroups is employed depending on soft relation. It is a more generalized form than the roughness carried out by congruence relations and set-valued mapping with bipolar fuzzy sets. Soft relations are used to carry out this kind of roughness under aftersets and foresets. Additionally, we will explore the roughness in the bipolar fuzzy substructures by using a soft compatible relation. The rough bipolar fuzzy ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in semigroups are further extensions of the concept. Furthermore, while developing the approximation space of bipolar fuzzy substructures, soft compatible and soft complete relations are required. We prove that every bipolar fuzzy ideal is an upper rough bipolar fuzzy ideal under compatible and soft complete relations but converse does not hold. Moreover, an algorithm is being established based on soft relations that help to make the best decision via bipolar fuzzy sets.