A Study on Rough Fuzzy Bipolar Soft Ideals in Semigroups

Abstract

The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs.