An efficient scheme to study the approximation of BS ideals in semigroups

Rough set (RS) theory is a valuable mathematical tool to address incomplete knowledge. In contrast, BS sets (BSSs) can handle the vagueness and the bipolarity of the data in a variety of scenarios. The main goal of the current article is to analyze the concept of rough BS (BS) ideals in semigroup (Sg) by combining RS theory and BSSs. To this end, the ideas of BS left ideals (BS-LIs), BS right ideals (BS-RIs), BS two-sided ideals (BS-2SIs), BS interior ideals (BS-IIs) and BS bi-ideals (BS-BIs) over the Sg ? are developed. Some significant properties associated with these notions are highlighted with several concrete illustrations. Furthermore, it has been noted that the congruence relation (Cr) and complete congruence relation (CCr) are found to be essential for creating rough approximations of BS ideals. As a result, their affiliated features are evaluated through Cr and CCr.