Roughness of \(( \alpha ,\beta )\)-bipolar fuzzy ideals in semigroups

Semigroup (SG) is a prominent algebraic structure having an associative binary operation. The theories of fuzzy sets (FSs) and rough sets (RSs) are invented to combat the uncertainty and vagueness of the data. Furthermore, the notion of the bipolar fuzzy set (BFS) is one of the significant generalizations that can accommodate the fuzziness, uncertainty and bipolarity of the data in real-life dilemmas. This article aims to study the roughness of \((\alpha , \beta )\) $(\alpha ,\beta )$ -bipolar fuzzy ideals ( \((\alpha , \beta )-\mathcal {BFI}\) $(\alpha ,\beta )-\mathcal{BFI}$ s) in SGs. In this article, firstly, we will portray the characterization of SGs based on their \((\alpha , \beta )-\mathcal {BFI}\) $(\alpha ,\beta )-\mathcal{BFI}$ s, and then this idea is expanded to the approximations of \((\in ,\in \vee q)-\mathcal {BFI}\) $(\in ,\in \vee q)-\mathcal{BFI}$ s in SGs by defining a congruence relation (CR) on the SG. Besides, it becomes apparent that CR and complete CR (CCR) are crucial in constructing rough approximations of \((\alpha , \beta )\) $(\alpha ,\beta )$ -BFIs. Consequently, their respective features are analyzed via CR and CCR.