Incorporating complex bipolar fuzzy set with subrings and application in decision making

A complex bipolar fuzzy set ( $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$) is an extension of a complex fuzzy set and a $\mathbb{B}\mathbb{F}$ set with a wide range of values. A $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ is differentiated from a $\mathbb{B}\mathbb{F}$ set by the incorporation of negative and positive membership functions to the unit circle in the complex plane, which empowers one to handle the vagueness more effectively. We aim to generalize the notions of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ by proposing a general algebraic structure to dealing with complex bipolar ( $\mathbb{C}\mathbb{B}$) fuzzy data by integrating the idea of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ and subrings. The structure of $\mathbb{C}\mathbb{B}$ fuzzy subrings, such as the $\mathbb{C}\mathbb{B}$ fuzzy ring isomorphism, $\mathbb{C}\mathbb{B}$ fuzzy quotient ring, and $\mathbb{C}\mathbb{B}$ fuzzy ring homomorphism, are examined in this paper. We develop the ( $\delta ,\alpha ;\sigma ,\beta$)-cut of a $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ and explore its algebraic interpretations. Additionally, we describe the $\mathbb{C}\mathbb{B}$ fuzzy support set and demonstrate some significant characteristics associated with this idea. Furthermore, we use the concept of a naturally occurring complex ring homomorphism to describe a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism. Additionally, we prove a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism between the $\mathbb{C}\mathbb{B}$ fuzzy subrings of the ring and the $\mathbb{C}\mathbb{B}$ fuzzy subring of the complex quotient ring. We demonstrate a strong connection among two $\mathbb{C}\mathbb{B}$ fuzzy subrings of complex quotient rings under a specific $\mathbb{C}\mathbb{B}$ fuzzy surjective homomorphism. We construct a complex fuzzy isomorphism between both associated $\mathbb{C}\mathbb{B}$ fuzzy subrings. Furthermore, we introduce three basic results of the $\mathbb{C}\mathbb{B}$ fuzzy isomorphism to explain the relationship between two $\mathbb{C}\mathbb{B}$ fuzzy subrings. Finally, we use a complex bipolar fuzzy subring in decision making.