An Innovative Perspective on Bipolar Fuzzy Fantastic Ideals in BCK/BCI-Algebras

In this paper, we propose the concept of $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy ideals in BCK/BCI-algebras. We show that an $(\in, \in \vee \check{q_{\varphi}})$-bipolar fuzzy ideal is an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy ideal. For a BCK/BCI-algebra, it has been shown that an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}})$-bipolar fuzzy ideal is an $(\in, \in \vee \check{q})$-bipolar fuzzy ideal of $\check{\aleph}$, but not conversely, and then an example is given. We introduce the concept of $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideals in BCK/BCI-algebras. It has been shown that an $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideal is an $(\in, \in)$-bipolar fuzzy ideal in BCK/BCI-algebras. Furthermore, the connection between $(\in, \in \vee (\varphi^{\divideontimes}, \check{q_{\varphi}}))$-bipolar fuzzy fantastic ideals and fantastic ideals are established.