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

Abstract

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