Generalized Intuitionistic Fuzzy Ideals of BCK∕BCI-algebras Based on 3-valued Logic and Its Computational Study

Abstract

On the basis of cut sets of the grade of membership of fuzzy point $x_a$ to belongingness $(\in )$, or quasi-coincident $\left(q\right)$, or belongingness and quasi-coincident$(\in \wedge q)$, or belongingness or quasi-coincident $(\in \vee q)$ to an intuitionistic fuzzy set $A$ of $X$, an $(\alpha ,\beta )$-intuitionistic fuzzy ideal of $X$ is introduced by applying the Lukasiewicz $3$-valued logic, where $\alpha ,\beta \in \{\in ,q,\in \wedge q,\in \vee q\}$ and $\alpha \ne \in \wedge q$. It is shown that an intuitionistic fuzzy set of $X$ is an $(\in ,\in )$(or $(\in ,\in \vee q)$ or $(\in \wedge q,\in )$)-intuitionistic fuzzy ideal of $X$ if and only if $A$ denote an intuitionistic fuzzy ideal with thresholds $(0,1)$ (or $(0,0.5)$ or $(0.5,1)$) of $X$ respectively. It is observed that $A$ denote an $(\in ,\in )$ (or $(\in \wedge q,\in )$ or $(\in ,\in \vee q)$)-intuitionistic fuzzy ideal of $X$ if and only if for any $p\in (0,1]$ (or $p\in (0,0.5]$ or $p\in (0.5,1]$), then $A_p$ served as fuzzy ideal of $X$ respectively. It provided that an intuitiostic fuzzy set is an intuitionistic fuzzy ideal of $X$ with thresholds $(s,t)$ if and only if for any $p\in (s,t]$, then the cut set $A_p$ appear as fuzzy ideal of $X$.

Keywords:

• BCK/BCI-algebras
• Intuitionistic fuzzy ideal
• Fuzzy points
• Cut sets of intuitionistic fuzzy sets
• Lukasiewicz implication operator