More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras

Abstract

The concepts of interval-valued $\left(\in ,\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy subalgebras, interval-valued $\left(\in ,\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideals, and interval-valued $\left(\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right),\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideals are introduced, and related properties are studied. Many examples are given in support of these new notions. Furthermore, interval-valued $\left(\in ,\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy commutative ideals are defined, and some important properties are discussed. For a BCK-algebra $X$ , it is proved that every interval-valued $\left(\in ,\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy commutative ideal of BCK-algebra $X$ is an interval-valued $\left(\in ,\in \vee \left(\widetilde{{\kappa }^{\ast }},q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideal of $X$ , but the converse need not be true, in general, and then a counterexample is constructed.