Fuzzy Set Theoretic Approach to Generalized Ideals in BCK/BCI-Algebras

Abstract

This paper deals with the study of generalizations of fuzzy subalgebras and fuzzy ideals in BCK/BCI-algebras. In fact, the notions of $\left(\in ,\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -fuzzy subalgebras, $\left(\in ,\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideals, and $\left(\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right),\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideals in BCK/BCI-algebras are introduced. Some examples are provided to demonstrate the logic of the concepts used in this paper. Moreover, some characterizations of these notions are discussed. In addition, the concept of $\left(\in ,\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -fuzzy commutative ideals is introduced, and several significant characteristics are discussed. It is shown that for a BCK-algebra $\mathcal{A}$ , every $\left(\in ,\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -commutative ideal of a BCK-algebra is an $\left(\in ,\in \vee \left({\tilde{\kappa }}^{\ast },q_{\tilde{\kappa }}\right)\right)$ -fuzzy ideal, but the converse does not hold in general; a counter example is constructed.