Generalizations of Fuzzy q-Ideals of BCI-Algebras

Abstract

In this paper, we introduce the notion of $\left(\in ,\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy $q$ -ideals of BCI-algebras to propose a more general form of fuzzy $q$ -ideals of BCI-algebras. We prove that $\left(\in ,\in \vee q\right)$ -fuzzy $q$ -ideals and $\left(\in \vee \left({\kappa }^{\ast },q_{\kappa }\right),\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy $q$ -ideals are $\left(\in ,\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy $q$ -ideals, but the converse assertion is not valid and examples are given to support this. It is proved that every $\left(\in ,\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy $q$ -ideal is an $\left(\in ,\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy ideal, but the converse need not be true in general and an example is provided. In addition, correspondence between $\left(\in ,\in \vee \left({\kappa }^{\ast },q_{\kappa }\right)\right)$ -fuzzy $q$ -ideals and $q$ -ideals of BCI-algebras is considered.