Interval-Valued \(m\)-Polar Fuzzy Positive Implicative Ideals in \(BCK\)-Algebras

Abstract

In this paper, the notion of interval-valued $m$ -polar fuzzy positive implicative ideals in BCK-algebras is presented. Then, the relationships between interval-valued $m$ -polar fuzzy positive implicative ideals and interval-valued $m$ -polar fuzzy ideals are investigated. After that, the concepts of interval-valued $m$ -polar $\left(\in ,\in \vee q_{\tilde{\kappa }}\right)$ -fuzzy positive implicative ideals and interval-valued $m$ -polar $\left(\in ,\in \vee q_{\tilde{\kappa }}\right)$ -fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valued $m$ -polar $\left(\in ,\in \vee q_{\tilde{\kappa }}\right)$ -fuzzy positive implicative ideals are interval-valued $m$ -polar $\left(\in ,\in \vee q_{\tilde{\kappa }}\right)$ -fuzzy ideals, but the converse need not be true in general and an example is given in this aim.