$(\in,\in\vee q)$-bipolar Fuzzy $BCK/BCI$-algebras

Abstract

In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of $(\in,\in\vee q)$-bipolar fuzzy subalgebras and ideals of $BCK/BCI$-algebras are introduced and their related properties are investigated by some examples. We study bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-ideals by their level subalgebras and level ideals. We also provide the relationship between $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-subalgebras, and $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-ideals and bipolar fuzzy $BCK/BCI$-ideals by counter examples.