A new form of fuzzy set and its application in $BCK$-algebras and $BCI$-algebras

Abstract

The notion of the J-operator in the closed interval $[0,1]$ is introduced and several properties are investigated. Using the J-operator, a new fuzzy set called the $Y_J^{\varepsilon}$-fuzzy set is established and it is applied it to subalgebras in $BCK/BCI$-algebras. The concept of the $Y_J^{\varepsilon}$-fuzzy subalgebra is introduced and its properties are discussed. Conditions for a fuzzy set to be a $Y_J^{\varepsilon}$-fuzzy subalgebra are provided, and the relationship between the fuzzy subalgebra and the $Y_J^{\varepsilon}$-fuzzy subalgebra is discussed.