Ideals of BCK-algebras and BCI-algebras Based on a New Form of Fuzzy Set

Abstract

Ideals in BCK/BCI algebra based on $Y_J^{\varepsilon}$-fuzzy sets are studied. The fundamental properties of the level set of $Y_J^{\varepsilon}$-fuzzy sets are investigate first. The concept of (closed) $Y_J^{\varepsilon}$-fuzzy ideals in BCK/BCI-algebras is introduces, and several properties are investigated. The relationship between $Y_J^{\varepsilon}$-fuzzy ideal and $Y_J^{\varepsilon}$-fuzzy subalgebra are discussed, and also the relationship between $Y_J^{\varepsilon}$-fuzzy ideal and fuzzy ideal is identified. The characterization of (closed) $Y_J^{\varepsilon}$-fuzzy ideal using the Y-level set is established. The necessary and sufficient conditions for $Y_J^{\varepsilon}$-fuzzy ideal to be closed is explored, and conditions for $Y_J^{\varepsilon}$-fuzzy subalgebra to be $Y_J^{\varepsilon}$-fuzzy ideal are provided.