Some kinds of (∈,∈∨q)(∈,∈∨q)-interval-valued fuzzy ideals of BCI-algebras

In this paper, we first introduce the notions of (positive implicative, implicative and commutative) interval-valued fuzzy ideals of BCI-algebras, which are generalizations of (positive implicative, implicative and commutative) fuzzy ideals, respectively, and investigate some of their related properties. The concept of quasi-coincidence of an interval-valued fuzzy point within an interval-valued fuzzy set is introduced. In fact, the concept is a generalized concept of quasi-coincidence of a fuzzy point within a fuzzy set. By using this new idea, we further introduce the notions of (positive implicative, implicative and commutative) (∈,∈∨q)(∈,∈∨q)-interval-valued fuzzy ideals of BCI-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval-valued fuzzy ideals are derived. The relationship among these generalized interval-valued fuzzy ideals of BCI-algebras is also considered.