FUZZY p-IDEALS OF BCI-ALGEBRAS WITH DEGREES IN THE INTERVAL (0, 1

Abstract

Abstract. The notion of an enlarged p-ideal and a fuzzy p-ideal in BCI-algebras with degree are introduced. Related properties of them are in-vestigated. 1. IntroductionThe concept of a fuzzy set is applied to generalize some of the basic conceptsof general topology ([1]). Rosenfeld ([6]) constituted a similar application tothe elementary theory of groupoids and groups. Xi ([7]) applied to the conceptof fuzzy set to BCK-algebras. Y. B. Jun and J. Meng ([4]) introduced of fuzzyp-ideals in BCI-algebras and studied their properties.In this paper, we introduce the notion of an enlarged p-ideal and a fuzzyp-ideal in BCI-algebras with degree. We study related properties of them.2. PreliminariesWe review some definitions and properties that will be useful in our results.By a BCI-algebra we mean an algebra (X,∗,0) of type (2,0) satisfying thefollowing conditions:(a1) (∀x,y,z ∈ X)(((x ∗y)∗(x ∗z))∗(z ∗y) = 0),(a2) (∀x,y ∈ X)((x ∗(x ∗y))∗y = 0),(a3) (∀x ∈ X)(x ∗x = 0),(a4) (∀x,y ∈ X)(x∗y = 0, y ∗x = 0 ⇒ x = y).If a BCI-algebra X satisfies the following identity:(a5) (∀x ∈ X)(0∗x = 0),then X is called a BCK-algebra.In any BCI-algebra X one can define a partial order “≤” by putting x ≤ yif and only if x ∗y = 0.A BCI-algebra X has the following properties: