ELSEVIER Fuzzy Sets and Systems 101 (1999) 153 158 FUZZY sets and systems Fuzzy H-ideals in BCI-algebras H.M. Khalid*, B. Ahmad Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakar~va UniversiO'. Multan 60800. Pakistan Received January 1996; received in revised form January 1997 Abstract In this paper, we define fuzzy H-ideals in BCI-algebras and study its several properties. ~,i~1999 Elsevier Science B.V. All rights reserved. Keywords: H-ideals; Fuzzy H-ideals; Fuzzy relations and cartesian product of fuzzy H-ideals in BCI-algebras 1. Introduction In 1965, Zadeh [9] introduced the concept of fuzzy subsets. Since then the researchers have applied this notion to several mathematical dis- ciplines. In 1991, Ougen [7] defined fuzzy subsets in BCK-algebras and investigated some proper- ties. In 1993, Jun [4] and Ahmad [1] applied it to BCI-algebras. Recently, Jun and Meng [5] introduced fuzzy p-ideals in BCI-algebras and studied several interesting properties. Recall that a set I is called a p-ideal if (i) 0eI, (ii) (x * z)*(y, z)e I, y ~ I imply x e l for all x, y, z e X. Note that every p-ideal is ideal [10]. For details of p-ideals and fuzzy p-ideals, we refer to [10, 5]. In this paper, we define another interesting class called fuzzy H-ideals and study their properties. For H- ideals, we refer to [6]. An algebra (X,*,0) of type (2,0) is called a BCI-algebra if it satisfies the following condi- tions: (1) (x,y),(x*z) <~ z,y, (2) x * ( x * y ) ~< y, (3) x ~< x, (4) x~<y,y~<ximplyx=y, (5) x ~< 0 implies x = 0, where x ~< y means x*y = 0. A subset A of a BCI-algebra X is called an ideal, if for any x, y in X, (6) 0 ~ A, (7) x,y, ycAimplyx~A [3]. Definition (Khalid and Ahmad [6]). A non- empty subset A of a BCI-algebra X is called an H-ideal of X if(i) 0cA, (ii) x,(y,z) eA, yeA imply x • z e A. *Corresponding author. Permanent address: Department of Mathematics, Azad Jammu and Kashmir University, Muzaffarabad, Pakistan. If we put z = 0, then it follows that A is an ideal. Thus, every H-ideal is an ideal. The following example shows that H-ideals [6] exist: 0165-0114/99/$ see front matter a' 1999 Elsevier Science B.V. All rights reserved. PII: S0165-01 14(97)00042-0