Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 461–470 FUZZY SET THEORY APPLIED TO IMPLICATIVE IDEALS IN BCK-ALGEBRAS Young Bae Jun and Seok Zun Song Abstract. As a continuation of [4], characterizations of fuzzy im- plicative ideals are given. An extension property for fuzzy implica- tive ideals is established. We prove that the family of fuzzy implica- tive ideals is a completely distributive lattice. Using level subsets of a BCK-algebra X with respect to a fuzzy set ¯ A in X, we construct a fuzzy implicative ideal of X containing ¯ A. 1. Introduction The study of BCK-algebras was initiated by K. Is´ eki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. For the general development of BCK-algebras, the ideal theory and its fuzzification play an important role. The notion of implicative ideals in a BCK-algebra is first introduced by K. Is´ eki [1] in 1975, and then the fuzzification of implicative ideals is discussed in [8] by O. G. Xi. In 1995, Y. B. Jun, S. M. Hong and E. H. Roh [4] developed the fuzzy implicative ideals in BCK-algebras. This paper is a continuation of the paper [4]. We first give characterizations of fuzzy implicative ideals, and establish an extension property for fuzzy implicative ideals. We prove that the family of fuzzy implicative ideals is a completely distributive lattice. Given a fuzzy set ¯ A in a BCK-algebra X , we construct a fuzzy implicative ideal of X containing ¯ A by using level subsets of X with respect to ¯ A. 2. Preliminaries By a BCK-algebra, we mean an algebra (X ; ∗, 0) of type (2,0) satis- fying the following conditions: Received August 2, 2002. 2000 Mathematics Subject Classification: 06F35, 03G25, 03B52. Key words and phrases: (fuzzy) ideal, (fuzzy) implicative ideal.