Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows

Based on the theory of falling shadows and fuzzy sets, the notion of a falling fuzzy implicative ideal of a BCK-algebra is introduced. Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals are given. Relations between fuzzy implicative ideals and falling fuzzy implicative ideals are provided. 1. Introduction and Preliminaries 1.1. Introduction In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [1] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [2] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. Falling shadow representation theory shows us the way of selection relaid on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated in [3]. Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Jun and Park [6] discussed the notion of a falling fuzzy subalgebra/ideal of a BCK/BCI-algebra. Jun and Kang [7, 8] also considered falling fuzzy positive implicative ideals and falling fuzzy commutative ideals. In this paper, we establish a theoretical approach to define a fuzzy implicative ideal in a BCK-algebra based on the theory of falling shadows. We consider relations between fuzzy implicative ideals and falling fuzzy implicative ideals. We provide relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals. 1.2. Basic Results on BCK-Algebras and Fuzzy Aspects A BCK/BCI-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers. An algebra of type is called a BCI-algebra if it satisfies the following conditions: (i) , (ii) , (iii) , (iv) . If a BCI-algebra satisfies the following identity: (v) , then is called a -algebra. Any BCK-algebra satisfies the following axioms: (a1) , (a2) , (a3) , where if and only if .A subset of a BCK-algebra is called an ideal of if it satisfies the following: (b1) , (b2) . Every ideal of a BCK-algebra has the following assertion: A subset of a BCK-algebra is called a positive implicative ideal of if it