Abstract:
We consider the intuitionistic fuzzification of the concept ofsubalgebras and ideals in BCK-algebras, and investigate some oftheir properties. We introduce the notion of equivalencerelations on the family of all intuitionistic fuzzy ideals of aBCK-algebra and investigate some related properties.

Abstract:
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

Abstract:
We consider the fuzzification of the notion of implicative hyper BCK-ideals, and then investigate several properties. Using the concept of level subsets, we give a characterization of a fuzzy implicative hyper BCK-ideal. We state a relation between a fuzzy hyper BCK-ideal and a fuzzy implicative hyper BCK-ideal. We establish a condition for a fuzzy hyper BCK-ideal to be a fuzzy implicative hyper BCK-ideal. Finally, we introduce the notion of hyper homomorphisms of hyper BCK-algebras, and discuss related properties.

Abstract:
In this paper we ?rst de?ne the category of fuzzy hyper BCK- algebras. After that we show that the category of hyper BCK-algebras has equalizers, coequalizers, products. It is a consequence that this category is complete and hence has pullbacks.

Abstract:
We consider the fuzzification of the notion of fuzzy multiply positive implicative hyper BCK-ideals of BCK-algebras and then some related results are obtained. Using the concept of level subsets, we give a characterization of a fuzzy multiply positive implicative hyper BCK-ideal. We state a relation between a fuzzy hyper BCK-ideal and a fuzzy multiply positive implicative hyper BCK-ideal. Moreover, we introduce the notions of Noetherian hyper BCK-algebras and hyper homomorphisms of hyper BCK-algebras and investigate some related properties. Finally, we introduce the concept of hyper normalization of hyper BCK-algebras and discuss related properties.

Abstract:
We introduce the notion of sensible fuzzy ideals of BCK-algebraswith respect to a t-conorm and investigate some of theirproperties. We give the conditions for a sensible fuzzy subalgebrawith respect to a t-conorm to be a sensible fuzzy ideal withrespect to a t-conorm. Some properties of the direct product andS-product of fuzzy ideals of BCK-algebras with respect to at-conorm are also discussed.

Abstract:
We consider the fuzzification of the notion of an n-fold positive implicative ideal. We give characterizations of an n-fold fuzzy positive implicative ideal. We establish the extension property for n-fold fuzzy positive implicative ideals, and state a characterization of PIn-Noetherian BCK-algebras. Finally we study the normalization of n-fold fuzzy positive implicative ideals.

Abstract:
Under the semantic frame of $L^*$-lattice valued logic, we define the concept of intuitional fuzzifying BCK-algebra on $L^*$-lattice valued logic by using $\L$ukasiewicz implication operator as tool. In BCK-algebra, the concepts of ideals, positive implicative ideals and implicative ideals have ever been depicted by classical set theory, but now, they are redefined by a unary predicate calculus on $L^*$-lattice valued logic, and their properties and relations among them are discussed.

Abstract:
The notion of -subalgebras of several types is introduced, and related properties are investigated. Conditions for an -structure to be an -subalgebra of type are provided, and a characterization of an -subalgebra of type is considered. 1. Introduction A (crisp) set in a universe can be defined in the form of its characteristic function yielding the value for elements belonging to the set and the value for elements excluded from the set So far, most of the generalization of the crisp set have been conducted on the unit interval and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point into the interval Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [1] introduced a new function which is called negative-valued function, and constructed -structures. They applied -structures to BCK/BCI-algebras, and discussed -subalgebras and -ideals in BCK/BCI-algebras. Jun et al. [2] considered closed ideals in BCH-algebras based on -structures. To obtain more general form of an -subalgebra in BCK/BCI-algebras, we define the notions of -subalgebras of types , and and investigate related properties. We provide a characterization of an -subalgebra of type We give conditions for an -structure to be an -subalgebra of type 2. Preliminaries Let be the class of all algebras with type . By a BCI-algebra we mean a system in which the following axioms hold: (i) (ii) (iii) (iv) for all If a BCI-algebra satisfies for all then we say that is a BCK-algebra. We can define a partial ordering by In a BCK/BCI-algebra , the following hold: for all A nonempty subset of a BCK/BCI-algebra is called a subalgebra of if for all For our convenience, the empty set is regarded as a subalgebra of We refer the reader to the books [3, 4] for further information regarding BCK/BCI-algebras. For any family of real numbers, we define Denote by the collection of functions from a set to We say that an element of is a negative-valued function from to (briefly, -function on ). By an -structure we mean an ordered pair of and an -function on In what follows, let denote a BCK/BCI-algebra and an -function on unless otherwise specified. Definition 2.1 (see [1]). By a subalgebra of based on -function (briefly, -subalgebra of ), we mean an -structure in which satisfies the following assertion: For any -structure and the set is called a

Abstract:
In this paper, we introduce the notion of a BCK-topological module in a natural way and establish that every decreasing sequence of submodules on a BCK-module M over bounded commutative BCK-algebra X is indeed a BCK- topological module. We have defined the notion of compatible and strict BCK- module homomorphisms, and establish that a strict BCK-module homomorphism is an open as well as a continuous mapping. Also, we establish the necessary and sufficient condition for a compatible mapping to be strict.