In this note first we define the notions of fuzzy positive implicative hyper BCK-ideals of types 1,2,3 and 4. Then we prove some theorems which characterize the above notions according to the level subsets. Also we obtain the relationships among these notions, fuzzy (strong, weak, reflexive) hyper BCK-ideals and fuzzy positive implicative hyper BCK-ideals of types 5,6,7 and 8. Then, we define the notions of fuzzy (weak) implicative hyper BCK-ideals and we obtain some related results. Finally, by considering the product of two hyper BCK-algebras we give some theorems which show that how the projections of a fuzzy (positive implicative, implicative) hyper BCK-ideal is again a fuzzy (positive implicative, implicative) hyper BCK-ideal.

In BCK-algebras, the notion of Ł ukasiewicz fuzzy positive implicative ideal is introduced, and several properties are investigated. The relationship between Ł ukasiewicz fuzzy ideal and Ł ukasiewicz fuzzy positive implicative ideal is discussed, and characterizations of a Ł ukasiewicz fuzzy positive implicative ideal are considered. Conditions for a Ł ukasiewicz fuzzy ideal to be a Ł ukasiewicz fuzzy positive implicative ideal are provided, and conditions for the 2-set, q-set and O-set to be positive implicative ideals are explored.

In this paper, by considering soft set theory, lattice theory and filter theory of EQ-algebras, we introduce the concepts of intuitionistic soft positive implicative L-prefilter and intuitionistic soft implicative L-prefilter. Next, we give some examples and equivalent conditions to characterize them. Furthermore, the relationships between these two notions are investigated.

The notion of a positive implicative equality algebras are defined, and related properties are studied. Characterizations of a positive implicative equality algebra is investigated. Conditions for an equality algebra to be positive implicative are provided. Equality algebra with some types is considered, and several properties are investigated. Using equality algebra with some types, we characterize a commutative equality algebra and a positive implicative algebra.

Lukasiewicz fuzzy set is applied to positive implicative filter of BE-algebra. The notion of positive implicative Lukasiewicz fuzzy BE-filters is introduced, and its properties are investigated. The relationship between fuzzy positive implicative BE-filter and positive implicative Lukasiewicz fuzzy BE-filter is discussed, and conditions under which Lukasiewicz fuzzy BE-filter can be positive implicative Lukasiewicz fuzzy BE-filter are explored. Characterizations of positive implicative Lukasiewicz fuzzy BE-filter are provided. Conditions for Lukasiewicz fuzzy set to be positive implicative Lukasiewicz fuzzy BE-filter are considered. Conditions are found where E-set, q-set, and O-set of the Lukasiewicz fuzzy set can be positive implicative BE-filter.

Molodtsov proposed soft set theory to address uncertainty in a parameter manner, and Atanassov introduced intuitionistic fuzzy set which is very useful in providing a flexible model to elaborate uncertainty and vagueness involved in decision making. Using intuitive fuzzy set and soft set, Jun et al. introduced a new notion, “ soju structure” , as a kind of hybrid structure. The concept of implicative soju ideal in BCK-algebra will be introduced and several properties will be investigated in this article. We will discuss the relationship between soju subalgebra, soju ideal, positive implicative soju ideal, commutative soju ideal and implicative soju ideal. Also, characterizations of implicative soju ideal will be established.

The notion of fuzzy points is used to study implicative filters in equality algebras. The concept of implicative (∈, ∈)-fuzzy  filters is introduced, and its properties are investigated. The relationship between (∈, ∈)-fuzzy filters, implicative (∈, ∈)- fuzzy filters and positive implicative (∈, ∈)-fuzzy filters are established. Characterizations of implicative (∈, ∈)-fuzzy filters  are discussed, and the conditions for the (positive implicative) (∈, ∈)-fuzzy filter to be an implicative (∈, ∈)-fuzzy filter are  explored. The extension property for the implicative (∈, ∈)- fuzzy filter is established. The properties associated with ∈tset  and qt-set of a fuzzy set Ψ in E and also look for conditions under which they can become implicative filters are  examined.

In this article, we introduce a new category of fuzzy BCK-filters, i.e., fuzzy implicative BCK-filters in bounded BCK-algebras and show that every fuzzy implicative BCK-filter is a fuzzy BCK-filter, while the opposite of this article in the case It is not established at all. We studied the properties of a fuzzy BCK-filter and introduced (x) (θ) (y µ) which is a µ fuzzy BCK-filter and we show that this is the smallest fuzzy BCK-filter containing µ. In the following, the relationship between We examine the fuzzy BCK-filters and the fuzzy BCK-algebras and show that in the torsional BCK-algebra X every fuzzy BCK-filter µ in X is a fuzzy BCK-filter if and only if X is a BCK-algebra.