y. b. jun

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.

Using the fuzzy points, the concept of (∈, ∈)-fuzzy filters of equality algebras is introduced, and related properties are investigated. Characterizations of (∈, ∈)-fuzzy filters are explored, and the conditions under which ∈t-set, qt-set, and positive set can be filters are investigated.

The Jun’s YεJ -fuzzy set is applied to the Sheffer stroke BL-algebra, and the notions of an (∈, ∈)- YεJ -fuzzy quasisubalgebra and an (∈, ∈ ∨q)-YεJ -fuzzy quasi-subalgebra in Sheffer stroke BL-algebras are introduced, and several  properties are explored. Characterizations of (∈, ∈)- YεJ - fuzzy quasi-subalgebras and (∈, ∈ ∨q)- YεJ -fuzzy  quasisubalgebras are investigated, and the relationships between the fuzzy quasi-subalgebra, the (∈, ∈)- YεJ -fuzzy quasi- subalgebra and the (∈, ∈ ∨q)- YεJ -fuzzy quasisubalgebra are considered. Conditions for a fuzzy set to be an (∈, ∈∨q)- YεJ -fuzzy quasi-subalgebra are displayed.

The concepts of SR-fuzzy quasi-subalgebras and (ultra) SR-fuzzy filters over Sheffer stroke BL-algebras are introduced for the study of quasi-subalgebras and (ultra) filters using the SR-fuzzy set, and various properties are investigated accordingly. The relationship between SR-fuzzy quasi-subalgebras and SR-fuzzy filters is considered, and the characterizations of an (ultra) SR-fuzzy filter are discussed.

The concepts of Dokdo sub-hoops and Dokdo filters are introduced‎, ‎and their properties are investigated.‎‎ The relationship between Dokdo sub-hoops and Dokdo filters is discussed‎, ‎and‎‎ characterizations of Dokdo sub-hoops and Dokdo filters are established.‎

Lukasiewicz fuzzy set is applied to positive implicative filter of BE-algebra. The notion of positive implicative Lukasiewiczfuzzy 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.

The notion of (closed) \L{}ukasiewicz fuzzy ideal is introduced, and several properties are investigated.The relationship between \L{}ukasiewicz fuzzy subalgebra and \L{}ukasiewiczfuzzy ideal is discussed, and characterization of a \L{}ukasiewicz fuzzy ideal is considered.Conditions for a \L{}ukasiewicz fuzzy subalgebra to be a \L{}ukasiewicz fuzzy ideal are provided, andconditions for the $\in$-set, $q$-set and $O$-set to be ideals are explored.

After introducing the concept of single valued neutrosophic ideal in a pseudo MV-algebra, its properties are examined. The various conditions under which a single valued neutrosophic set can be a single valued neutrosophic ideal are examined. Characterizations of a single valued neutrosophic ideal of a pseudo MV-algebra are considered.

With the aim of applying the Łukasiewicz fuzzy set to commutative ideal in BCI-algebras, the concept of Łukasiewicz fuzzy commutative ideal is introduced, and its properties are investigated. The relationship between a Łukasiewicz fuzzy ideal and a Łukasiewicz fuzzy commutative ideal are discussed. After providing an example of a Łukasiewicz fuzzy ideal, not a Łukasiewicz fuzzy commutative ideal, conditions under which a Łukasiewicz fuzzy ideal can be a Łukasiewicz fuzzy commutative ideal are explored. Characterizations of Łukasiewicz fuzzy commutative ideals are displayed. Conditions under which $\in$-set, q-set, and O-set can be commutative ideals are found.

A Dokdo structure is used to study implicative ideals in BCK-algebras. The notion of Dokdo implicative ideals in BCK-algebras is introduced, and the relevant properties are investigated. The relationship between Dokdo subalgebras, Dokdo ideals, and Dokdo implicative ideals are discussed, and conditions that allow a Dokdo ideal and a Dokdo subalgebra to be a Dokdo implicative ideal are provided.

The notion of commutative falling intuitionistic fuzzy ideal of a BCK-algebra is introduced and related properties are investigated. We verify that every commutative intuitionistic fuzzy ideal is a commutative falling intuitionistic fuzzy ideal, and provide example to show that a commutative falling intuitionistic fuzzy ideal is not a commutative intuitionistic fuzzy ideal. Relations between a falling intuitionistic fuzzy ideal and a commutative falling intuitionistic fuzzy ideal are considered, and a condition for a falling intuitionistic fuzzy ideal to be a commutative falling intuitionistic fuzzy ideal is provided.

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

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.

In this paper, first we define the concept of nilpotent element on a hoop H, study some properties of them and investigate the relation with ultra deductive systems. Then by using this notion, we introduce cyclic hoops and prove that every cyclic hoop has a unique generator and is a local MV-algebra. In the follows, we introduce the notion of Boolean elements on hoops and investigate some of their properties and relation among Boolean elements with ultra deductive systems and nilpotent elements. Finally, we introduce a functor between the category of hoops and category of Boolean elements of them.

Further properties on (belligerent) GE-filters are discussed, and the quotient GEalgebra via a GE-filter is established. Conditions for the set →c to be a belligerent GE-filterare provided. The extension property of belligerent GE-filter is composed. The notions of abalanced element, a balanced GE-filter, an antisymmetric GE-algebra and a voluntary GE-filterare introduced, and its properties are examined. The relationship between a GE-subalgebraand a GE-filter is established. Conditions for every element in a GE-algebra to be a balancedelement are provided. The conditions necessary for a GE-filter to be a voluntary GE-filter areconsidered. The GE-filter generated by a given subset is established, and its shape is identified

In this paper, we defined the notions of (∈, ∈)-fuzzy implicative deductive systems and (∈, ∈ ∨ q)-fuzzy implicative deductive systems of hoops and studied some traits and tried to define some definitions that are equivalent to them. Thus by using the notion of (∈, ∈)- fuzzy deductive system of hoop, we defined a new congruence relation on hoop and show that the algebraic structure that is made by it is a Brouwerian semilattice, Heyting algebra and Wajesberg hoop.

In the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37], Ghosh and Samanta introduced the concept of hyperfuzzy sets as a generalization of fuzzy sets and interval-valued fuzzy sets, and applied it to group theory. The aim of this manuscript is to study N -structures in BCK/BCI-algebras induced from hyperfuzzy structures.

Based on subalgebras and filters in hoops, the notions of S-quasi-valuation maps and F-quasi-valuation maps are introduced, and related properties are investigated. Relations between S-quasi-valuation maps and F-quasi-valuation maps are introduced. Using F-quasi-valuation map, a (pseudo) metric space is indued, and we show that the operations “⊙”, “→” and “∧” in a hoop are uniformly continuous.