young bae jun

The fuzzy set is a fantastic tool for expressing hesitancy and dealing with uncertainty in real-world circumstances. Soft set theory has recently been developed to deal with practical problems. The soft and fuzzy sets were combined by Jun et al. to generate hybrid structures. The idea of hybrid ideals on a distributive lattice is discussed in this work. The relation between hybrid congruences and hybrid ideals on a distributive lattice is also examined. In addition, the product of hybrid ideals and its numerous results are discussed.

By applying the concept of the Lukasiewicz fuzzy set to the filter in hoops, the Lukasiewicz fuzzy filter is introduced and its properties are investigated.The relationship between fuzzy filter and Lukasiewicz fuzzy filter is discussed.Conditions for the Lukasiewicz fuzzy set to be a Lukasiewicz fuzzy filter are provided, andcharacterizations of Lukasiewicz fuzzy filter are displayed.The conditions under which the three subsets, $\in$-set, q-set, and O-set, will be filter are explored.

The notions of (transitive, commutative, antisymmetric) bordered GE-algebras are introduced,and their properties are investigated. Relations between a commutative bordered GE-algebra and anantisymmetric bordered GE-algebra are considered, and also relations between a commutative borderedGE-algebra and a transitive bordered GE-algebra are discussed. Relations between a bordered GE-algebra and a bounded Hilbert algebra are stated, and the conditions under which every bordered GE-algebra (resp., bounded Hilbert algebra) can be a bounded Hilbert algebra (resp., bordered GE-algebra) are found. The concept of duplex bordered GE-algebras is introduced, and its properties are investigated. Relations between an antisymmetric bordered GE-algebra and a duplex bordered GE-algebra are discussed, and the conditions under which an antisymmetric bordered GE-algebra can be a duplex GE-algebra are established. A characterization of a duplex bordered GE-algebra is provided. A new bordered GE-algebra called cross bordered GE-algebra which is wider than duplex bordered GE-algebra is introduced, and its properties are investigated. Relations between a duplex bordered GE-algebra and a cross bordered GE-algebra are considered.

In this paper, we introduce the notion of hybrid bi-ideals in semigroups and investigate some of their important properties. We also give various equivalent conditions for a semigroup to be regular and hybrid structures to be hybrid bi-ideals of S.

The notion of fuzzy ideal in bounded equality algebras is defined, and several properties are studied. Fuzzy ideal generated by a fuzzy set is established, and a fuzzy ideal is made by using the collection of ideals. Characterizations of fuzzy ideal are discussed. Conditions for a fuzzy ideal to attains its infimum on all ideals are provided. Homomorphic image and preimage of fuzzy ideal are considered. Finally, quotient structures of equality algebra induced by (fuzzy) ideal are studied.

The notion of a (limited) commutative $T\&F$-ideal in BCK-algebras and BCI-algebras is introduced, and their properties are investigated. A relationship between a $T\&F$-ideal and a commutative $T\&F$-ideal in BCK-algebras and BCI-algebras is established, and examples to show that any $T\&F$-ideal may not be commutative are given. Proper conditions for a $T\&F$-ideal to be commutative are provided. Using a commutative ideal of a BCK-algebra and a BCI-algebra, a commutative $T\&F$-ideal is established. The closed $T\&F$-ideal in a BCI-algebra is introduced, and a condition for a closed $T\&F$-ideal to be commutative is discussed. Characterization of a commutative $T\&F$-ideal in a BCI-algebra is considered.

In this paper, the properties of GE-ideals of transitive bordered GE-algebra are studied and characterizations of GE-ideals are given. We have observed that the set of all GE-ideals of a transitive bordered GE-algebra forms a complete lattice. The notion of bordered GE-morphism is introduced and established fundamental bordered GE-morphism theorem. A congruence relation on a bordered GE-algebra with respect to GE-ideal is introduced and some bordered GE-morphism theorems are derived.

‎The idea of  Lukasiewicz $t$-conorm is used to construct the concept of  Lukasiewicz anti fuzzy sets based on a given anti fuzzy set‎, ‎and it is applied to BE-algebras‎. ‎The notion of  Lukasiewicz anti fuzzy BE-ideal is introduced‎, ‎and its properties are investigated‎. ‎The conditions under which  Lukasiewicz anti fuzzy set will be  Lukasiewicz anti fuzzy BE-ideal are explored‎, ‎and the relationship between anti fuzzy BE-ideal and  Lukasiewicz anti fuzzy BE-ideal are discussed‎. ‎Three types of subsets so called $\lessdot$-subset‎, ‎$\Upsilon$-subset‎, ‎and anti subset are constructed‎, ‎and the conditions under which they can be BE-ideals are explored‎.

The aim of this article is to apply a $(3,2)$-fuzzy set to the UP-subalgebras and UP-filters of UP-algebras. The concepts of $(3,2)$-fuzzy UP-subalgebra, $(3,2)$ -fuzzy near UP-filter and $(3,2)$ -fuzzy UP-filter in UP-algebras are introduced and  several properties, including their relations, are investigated. The conditions under which the $(3,2)$-fuzzy UP-subalgebra $($resp., $(3,2)$ -fuzzy near UP-filter$)$ can be the $(3,2)$-fuzzy near UP-filter $($resp., $(3,2)$-fuzzy UP-filter$)$ are searched. The characterizations of $(3,2)$-fuzzy UP-filter is provided and the relationship between intuitionistic fuzzy UP-subalgebra and $(3,2)$-fuzzy UP-subalgebra is discussed. We use fuzzy UP-subalgebra $($resp., fuzzy near UP-filter, fuzzy UP-filter$)$ to create a $(3,2)$-fuzzy UP-subalgebra $($resp., $(3,2)$-fuzzy near UP-filter, $(3,2)$-fuzzy UP-filter$)$.

In BCK-algebra, the concept of  a positive implicative $T\&F$-ideal is introduced, and further several properties are  investigated. The relationship between  $T\&F$-ideals and  positive implicative $T\&F$-ideals is established, and an example is given to reveal that  a $T\&F$-ideal is not a positive implicative $T\&F$-ideal. Various conditions under which  a $T\&F$-ideal can be a positive implicative $T\&F$-ideal are explored and various characterizations of a positive  implicative $T\&F$-ideal are studied. The extended property of a positive implicative $T\&F$-ideal is constructed.

The notions of $xi$-inside GE-derivation and $xi$-outside GE-derivation on a GE-algebra are introduced and its properties are investigated. Conditions for a self-map on GE-algebra to be a $xi$-inside GE-derivation and a $xi$-outside GE-derivation are provided. The $xi$-inside GE-derivation or the $xi$-outside GE-derivation $varrho$ are used to form two sets $X_{(varrho = xi)}$ and ${rm ker}(varrho)$, and GE-subalgebra and GE-filter are studied for these two sets.

The concept of very true GE-algebra using very true operator is introduced and its properties are studied to expand the scope of research of GE-algebras. The concepts of simple very true GE-algebra and very true GE-filter are introduced. The characterization of simple very true GE-algebra is discussed, and several properties on very true GE-filter are investigated. Using a very true GE-filter, the quotient very true GE-algebra is constructed, and the uniform and topological space are established.

A new sub-structure called (vivid) deductive system is introduced and their properties are examined. Conditions for a subset to be a deductive system are provided. The notion of upper GE-set is also introduced, and an example to show that any upper GE-set may not be a deductive system are supplied. Conditions for an upper GE-set to be a deductive system are provided. An upper GE-set is used to consider conditions for a subset to be a deductive system. The characterization of deductive system is established, and relationship between deductive system and vivid deductive system are created. Conditions for a deductive system to be a vivid deductive system are given, and the extension property for vivid deductive system is constructed.

The notion of commutative MBJ-neutrosophic ideal is introduced, and several properties are investigated. Relations between MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal are considered. Characterizations of commutative MBJ-neutrosophic ideal are discussed.

The notions of a crossing cubic ideal in a BCK/BCI-algebra, a closed crossing cubic ideal in a BCI-algebra, and a crossing cubic $circ$-subalgebra of a BCK-algebra with the condition (S) are introduced, and several properties are investigated. The relationship between them is established. Conditions for a crossing cubic structure to be a closed crossing cubic ideal are provided. Conditions under which crossing cubic ideals are closed are explored. Characterizations of crossing cubic ideals are discussed. The translation of crossing cubic subalgebras and crossing cubic ideals are studied. Conditions for the translation of a crossing cubic structure to be a crossing cubic subalgebra (ideal) are provided, and its characterization is established.

The notion of a neutrosophic quadruple BCI-commutative ideal in a neutrosophic quadruple BCI-algebra is introduced, and  several properties are investigated. Relations between a neutrosophic quadruple ideal and a neutrosophic quadruple BCI-commutative ideal are discussed, and conditions for the neutrosophic quadruple ideal to be a neutrosophic quadruple BCI-commutative ideal are given.  Conditions for the neutrosophic quadruple set to be a neutrosophic quadruple BCI-commutative ideal are provided, and  the extension property of a neutrosophic quadruple BCI-commutative ideal is considered.

Two dimensional event set is introduced, and it is applied to algebraic structures. Two dimensional BCK/BCI-eventful algebra, paired B-algebra and paired BCK/BCI-algebra are defined, and several properties are investigated. Conditions for two dimensional eventful algebra to be a B-algebra and a BCK/BCI-algebra are provided. The process of inducing a paired B-algebra using a group is discussed. Using two dimensional BCI-eventful algebra, a commutative group is established.

Commutative neutrosophic quadruple ideals and BCKalgebras are discussed, and related properties are investigated. Conditions for the neutrosophic quadruple BCKalgebra to be commutative are considered. Given subsets A and B of a neutrosophic quadruple BCK-algebra, conditions for the set NQ (A; B) to be a commutative ideal of a neutrosophic quadruple BCK-algebra are provided.

In this paper, we apply m-polar fuzzy set to hyper BCKalgebra. We introduce the notions of k-polar fuzzy hyper BCK-ideal, k-polar fuzzy weak hyper BCK-ideal, k-polar fuzzy s-weak hyper BCK-ideal, k-polar fuzzy strong hyper BCK-ideal and k-polar fuzzy reflexive hyper BCKideal, and investigate related properties and their relations. We discuss k-polar fuzzy (weak, s-weak, strong, reflexive) hyper BCK-ideal in relation to k-polar level set

ýIn the classical group theory there isý an open questioný: ýIs every torsion free n-Engel group (for n ≥ 4)ý, nilpotent?ý. ýTo answer the questioný, ýTraustasoný [11] showed that with some additional conditions allý ý4-Engel groups are locally nilpotentý. ýHereý, ýwe gave some partialý answer to this question on Engel fuzzy subgroupsý. ýWe show that if μ is a normal 4-Engel fuzzyý subgroup of group Gý, ýx,y in G and a =yxý, ýthen μ| is a generalized nilpotent of class atý most 2ý. ýAlso we define a torsion free fuzzy subgroup and showý ýthat if μ is a 4-Engel torsion free fuzzy subgroup of Gý, ýthen μ| is a generalized nilpotent of class at most 4ý, ýfor conjugate elements a,y in Gý.

Coupled N-structures are introduced, and its application is discussed in BCK/BCI- algebras. The notions of a coupled N-subalgerba, a coupled N-ideal and a coupled NC- ideal are introduced, and their relations are investigated. Characterizations of a coupled N-ideal and a coupled NC-ideal are discussed. Conditions for a coupled N-subalgerba to be a coupled N-ideal are considered.

In this paper, we first introduce the notions of interval-valued fuzzy (p-,q- and a-) ideals of BCI algebras and investigate some of their related properties. Further, we introduce the notions of ()-interval-valued fuzzy (p-,q- and a-) ideals of BCI algebras and investigate some of their related properties. Some characterization theorems of these generalized intervalvalued fuzzy ideals are derived. The relationship among these generalized interval-valued fuzzy ideals of BCI algebras is also considered.