Regular (ϵ,ϵνqk) - Fuzzy Duo Ordered Semigroups

2013 ◽  
Vol 756-759 ◽  
pp. 3084-3088
Author(s):  
Qi Cheng ◽  
Feng Lian Yuan ◽  
Yun Qiang Yin ◽  
Qing Yan Chen
Keyword(s):  
Fuzzy Set ◽  
Ordered Semigroup ◽  
Ordered Semigroups ◽  
Fuzzy Point ◽  
Fuzzy Ideal ◽  
Characterization Theorems ◽  
The Ideal

In this paper, the ideal of quasi-coincidence of a fuzzy point with a fuzzy set is generalized and the concept of an - fuzzy ideal (bi-ideal, quasi-ideal) of an ordered semigroup is introduced. The the notion of - fuzzy duo ordered semigroups is introduced and some characterization theorems are presented in terms of - fuzzy ideals.

Characterizations of Fuzzy Sublattices Based on Fuzzy Point

2020 ◽  
pp. 105-127
Author(s):  
Chiranjibe Jana ◽  
Faruk Karaaslan
Keyword(s):  
Fuzzy Set ◽  
Cartesian Product ◽  
Inverse Image ◽  
Fuzzy Point ◽  
Fuzzy Ideal ◽  
The Relationship

In a lattice 𝔏, the authors used the concept of belongingness and quasi-coincidence of fuzzy point to a fuzzy set, and by this notion, (∈,∈∨q)-fuzzy sublattice, (∈,∈∨q)-fuzzy ideal, cartesian product of (∈,∈∨q)-fuzzy sublattice, (∈,∈∨q)-fuzzy complemented sublattice, and cartesian product of (∈,∈∨q)-fuzzy complemented sublattice are introduced, and their properties are briefly studied. The relationship between fuzzy sublattice and (∈,∈∨q)-fuzzy sublattice, fuzzy ideal and (∈,∈∨q)-fuzzy ideal of L are established. The authors prove that the cartesian product of two (∈,∈∨q)-fuzzy ideals of a lattice is not necessarily a fuzzy ideal of a lattice. The theory of image and inverse image of an (∈,∈∨q)-fuzzy sublattice and (∈,∈∨q)-fuzzy ideal, an (∈,∈∨q)-fuzzy complemented sublattice, and (∈,∈∨q)-fuzzy complemented ideal of 𝔏 on the basis of homomorphism of lattices are also significantly established.

scholarly journals Ordered Semigroups Based on ∈,∈∨qkδ-Fuzzy Ideals

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Faiz Muhammad Khan ◽  
Nie Yufeng ◽  
Hidayat Ullah Khan ◽  
Bakht Muhammad Khan
Keyword(s):  
Applied Sciences ◽  
Characteristic Function ◽  
Ideal Theory ◽  
Ordered Semigroup ◽  
Algebraic Structures ◽  
Ordered Semigroups ◽  
Central Focus ◽  
Fuzzy Ideal ◽  
New Type

A new trend of using fuzzy algebraic structures in various applied sciences is becoming a central focus due to the accuracy and nondecoding nature. The aim of the present paper is to develop a new type of fuzzy subsystem of an ordered semigroup S. This new type of fuzzy subsystem will overcome the difficulties faced in fuzzy ideal theory of an ordered semigroup up to some extent. More precisely, we introduce ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals of S. These concepts are elaborated through appropriate examples. Further, we are bridging ordinary ideals and ∈,∈∨qkδ-fuzzy ideals of an ordered semigroup S through level subset and characteristic function. Finally, we characterize regular ordered semigroups in terms of ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals.

scholarly journals Characterizations of Ordered Semigroups by New Type of Interval Valued Fuzzy Quasi-Ideals

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Jian Tang ◽  
Xiangyun Xie ◽  
Yanfeng Luo
Keyword(s):  
Fuzzy Set ◽  
Ordered Semigroups ◽  
Fuzzy Point ◽  
New Type ◽  
Interval Valued

The concept of non-k-quasi-coincidence of an interval valued ordered fuzzy point with an interval valued fuzzy set is considered. In fact, this concept is a generalized concept of the non-k-quasi-coincidence of a fuzzy point with a fuzzy set. By using this new concept, we introduce the notion of interval valued(∈¯,∈¯∨qk~¯)-fuzzy quasi-ideals of ordered semigroups and study their related properties. In addition, we also introduce the concepts of prime and completely semiprime interval valued(∈¯,∈¯∨qk~¯)-fuzzy quasi-ideals of ordered semigroups and characterize bi-regular ordered semigroups in terms of completely semiprime interval valued(∈¯,∈¯∨qk~¯)-fuzzy quasi-ideals. Furthermore, some new characterizations of regular and intra-regular ordered semigroups by the properties of interval valued(∈¯,∈¯∨qk~¯)-fuzzy quasi-ideals are given.

BI-IDEALS OF ORDERED SEMIGROUPS BASED ON THE INTERVAL-VALUED FUZZY POINT

2016 ◽  
Vol 78 (2) ◽  
Author(s):  
Hidayat Ullah Khan ◽  
Nor Haniza Sarmin ◽  
Asghar Khan ◽  
Faiz Muhammad Khan
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Real World ◽  
Fuzzy Set Theory ◽  
Fuzzy Subset ◽  
Ordered Semigroups ◽  
Fuzzy Point ◽  
Interval Valued ◽  
Real World Problems

Interval-valued fuzzy set theory (advanced generalization of Zadeh's fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with () relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued -fuzzy bi-ideals are described. It is shown that an interval-valued -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi-ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued -fuzzy bi-ideal are considered. 

Uni-Hesitant Fuzzy Set Approach to the Ideal Theory of BCK-Algebras

2020 ◽  
pp. 128-139
Author(s):  
G. Muhiuddin
Keyword(s):  
Level Set ◽  
Fuzzy Set ◽  
Theoretical Approach ◽  
Ideal Theory ◽  
Hesitant Fuzzy Set ◽  
Commutative Ideal ◽  
Fuzzy Ideal ◽  
The Ideal

In this chapter, the author studies the uni-hesitant fuzzy set-theoretical approach to the ideals of BCK-algebras. For a hesitant fuzzy set H on S and a subset of [0,1], the set L(H;ʎ):={x∈S|xH⊆ʎ}, is called the uni-hesitant level set of H. Moreover, the author discusses the relations between uni-hesitant fuzzy commutative ideals and uni-hesitant fuzzy ideals. Further, he considered the characterizations of uni-hesitant fuzzy commutative ideals in BCK-algebras. Finally, he proved some conditions for a uni-hesitant fuzzy ideal to be a uni-hesitant fuzzy commutative ideal.

scholarly journals New fuzzy generalized bi Γ-ideals of the type (∈,∈∨qk) in ordered Γ-semigroups

2017 ◽  
Vol 13 (4) ◽  
pp. 6666-670
Author(s):  
Ibrahim Gambo ◽  
Nor Haniza Sarmin ◽  
Hidayat Ullah Khan ◽  
Faiz Muhammad Khan
Keyword(s):  
Characteristic Function ◽  
Real Number ◽  
Fuzzy Set ◽  
Unit Interval ◽  
Membership Functions ◽  
Fuzzy Subset ◽  
Ordered Semigroups ◽  
The Past ◽  
Fuzzy Point

A fuzzy subset A defined on a set X is represented as A = {(x, A (x), where x ∈ X}. It is not always possible for membership functions of type λA : X → [0,1] to associate with each point x in a set X a real number in the closed unit interval [0,1] without the loss of some useful information. The importance of the ideas of “belongs to” (∈) and “quasi coincident with” (q) relations between a fuzzy point and fuzzy set is evident from the research conducted during the past two decades. Ordered Γ-semigroup (generalization of ordered semigroups) play an important role in the broad study of ordered semigroups. In this paper, we provide an extension of fuzzy generalized bi Γ-ideals and introduce (∈,∈∨qk)-fuzzy generalized bi Γ-ideals of ordered Γ-semigroup. The purpose of this paper is to link this new concept with ordinary generalized bi Γ-ideals by using level subset and characteristic function.

Classification of Ordered Semigroups in Terms of Generalized Interval-Valued Fuzzy Interior Ideals

2016 ◽  
Vol 25 (2) ◽  
pp. 297-318
Author(s):  
Hidayat Ullah Khan ◽  
Nor Haniza Sarmin ◽  
Asghar Khan ◽  
Faiz Muhammad Khan
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Decision Making Process ◽  
Ordered Semigroup ◽  
New Approach ◽  
Ordered Semigroups ◽  
Applied Fields ◽  
Interval Valued

AbstractSeveral applied fields dealing with decision-making process may not be successfully modeled by ordinary fuzzy sets. In such a situation, the interval-valued fuzzy set theory is more applicable than the fuzzy set theory. Using a new approach of “quasi-coincident with relation”, which is a central focused idea for several researchers, we introduced the more general form of the notion of (α,β)-fuzzy interior ideal. This new concept is called interval-valued$( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$-fuzzy interior ideal of ordered semigroup. As an attempt to investigate the relationships between ordered semigroups and fuzzy ordered semigroups, it is proved that in regular ordered semigroups, the interval-valued$( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$-fuzzy ideals and interval-valued$( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$-fuzzy interior ideals coincide. It is also shown that the intersection of non-empty class of interval-valued$( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$-fuzzy interior ideals of an ordered semigroup is also an interval-valued$( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$-fuzzy interior ideal.

A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

2021 ◽  
Vol 10 (3) ◽  
pp. 1-17
Author(s):  
Debabrata Mandal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Cartesian Product ◽  
Intuitionistic Fuzzy Set ◽  
Ideal Theory ◽  
Hesitant Fuzzy Set ◽  
Neutrosophic Set ◽  
Intuitionistic Fuzzy ◽  
Interval Valued ◽  
The Ideal

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.

scholarly journals Bipolar Fuzzy Implicative Ideals of BCK-Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
G. Muhiuddin ◽  
D. Al-Kadi
Keyword(s):  
Fuzzy Set ◽  
Extension Property ◽  
Fuzzy Ideal

The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.

Fuzzy roughness via ideals

2020 ◽  
Vol 39 (5) ◽  
pp. 6869-6880
Author(s):  
S. H. Alsulami ◽  
Ismail Ibedou ◽  
S. E. Abbas
Keyword(s):  
Fuzzy Set ◽  
Closure Operator ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Fuzzy Connectedness ◽  
Fuzzy Approximation ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Interior Operator

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

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