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.

scholarly journals A p-Ideal in BCI-Algebras Based on Multipolar Intuitionistic Fuzzy Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 993
Author(s):  
Jeong-Gon Lee ◽  
Mohammad Fozouni ◽  
Kul Hur ◽  
Young Bae Jun
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Finite Degree ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Ideal ◽  
The Relationship

In 2020, Kang, Song and Jun introduced the notion of multipolar intuitionistic fuzzy set with finite degree, which is a generalization of intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. In this paper, we used this notion to study p-ideals of BCI-algebras. The notion of k-polar intuitionistic fuzzy p-ideals in BCI-algebras is introduced, and several properties were investigated. An example to illustrate the k-polar intuitionistic fuzzy p-ideal is given. The relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal is displayed. A k-polar intuitionistic fuzzy p-ideal is found to be k-polar intuitionistic fuzzy ideal, and an example to show that the converse is not true is provided. The notions of p-ideals and k-polar ( ∈ , ∈ ) -fuzzy p-ideal in BCI-algebras are used to study the characterization of k-polar intuitionistic p-ideal. The concept of normal k-polar intuitionistic fuzzy p-ideal is introduced, and its characterization is discussed. The process of eliciting normal k-polar intuitionistic fuzzy p-ideal using k-polar intuitionistic fuzzy p-ideal is provided.

scholarly journals Foldness of Bipolar Fuzzy Sets and Its Application in BCK/BCI-Algebras

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1036
Author(s):  
Young Bae Jun ◽  
Seok-Zun Song
Keyword(s):  
Information Processing ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Negative Information ◽  
Positive Information ◽  
Fuzzy Ideal ◽  
Recent Trends ◽  
Modern Information ◽  
Bipolar Fuzzy Sets

Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for example, already observed or experienced, indicates what is guaranteed to be possible, and negative information indicates that it is impossible, prohibited, or certainly false. The purpose of this paper is to apply the bipolar fuzzy set to BCK/BCI-algebras. The notion of (translated) k-fold bipolar fuzzy sets is introduced, and its application in BCK/BCI-algebras is discussed. The concepts of k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are introduced, and related properties are investigated. Characterizations of k-fold bipolar fuzzy subalgebra/ideal are considered, and relations between k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are displayed. Extension of k-fold bipolar fuzzy subalgebra is discussed.

Generalized Fuzzy α-Ideals of BCI-Algebras

2013 ◽  
Vol 321-324 ◽  
pp. 2794-2797
Author(s):  
Xiao Zhang ◽  
Shao Quan Sun
Keyword(s):  
Fuzzy Set ◽  
Fuzzy Ideal

In this Paper, the Concept of Generalized Fuzzy α-Ideal of BCI-Algebra is Introduced. the Following Results are Obtained: for a BCI-Algebra X, any Generalized Fuzzy α-Ideal of X must be a Generalized Fuzzy Ideal; a Fuzzy Set A of X is a Generalized Fuzzy α-Ideal of X if and only if for all t∈[λ,μ], At is either Empty or an α-Ideal of X; Suppose that A and B are Generalized Fuzzy α-Ideals of X, then so are A∩B and A×B. also, a Characterizations of Generalized Fuzzy α-Ideal is Given.

scholarly journals (Fuzzy) Ideals of BN-Algebras

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Grzegorz Dymek ◽  
Andrzej Walendziak
Keyword(s):  
Fuzzy Set ◽  
Fuzzy Ideal ◽  
One To One ◽  
Normal Ideals

The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained.

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 On Fuzzy Ideals ofBL-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Fuzzy Set ◽  
Representation Theorem ◽  
Prime Ideals ◽  
Converse Implication ◽  
Fuzzy Ideal ◽  
Stone Representation

In this paper we investigate further properties of fuzzy ideals of aBL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of aBL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime idealωis a fuzzy irreducible ideal if and only ifω0=1and|Im⁡(ω)|=2. We give the Krull-Stone representation theorem of fuzzy ideals inBL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of aBL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

scholarly journals Sup-hesitant fuzzy quasi-associative ideals of BCI-algebras

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4189-4197
Author(s):  
Habib Harizavi ◽  
Young Jun
Keyword(s):  
Extension Property ◽  
Fuzzy Ideal

The notion of Sup-hesitant fuzzy quasi-associative ideal in BCI-algebras is introduced, and related properties are investigated. Characterizations of Sup-hesitant fuzzy quasi-associative ideal are provided. Relations between Sup-hesitant fuzzy ideal and Sup-hesitant fuzzy quasi-associative ideal are displayed. Conditions for a Sup-hesitant fuzzy ideal to be a Sup-hesitant fuzzy quasi-associative ideal are provided. Extension property for Sup-hesitant fuzzy quasi-associative ideal is established.

Folding Theory for Fantastic Filters in BL-Algebras

2011 ◽  
Vol 2 (4) ◽  
pp. 32-42 ◽  
Author(s):  
Celestin Lele
Keyword(s):  
Level Set ◽  
Fuzzy Set ◽  
Extension Property ◽  
Fuzzy Filter ◽  
Fantastic Filter

In this paper, the author examines the notion of n-fold fantastic and fuzzy n-fold fantastic filters in BL-algebras. Several characterizations of fuzzy n-fold fantastic filters are given. The author shows that every n-fold (fuzzy n-fold) fantastic filter is a filter (fuzzy filter), but the converse is not true. Using a level set of a fuzzy set in a BL-algebra, the author gives a characterization of fuzzy n-fold fantastic filters. Finally, the author establishes the extension property for n-fold and fuzzy n-fold fantastic filters in BL-algebras. The author also constructs some algorithms for folding theory applied to fantastic filters in BL-algebras.

Folding Theory for Fantastic Filters in BL-Algebras

2013 ◽  
pp. 261-271
Author(s):  
Celestin Lele
Keyword(s):  
Level Set ◽  
Fuzzy Set ◽  
Extension Property ◽  
Fuzzy Filter ◽  
Fantastic Filter

In this paper, the author examines the notion of n-fold fantastic and fuzzy n-fold fantastic filters in BL-algebras. Several characterizations of fuzzy n-fold fantastic filters are given. The author shows that every n-fold (fuzzy n-fold) fantastic filter is a filter (fuzzy filter), but the converse is not true. Using a level set of a fuzzy set in a BL-algebra, the author gives a characterization of fuzzy n-fold fantastic filters. Finally, the author establishes the extension property for n-fold and fuzzy n-fold fantastic filters in BL-algebras. The author also constructs some algorithms for folding theory applied to fantastic filters in BL-algebras.

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