scholarly journals On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 998
Author(s):  
Alaa Altassan ◽  
Muhammad Haris Mateen ◽  
Dragan Pamucar
Keyword(s):  
Fuzzy Set ◽  
Significant Relationship ◽  
Quotient Ring ◽  
Surjective Homomorphism ◽  
Support Set ◽  
Fuzzy Ideal ◽  
Quotient Rings ◽  
Fundamental Theorems ◽  
The Given

In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.

scholarly journals A Certain Structure of Bipolar Fuzzy Subrings

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1397
Author(s):  
Hanan Alolaiyan ◽  
Muhammad Haris Mateen ◽  
Dragan Pamucar ◽  
Muhammad Khalid Mahmmod ◽  
Farrukh Arslan
Keyword(s):  
Fuzzy Set ◽  
Quotient Ring ◽  
Ring Homomorphism ◽  
Universal Relation ◽  
Support Set ◽  
Ring Isomorphism ◽  
Quotient Rings ◽  
Fundamental Theorems ◽  
Fuzzy Ring

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

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.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

Quotient Rings of a Class of Lattice-Ordered-Rings

1973 ◽  
Vol 25 (3) ◽  
pp. 627-645 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Identity Element ◽  
Quotient Ring ◽  
Ring Extension ◽  
Qf Ring ◽  
Left Quotient ◽  
Quotient Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

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.

scholarly journals Maximal quotient rings of prime group algebras. II Uniform right ideals

1977 ◽  
Vol 24 (3) ◽  
pp. 339-349 ◽  
Author(s):  
John Hannah
Keyword(s):  
Group Algebra ◽  
Quotient Ring ◽  
Group Algebras ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Residually Finite ◽  
Zero Characteristic ◽  
Quotient Rings

AbstractSuppose KG is a prime nonsingular group algebra with uniform right ideals. We show that G has no nontrivial locally finite normal subgroups. If G is soluble or residually finite, or if K has zero characteristic and G is linear, then the maximal right quotient ring of KG is simple Artinian.

A Fuzzy Adaptive K-SVD Dictionary Algorithm for Face Recogntion

2013 ◽  
Vol 347-350 ◽  
pp. 3797-3803 ◽  
Author(s):  
Xiao Ning Song ◽  
Zi Liu
Keyword(s):  
Iterative Method ◽  
Fuzzy Set ◽  
Sparse Coding ◽  
Sparse Representations ◽  
Experimental Results ◽  
Sparse Signal ◽  
Signal Representations ◽  
The Given ◽  
Class Assignment ◽  
Fuzzy Adaptive

Sparse representations using overcomplete dictionaries has concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting one from a prespecified set of linear transforms or adapting the dictionary to a set of training signals. The K-SVD algorithm is an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data. However, the existing K-SVD algorithm is employed to dwell on the concept of a binary class assignment meaning that the multi-classes samples are assigned to the given classes definitely. The work proposed in this paper provides a novel fuzzy adaptive way to adapting dictionaries in order to achieve the fuzzy sparse signal representations, the update of the dictionary columns is combined with an update of the sparse representations by incorporated a new mechanism of fuzzy set, which is called fuzzy K-SVD. Experimental results conducted on the ORL and Yale face databases demonstrate the effectiveness of the proposed method.

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