scholarly journals Certain Notions of Neutrosophic Topological K-Algebras

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 234 ◽  
Author(s):  
Muhammad Akram ◽  
Hina Gulzar ◽  
Florentin Smarandache ◽  
Said Broumi
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Research Article

The concept of neutrosophic set from philosophical point of view was first considered by Smarandache. A single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. In this research article, we apply the notion of single-valued neutrosophic sets to K-algebras. We introduce the notion of single-valued neutrosophic topological K-algebras and investigate some of their properties. Further, we study certain properties, including C 5 -connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

Neutrosophic Sets and Logic

2017 ◽  
pp. 18-63
Author(s):  
Mumtaz Ali ◽  
Florentin Smarandache ◽  
Luige Vladareanu
Keyword(s):  
Fuzzy Sets ◽  
Approximation Theory ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Hybrid Structures ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Basic Concepts ◽  
Interval Valued

Neutrosophic sets and Logic plays a significant role in approximation theory. It is a generalization of fuzzy sets and intuitionistic fuzzy set. Neutrosophic set is based on the neutrosophic philosophy in which every idea Z, has opposite denoted as anti(Z) and its neutral which is denoted as neut(Z). This is the main feature of neutrosophic sets and logic. This chapter is about the basic concepts of neutrosophic sets as well as some of their hybrid structures. This chapter starts with the introduction of fuzzy sets and intuitionistic fuzzy sets respectively. The notions of neutrosophic set are defined and studied their basic properties in this chapter. Then we studied neutrosophic crisp sets and their associated properties and notions. Moreover, interval valued neutrosophic sets are studied with some of their properties. Finally, we presented some applications of neutrosophic sets in the real world problems.

scholarly journals On the Neutrosophic, Pythagorean and Some Other Novel Fuzzy Sets Theories Used in Decision Making: Invitation to Discuss

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1485
Author(s):  
Pavel Sevastjanov ◽  
Ludmila Dymova ◽  
Krzysztof Kaczmarek
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Short Paper ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Dempster Shafer Theory ◽  
Methodological Problems ◽  
Shafer Theory ◽  
Sets Theory

In this short paper, a critical analysis of the Neutrosophic, Pythagorean and some other novel fuzzy sets theories foundations is provided, taking into account that they actively used for the solution of the decision-making problems. The shortcomings of these theories are exposed. It is stated that the independence hypothesis, which is a cornerstone of the Neutrosophic sets theory, is not in line with common sense and therefore leads to the paradoxical results in the asymptotic limits of this theory. It is shown that the Pythagorean sets theory possesses questionable foundations, the sense of which cannot be explained reasonably. Moreover, this theory does not completely solve the declared problem. Similarly, important methodological problems of other analyzed theories are revealed. To solve the interior problems of the Atanassov’s intuitionistic fuzzy sets and to improve upon them, this being the reason most of the criticized novel sets theories were developed, an alternative approach based on extension of the intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory is proposed. No propositions concerned with the improvement of the Cubic sets theory and Single-Valued Neutrosophic Offset theory were made, as their applicability was shown to be very dubious. In order to stimulate discussion, many statements are deliberately formulated in a hardline form.

Rough Sets and Approximate Reasoning

2014 ◽  
pp. 180-228 ◽  
Author(s):  
B. K. Tripathy
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
Approximate Reasoning ◽  
Human Reasoning ◽  
Intuitionistic Fuzzy ◽  
Application Point ◽  
Future Work

Several models have been introduced to capture impreciseness in data. Fuzzy sets introduced by Zadeh and Rough sets introduced by Pawlak are two of the most popular such models. In addition, the notion of intuitionistic fuzzy sets introduced by Atanassov and the hybrid models obtained thereof have been very fruitful from the application point of view. The introduction of fuzzy logic and the approximate reasoning obtained through it are more realistic as they are closer to human reasoning. Equality of sets in crisp mathematics is too restricted from the application point of view. Therefore, extending these concepts, three types of approximate equalities were introduced by Novotny and Pawlak using rough sets. These notions were found to be restrictive in the sense that they again boil down to equality of sets and also the lower approximate equality is artificial. Keeping these points in view, three other types of approximate equalities were introduced by Tripathy in several papers. These approximate equalities were further generalised to cover the approximate equalities of fuzzy sets and intuitionistic fuzzy sets by him. In addition, considering the generalisations of basic rough sets like the covering-based rough sets and multigranular rough sets, the study has been carried out further. In this chapter, the authors provide a comprehensive study of all these forms of approximate equalities and illustrate their applicability through several examples. In addition, they provide some problems for future work.

scholarly journals Double-Valued Neutrosophic Sets, their Minimum Spanning Trees, and Clustering Algorithm

2018 ◽  
Vol 27 (2) ◽  
pp. 163-182 ◽  
Author(s):  
Ilanthenral Kandasamy
Keyword(s):  
Fuzzy Sets ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Distance Measure ◽  
Clustering Algorithms ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Inconsistent Information ◽  
Intuitionistic Fuzzy

AbstractNeutrosophy (neutrosophic logic) is used to represent uncertain, indeterminate, and inconsistent information available in the real world. This article proposes a method to provide more sensitivity and precision to indeterminacy, by classifying the indeterminate concept/value into two based on membership: one as indeterminacy leaning towards truth membership and the other as indeterminacy leaning towards false membership. This paper introduces a modified form of a neutrosophic set, called Double-Valued Neutrosophic Set (DVNS), which has these two distinct indeterminate values. Its related properties and axioms are defined and illustrated in this paper. An important role is played by clustering in several fields of research in the form of data mining, pattern recognition, and machine learning. DVNS is better equipped at dealing with indeterminate and inconsistent information, with more accuracy, than the Single-Valued Neutrosophic Set, which fuzzy sets and intuitionistic fuzzy sets are incapable of. A generalised distance measure between DVNSs and the related distance matrix is defined, based on which a clustering algorithm is constructed. This article proposes a Double-Valued Neutrosophic Minimum Spanning Tree (DVN-MST) clustering algorithm, to cluster the data represented by double-valued neutrosophic information. Illustrative examples are given to demonstrate the applications and effectiveness of this clustering algorithm. A comparative study of the DVN-MST clustering algorithm with other clustering algorithms like Single-Valued Neutrosophic Minimum Spanning Tree, Intuitionistic Fuzzy Minimum Spanning Tree, and Fuzzy Minimum Spanning Tree is carried out.

An Innovative Method to Unravel Neutroshopic Transportation Problem Using Harmonic Mean

2021 ◽  
Vol 10 (3) ◽  
pp. 55-66
Author(s):  
S. Krishna Prabha
Keyword(s):  
Fuzzy Sets ◽  
Transportation Problem ◽  
Intuitionistic Fuzzy Sets ◽  
Harmonic Mean ◽  
Neutrosophic Sets ◽  
Transportation Problems ◽  
Numerical Example ◽  
Intuitionistic Fuzzy ◽  
Innovative Method ◽  
Optimal Resolution

As a simplification of fuzzy sets and intuitionistic fuzzy sets to symbolize hesitant, conflicting, and curtailed information about factual world tribulations, neutrosophic sets have been established. There are many existing techniques accessible to solve transportation problems in neutrosophic environment. Among those existing routines, the harmonic mean scheme is applied to obtain the optimal resolution to neutrosophic transportation problem. A numerical example is publicized that the proposed technique gives an improved estimate when compared with the existing techniques.

scholarly journals Rough Atanassov’s Intuitionistic Fuzzy Sets Model over Two Universes and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Shuqun Luo ◽  
Weihua Xu
Keyword(s):  
Fuzzy Sets ◽  
Rough Set ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
The Novel ◽  
Intuitionistic Fuzzy ◽  
Two Universes ◽  
Novel Model ◽  
Atanassov’S Intuitionistic Fuzzy Sets

Recently, much attention has been given to the rough set models based on two universes. And many rough set models based on two universes have been developed from different points of view. In this paper, a novel model, that is, rough Atanassov’s intuitionistic fuzzy sets model over two different universes, is firstly proposed from Atanassov’s intuitionistic point of view. We study some important properties of approximation operators and investigate the rough degree in the novel model. Furthermore, an illustrated example is employed to demonstrate the conceptual arguments of the model. Finally, rough Atanassov’s intuitionistic fuzzy sets approach to decision is presented in the generalized approximation space over two universes by considering the problem about how to arrange patients to see the doctor reasonably, from which it can be found that the method is valuable and useful in real life.

Refined Neutrosophic Sets and Refined Neutrosophic Soft Sets

2016 ◽  
pp. 321-343 ◽  
Author(s):  
Irfan Deli
Keyword(s):  
Fuzzy Sets ◽  
Medical Diagnosis ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Soft Sets ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Formal Framework ◽  
Fuzzy Soft Sets ◽  
Intuitionistic Fuzzy Soft Sets

Refined neutrosophic sets (RNS) are a generalization of a neutrosophic sets, intuitionistic fuzzy sets, fuzzy sets, intuitionistic fuzzy multi-sets and fuzzy multi-sets. Similarly, refined neutrosophic soft sets (RNSS) are a generalization of a neutrosophic soft sets, intuitionistic fuzzy soft sets, fuzzy soft sets, intuitionistic fuzzy soft multi-sets and fuzzy soft multi-sets. These sets are a powerful general formal framework that has been proposed to present uncertainty, imprecise, incomplete, inaccurate and inconsistent information which exist in real life. This chapter will survey concept of RNS and concept of RNSS with basic definitions and will present an efficient approach for both RNS and RNSS. Also, the chapter will introduce an application of RNS in medical diagnosis problem, pattern recognition and an application of RNSS in decision making to illustrate the advantage of the proposed approach.

scholarly journals On Neutrosophic Offuninorms

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1136 ◽  
Author(s):  
Erick González Caballero ◽  
Florentin Smarandache ◽  
Maikel Leyva Vázquez
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Theory ◽  
Intuitionistic Fuzzy Sets ◽  
Membership Functions ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Wide Range ◽  
The Relationship ◽  
Interval Valued ◽  
Important Kind

Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been generalized to others—e.g., intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, or neutrosophic sets—then uninorm generalizations have emerged in those novel frameworks. Neutrosophic sets contain the notion of indeterminacy—which is caused by unknown, contradictory, and paradoxical information—and thus, it includes, aside from the membership and non-membership functions, an indeterminate-membership function. Also, the relationship among them does not satisfy any restriction. Along this line of generalizations, this paper aims to extend uninorms to the framework of neutrosophic offsets, which are called neutrosophic offuninorms. Offsets are neutrosophic sets such that their domains exceed the scope of the interval [0,1]. In the present paper, the definition, properties, and application areas of this new concept are provided. It is necessary to emphasize that the neutrosophic offuninorms are feasible for application in several fields, as we illustrate in this paper.

Reverse triple I method based on single valued neutrosophic fuzzy inference

2020 ◽  
Vol 39 (5) ◽  
pp. 7071-7083
Author(s):  
Ruirui Zhao ◽  
Minxia Luo ◽  
Shenggang Li
Keyword(s):  
Fuzzy Sets ◽  
Similarity Measure ◽  
Fuzzy Inference ◽  
Intuitionistic Fuzzy Sets ◽  
Neutrosophic Sets ◽  
Inconsistent Information ◽  
Intuitionistic Fuzzy ◽  
Triple I Method

The theory of single valued neutrosophic sets, which is a generalization of intuitionistic fuzzy sets, is more capable of dealing with inconsistent information in practice. In this paper, we propose reverse triple I method under single valued neutrosophic environment. Firstly, we give the definitions of single valued neutrosophic t-representation t-norms and single valued neutrosophic residual implications. Secondly, we develop a formula for calculating single valued neutrosophic residual implications. Then we propose reverse triple I method based on left-continuous single valued neutrosophic t-representation t-norms and its solutions. Lastly, we discuss the robustness of reverse triple I method based on the proposed similarity measure.

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