Special Issue "Fuzzy Mathematics"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 June 2018)

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Guest Editor
Prof. Dr. Etienne E. Kerre

Department of Applied Mathematics & Computer Science, Ghent university, Krijgslaan 281-S9, B-9000 Ghent, Belgium
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Interests: fuzzy set theory; fuzzy topology; fuzzy relational calculus and its application to information retrieval, medical diagnosis and databases; fuzzy numbers; extensions and alternatives of fuzzy set theory
Guest Editor
Prof. Dr. John Mordeson

Center for Mathematics of Uncertainty, Department of Mathematics, Creighton University, Omaha, NE 68178, USA
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Interests: fuzzy graph theory; fuzzy algebraic structures; fuzzy soft sets; applications of fuzzy mathematics to human trafficking, illegal immigration and social choice

Special Issue Information

Dear Colleagues,

Exactly today, 7 September, 2017, the founder of fuzzy set theory, Prof. Lotfi Zadeh, passed away at the age of 97. We would like to dedicate this special issue to our scientific father. In his 1965 seminal paper, entitled “Fuzzy sets” Zadeh extended Cantor’s binary set theory to a gradual model by introducing degrees of belonging and relationship. Very soon, this extension has been applied to almost all domains of contemporary mathematics giving birth to new disciplines, such as fuzzy topology, fuzzy arithmetic, fuzzy algebraic structures, fuzzy differential calculus, fuzzy geometry, fuzzy relational calculus, fuzzy databases and fuzzy decision making. In the beginning, mostly direct fuzzyfications of the classical mathematical domains have been launched by simply changing Cantor’s set-theoretic operations by Zadeh’s max-min extensions. The 1980s were characterized by an explosion of the possible fuzzyfications due to the discovery of triangular norms and co-norms. Starting from the nineties more profound analysis has been performed by studying the axiomatization of fuzzy structures and searching for links between the different models to represent imprecise and uncertain information. It is our aim to have in this Special Issue a healthy mix of excellent state-of-the-art papers, as well as brand-new material that can serve as a starting point for newcomers in the field to further develop this wonderful domain of fuzzy mathematics.

Prof. Dr. Etienne E. Kerre
Prof. Dr. John Mordeson
Guest Editors

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Keywords

  • fuzzy set theory
  • fuzzy algebraic structures
  • fuzzy metric spaces
  • fuzzy topological spaces
  • fuzzy relational calculus and applications
  • fuzzy geometry
  • fuzzy decision making
  • fuzzy projective spaces
  • fuzzy number theory
  • fuzzy analysis

Published Papers (16 papers)

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Research

Open AccessFeature PaperArticle L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New?
Mathematics 2018, 6(9), 146; https://doi.org/10.3390/math6090146
Received: 23 June 2018 / Revised: 19 August 2018 / Accepted: 20 August 2018 / Published: 23 August 2018
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Abstract
We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and [...] Read more.
We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy set. Finally we also discuss some questionable notations, in particular, those of “intuitionistic” and “Pythagorean” fuzzy sets. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
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Open AccessArticle The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models
Mathematics 2018, 6(8), 133; https://doi.org/10.3390/math6080133
Received: 30 May 2018 / Revised: 12 July 2018 / Accepted: 24 July 2018 / Published: 2 August 2018
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Abstract
In this paper, several portfolio choice models are studied: a purely possibilistic model in which the return of the risky is a fuzzy number, and four models in which the background risk appears in addition to the investment risk. In these four models, [...] Read more.
In this paper, several portfolio choice models are studied: a purely possibilistic model in which the return of the risky is a fuzzy number, and four models in which the background risk appears in addition to the investment risk. In these four models, risk is a bidimensional vector whose components are random variables or fuzzy numbers. Approximate formulas of the optimal allocation are obtained for all models, expressed in terms of some probabilistic or possibilistic moments, depending on the indicators of the investor preferences (risk aversion, prudence). Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets
Mathematics 2018, 6(7), 123; https://doi.org/10.3390/math6070123
Received: 30 May 2018 / Revised: 4 July 2018 / Accepted: 4 July 2018 / Published: 13 July 2018
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Abstract
The definition of the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets is given, and some of its basic properties are studied. Full article
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Open AccessArticle On Generalized Roughness in LA-Semigroups
Mathematics 2018, 6(7), 112; https://doi.org/10.3390/math6070112
Received: 30 May 2018 / Revised: 16 June 2018 / Accepted: 25 June 2018 / Published: 27 June 2018
Cited by 1 | PDF Full-text (235 KB) | HTML Full-text | XML Full-text
Abstract
The generalized roughness in LA-semigroups is introduced, and several properties of lower and upper approximations are discussed. We provide examples to show that the lower approximation of a subset of an LA-semigroup may not be an LA-subsemigroup/ideal of LA-semigroup under a set valued [...] Read more.
The generalized roughness in LA-semigroups is introduced, and several properties of lower and upper approximations are discussed. We provide examples to show that the lower approximation of a subset of an LA-semigroup may not be an LA-subsemigroup/ideal of LA-semigroup under a set valued homomorphism. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle The Emergence of Fuzzy Sets in the Decade of the Perceptron—Lotfi A. Zadeh’s and Frank Rosenblatt’s Research Work on Pattern Classification
Mathematics 2018, 6(7), 110; https://doi.org/10.3390/math6070110
Received: 25 May 2018 / Revised: 16 June 2018 / Accepted: 19 June 2018 / Published: 26 June 2018
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Abstract
In the 1950s, the mathematically oriented electrical engineer, Lotfi A. Zadeh, investigated system theory, and in the mid-1960s, he established the theory of Fuzzy sets and systems based on the mathematical theorem of linear separability and the pattern classification problem. Contemporaneously, the psychologist, [...] Read more.
In the 1950s, the mathematically oriented electrical engineer, Lotfi A. Zadeh, investigated system theory, and in the mid-1960s, he established the theory of Fuzzy sets and systems based on the mathematical theorem of linear separability and the pattern classification problem. Contemporaneously, the psychologist, Frank Rosenblatt, developed the theory of the perceptron as a pattern recognition machine based on the starting research in so-called artificial intelligence, and especially in research on artificial neural networks, until the book of Marvin L. Minsky and Seymour Papert disrupted this research program. In the 1980s, the Parallel Distributed Processing research group requickened the artificial neural network technology. In this paper, we present the interwoven historical developments of the two mathematical theories which opened up into fuzzy pattern classification and fuzzy clustering. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
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Open AccessArticle Fuzzy Semi-Metric Spaces
Mathematics 2018, 6(7), 106; https://doi.org/10.3390/math6070106
Received: 27 May 2018 / Revised: 15 June 2018 / Accepted: 19 June 2018 / Published: 22 June 2018
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Abstract
The T1-spaces induced by the fuzzy semi-metric spaces endowed with the special kind of triangle inequality are investigated in this paper. The limits in fuzzy semi-metric spaces are also studied to demonstrate the consistency of limit concepts in the induced topologies. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making
Mathematics 2018, 6(6), 92; https://doi.org/10.3390/math6060092
Received: 16 May 2018 / Revised: 26 May 2018 / Accepted: 28 May 2018 / Published: 30 May 2018
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Abstract
The objective of this manuscript is to present a novel information measure for measuring the degree of fuzziness in intuitionistic fuzzy sets (IFSs). To achieve it, we define an (R,S) -norm-based information measure called the entropy to measure the [...] Read more.
The objective of this manuscript is to present a novel information measure for measuring the degree of fuzziness in intuitionistic fuzzy sets (IFSs). To achieve it, we define an ( R , S ) -norm-based information measure called the entropy to measure the degree of fuzziness of the set. Then, we prove that the proposed entropy measure is a valid measure and satisfies certain properties. An illustrative example related to a linguistic variable is given to demonstrate it. Then, we utilized it to propose two decision-making approaches to solve the multi-attribute decision-making (MADM) problem in the IFS environment by considering the attribute weights as either partially known or completely unknown. Finally, a practical example is provided to illustrate the decision-making process. The results corresponding to different pairs of ( R , S ) give different choices to the decision-maker to assess their results. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle N-Hyper Sets
Mathematics 2018, 6(6), 87; https://doi.org/10.3390/math6060087
Received: 21 April 2018 / Revised: 16 May 2018 / Accepted: 21 May 2018 / Published: 23 May 2018
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Abstract
To deal with the uncertainties, fuzzy set theory can be considered as one of the mathematical tools by Zadeh. As a mathematical tool to deal with negative information, Jun et al. introduced a new function, which is called a negative-valued function, and constructed [...] Read more.
To deal with the uncertainties, fuzzy set theory can be considered as one of the mathematical tools by Zadeh. As a mathematical tool to deal with negative information, Jun et al. introduced a new function, which is called a negative-valued function, and constructed N -structures in 2009. Since then, N -structures are applied to algebraic structures and soft sets, etc. Using the N -structures, the notions of (extended) N -hyper sets, N -substructures of type 1, 2, 3 and 4 are introduced, and several related properties are investigated in this research paper. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle t-Norm Fuzzy Incidence Graphs
Mathematics 2018, 6(4), 62; https://doi.org/10.3390/math6040062
Received: 27 February 2018 / Revised: 2 April 2018 / Accepted: 16 April 2018 / Published: 20 April 2018
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Abstract
It is the case that, in certain applications of fuzzy graphs, a t-norm, instead of a minimum, is more suitable. This requires the development of a new theory of fuzzy graphs involving an arbitrary t-norm in the basic definition of a fuzzy graph. [...] Read more.
It is the case that, in certain applications of fuzzy graphs, a t-norm, instead of a minimum, is more suitable. This requires the development of a new theory of fuzzy graphs involving an arbitrary t-norm in the basic definition of a fuzzy graph. There is very little known about this type of fuzzy graph. The purpose of this paper is to further develop this type of fuzzy graph. We concentrate on the relatively new concept of fuzzy incidence graphs. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle Credibility Measure for Intuitionistic Fuzzy Variables
Mathematics 2018, 6(4), 50; https://doi.org/10.3390/math6040050
Received: 7 March 2018 / Revised: 22 March 2018 / Accepted: 26 March 2018 / Published: 2 April 2018
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Abstract
Credibility measures in vague environments are to quantify the approximate chance of occurrence of fuzzy events. This paper presents a novel definition about credibility for intuitionistic fuzzy variables. We axiomatize this credibility measure and to clarify, give some examples. Based on the notion [...] Read more.
Credibility measures in vague environments are to quantify the approximate chance of occurrence of fuzzy events. This paper presents a novel definition about credibility for intuitionistic fuzzy variables. We axiomatize this credibility measure and to clarify, give some examples. Based on the notion of these concepts, we provide expected values, entropy, and general formulae for the central moments and discuss them through examples. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle Hesitant Probabilistic Fuzzy Linguistic Sets with Applications in Multi-Criteria Group Decision Making Problems
Mathematics 2018, 6(4), 47; https://doi.org/10.3390/math6040047
Received: 3 February 2018 / Revised: 19 March 2018 / Accepted: 23 March 2018 / Published: 26 March 2018
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Abstract
Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill [...] Read more.
Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill structured and complex decision making problem. HPFLS provides a single framework where both stochastic and non-stochastic uncertainties can be efficiently handled along with hesitation. We have also proposed expected mean, variance, score and accuracy function and basic operations for HPFLS. Weighted and ordered weighted aggregation operators for HPFLS are also defined in the present study for its applications in multi-criteria group decision making (MCGDM) problems. We propose a MCGDM method with HPFL information which is illustrated by an example. A real case study is also taken in the present study to rank State Bank of India, InfoTech Enterprises, I.T.C., H.D.F.C. Bank, Tata Steel, Tata Motors and Bajaj Finance using real data. Proposed HPFLS-based MCGDM method is also compared with two HFL-based decision making methods. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessFeature PaperArticle Certain Algorithms for Modeling Uncertain Data Using Fuzzy Tensor Product Bézier Surfaces
Mathematics 2018, 6(3), 42; https://doi.org/10.3390/math6030042
Received: 31 January 2018 / Revised: 5 March 2018 / Accepted: 7 March 2018 / Published: 9 March 2018
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Abstract
Real data and measures are usually uncertain and cannot be satisfactorily described by accurate real numbers. The imprecision and vagueness should be modeled and represented in data using the concept of fuzzy numbers. Fuzzy splines are proposed as an integrated approach to uncertainty [...] Read more.
Real data and measures are usually uncertain and cannot be satisfactorily described by accurate real numbers. The imprecision and vagueness should be modeled and represented in data using the concept of fuzzy numbers. Fuzzy splines are proposed as an integrated approach to uncertainty in mathematical interpolation models. In the context of surface modeling, fuzzy tensor product Bézier surfaces are suitable for representing and simplifying both crisp and imprecise surface data with fuzzy numbers. The framework of this research paper is concerned with various properties of fuzzy tensor product surface patches by means of fuzzy numbers including fuzzy parametric curves, affine invariance, fuzzy tangents, convex hull and fuzzy iso-parametric curves. The fuzzification and defuzzification processes are applied to obtain the crisp Beziér curves and surfaces from fuzzy data points. The degree elevation and de Casteljau’s algorithms for fuzzy Bézier curves and fuzzy tensor product Bézier surfaces are studied in detail with numerical examples. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
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Open AccessArticle Numerical Methods for Solving Fuzzy Linear Systems
Mathematics 2018, 6(2), 19; https://doi.org/10.3390/math6020019
Received: 21 November 2017 / Revised: 25 January 2018 / Accepted: 29 January 2018 / Published: 1 February 2018
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Abstract
In this article, three numerical iterative schemes, namely: Jacobi, Gauss–Seidel and Successive over-relaxation (SOR) have been proposed to solve a fuzzy system of linear equations (FSLEs). The convergence properties of these iterative schemes have been discussed. To display the validity of these iterative [...] Read more.
In this article, three numerical iterative schemes, namely: Jacobi, Gauss–Seidel and Successive over-relaxation (SOR) have been proposed to solve a fuzzy system of linear equations (FSLEs). The convergence properties of these iterative schemes have been discussed. To display the validity of these iterative schemes, an illustrative example with known exact solution is considered. Numerical results show that the SOR iterative method with ω = 1.3 provides more efficient results in comparison with other iterative techniques. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
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Open AccessFeature PaperArticle Length-Fuzzy Subalgebras in BCK/BCI-Algebras
Mathematics 2018, 6(1), 11; https://doi.org/10.3390/math6010011
Received: 1 December 2017 / Revised: 3 January 2018 / Accepted: 5 January 2018 / Published: 12 January 2018
Cited by 1 | PDF Full-text (248 KB) | HTML Full-text | XML Full-text
Abstract
As a generalization of interval-valued fuzzy sets and fuzzy sets, the concept of hyperfuzzy sets was introduced by Ghosh and Samanta in the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37]. The [...] Read more.
As a generalization of interval-valued fuzzy sets and fuzzy sets, the concept of hyperfuzzy sets was introduced by Ghosh and Samanta in the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37]. The aim of this manuscript is to introduce the length-fuzzy set and apply it to B C K / B C I -algebras. The notion of length-fuzzy subalgebras in B C K / B C I -algebras is introduced, and related properties are investigated. Characterizations of a length-fuzzy subalgebra are discussed. Relations between length-fuzzy subalgebras and hyperfuzzy subalgebras are established. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle Hyperfuzzy Ideals in BCK/BCI-Algebras
Mathematics 2017, 5(4), 81; https://doi.org/10.3390/math5040081
Received: 17 November 2017 / Revised: 10 December 2017 / Accepted: 10 December 2017 / Published: 14 December 2017
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Abstract
The notions of hyperfuzzy ideals in BCK/BCI-algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy [...] Read more.
The notions of hyperfuzzy ideals in B C K / B C I -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy ideals are provided. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups
Mathematics 2017, 5(4), 71; https://doi.org/10.3390/math5040071
Received: 14 November 2017 / Revised: 21 November 2017 / Accepted: 23 November 2017 / Published: 27 November 2017
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Abstract
In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a (α1,α2;β1,β2)-BF subsemigroup. The notions of (α1,α2;β1, [...] Read more.
In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a ( α 1 , α 2 ; β 1 , β 2 ) -BF subsemigroup. The notions of ( α 1 , α 2 ; β 1 , β 2 ) -BF quasi(generalized bi-, bi-) ideals are discussed. Some inequalities of ( α 1 , α 2 ; β 1 , β 2 ) -BF quasi(generalized bi-, bi-) ideals are obtained. Furthermore, any regular semigroup is characterized in terms of generalized BF semigroups. Full article
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