Special Issue "New Trends in Fuzzy Set Theory and Related Items"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 April 2018)

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

Special Issue Editors

Guest Editor
Dr. Javier Fernandez

Department of Automatic and Computation, Public University of Navarra, Pamplona 31006, Spain
E-Mail
Interests: aggregation functions; theoretical aspects of fuzzy sets and their extensions; image processing; classification; decision making; bio-inspired algorithms; partial differential equations
Guest Editor
Dr. Esteban Indurain

Department of Mathematics, Public University of Navarre, Campus Arrosadía, Edificio las Encinas, Pamplona 31006, Spain
Website | E-Mail
Interests: ordered structures and their numerical representability; fuzzy sets and systems; functional equations; real analysis; general topology; mathematical social choice; mathematical economics; functional analysis
Guest Editor
Prof. Dr. Humberto Bustince

Department of Automatic and Computation, Public University of Navarra, Pamplona 31006, Spain
Website | E-Mail
Interests: aggregation functions; theoretical aspects of fuzzy sets and their extensions; image processing; classification; decision making; bio-inspired algorithms; partial differential equations

Special Issue Information

Dear Colleagues,

We have the intention of launching a Special Issue of Axioms. The central topic in the Special Issue will be “fuzzy set theory”. We would provide an opportunity to showcase recent developments in the many branches of both theoretical and practical studies in Mathematics, which are related to fuzzy set theory and/or its extensions and generalizations. Among the topics that this Special Issue will address, we may consider the following non-exhaustive list:

Fuzzy sets and systems; Fuzzy Logic; Linguistic labels; Fuzzy numbers; Functional equations; Aggregation functions and operators; Extensions of fuzzy sets; Ordered structures; Fuzzy relations; Miscellaneous applications of fuzzy sets and their extensions, etc.

Needless to say, the Special Issue is open to receiving further ideas, apart from the aforementioned topics.

In the hopes that this initiative are of interest, we encourage you to submit your current research to be included in the Special Issue.

Best regards,

Dr. Javier Fernandez
Dr. Esteban Indurain
Prof. Dr. Humberto Bustince
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 350 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fuzzy sets
  • Fuzzy logic
  • Labels
  • Aggregation operators
  • Functional equations
  • Ordered structures
  • Numerical representability
  • General topology
  • Social choice
  • Decision making;
  • Image processing
  • Type-2 fuzzy sets
  • Extensions of fuzzy sets
  • Fuzzy relations
  • Copulas

Published Papers (13 papers)

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Editorial

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Open AccessEditorial Introduction to Special Issue: New Trends in Fuzzy Set Theory and Related Items
Received: 10 May 2018 / Revised: 31 May 2018 / Accepted: 1 June 2018 / Published: 5 June 2018
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Abstract
We focus on the articles recently published in the special issue of Axioms devoted to “New Trends in Fuzzy Set Theory and Related Items”. Full article

Research

Jump to: Editorial

Open AccessArticle Quantiles in Abstract Convex Structures
Received: 14 May 2018 / Revised: 23 May 2018 / Accepted: 25 May 2018 / Published: 28 May 2018
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Abstract
In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex
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In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles. Full article
Open AccessArticle Graphs in an Intuitionistic Fuzzy Soft Environment
Received: 17 February 2018 / Revised: 9 March 2018 / Accepted: 14 March 2018 / Published: 23 March 2018
Cited by 7 | PDF Full-text (500 KB) | HTML Full-text | XML Full-text
Abstract
In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic
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In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic fuzzy soft graphs and strongyl edge irregular intuitionistic fuzzy soft graphs. We illustrate these novel concepts by several examples, and investigate some of their related properties. We present an application of intuitionistic fuzzy soft graph in a decision-making problem and also present our methods as an algorithm that is used in this application. Full article
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Open AccessArticle An Abstract Result on Projective Aggregation Functions
Received: 1 March 2018 / Revised: 13 March 2018 / Accepted: 19 March 2018 / Published: 20 March 2018
Cited by 1 | PDF Full-text (236 KB) | HTML Full-text | XML Full-text
Abstract
A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a
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A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a theorem obtained by Kim (1990) for real-valued aggregation functions defined on the n-dimensional Euclidean space in the context of measurement theory. In addition, two applications of the core theorem of the article are shown. The first is a simple extension of the main result to the context of multi-valued aggregation functions. The second offers a new characterization of projective bijection aggregators, thus connecting aggregation operators theory with social choice. Full article
Open AccessArticle Revision of the Kosiński’s Theory of Ordered Fuzzy Numbers
Received: 5 January 2018 / Revised: 20 February 2018 / Accepted: 26 February 2018 / Published: 2 March 2018
Cited by 1 | PDF Full-text (892 KB) | HTML Full-text | XML Full-text
Abstract
Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this
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Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosiński has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosiński’s addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski’s theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering. Full article
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Open AccessArticle Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem
Received: 9 August 2017 / Revised: 2 January 2018 / Accepted: 9 January 2018 / Published: 23 January 2018
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Abstract
In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In
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In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees. Full article
Open AccessArticle Cubic Interval-Valued Intuitionistic Fuzzy Sets and Their Application in BCK/BCI-Algebras
Received: 28 December 2017 / Revised: 12 January 2018 / Accepted: 15 January 2018 / Published: 23 January 2018
Cited by 1 | PDF Full-text (303 KB) | HTML Full-text | XML Full-text
Abstract
As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in BCK/BCI-algebra is considered. The notions of α-internal, β-internal, α-external and
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As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in B C K / B C I -algebra is considered. The notions of α -internal, β -internal, α -external and β -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of α -internal and α -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in B C K / B C I -algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed. Full article
Open AccessArticle Managing Interacting Criteria: Application to Environmental Evaluation Practices
Received: 5 December 2017 / Revised: 2 January 2018 / Accepted: 8 January 2018 / Published: 16 January 2018
Cited by 1 | PDF Full-text (576 KB) | HTML Full-text | XML Full-text
Abstract
The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve
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The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve traditional evaluation process in different facets. Firstly, different reviewers’ collectives related to the company’s activity are taken into account in the process to increase company internal efficiency and external legitimacy. Secondly, following the standard ISO 14031, two general categories of environmental performance indicators, management and operational, are considered. Thirdly, since the assumption of independence among environmental indicators is rarely verified in environmental context, an aggregation operator to bear in mind the relationship among such indicators in the evaluation results is proposed. Finally, this new model integrates quantitative and qualitative information with different scales using a multi-granular linguistic model that allows to adapt diverse evaluation scales according to appraisers’ knowledge. Full article
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Open AccessArticle On Indistinguishability Operators, Fuzzy Metrics and Modular Metrics
Received: 20 November 2017 / Revised: 8 December 2017 / Accepted: 12 December 2017 / Published: 15 December 2017
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Abstract
The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed
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The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed measurement or a certain degree of similarity can be only determined between the objects being compared. Since Trillas introduced such kind of operators, many authors have studied their properties and applications. In particular, an intensive research line is focused on the metric behavior of indistinguishability operators. Specifically, the existence of a duality between metrics and indistinguishability operators has been explored. In this direction, a technique to generate metrics from indistinguishability operators, and vice versa, has been developed by several authors in the literature. Nowadays, such a measurement of similarity is provided by the so-called fuzzy metrics when the degree of similarity between objects is measured relative to a parameter. The main purpose of this paper is to extend the notion of indistinguishability operator in such a way that the measurements of similarity are relative to a parameter and, thus, classical indistinguishability operators and fuzzy metrics can be retrieved as a particular case. Moreover, we discuss the relationship between the new operators and metrics. Concretely, we prove the existence of a duality between them and the so-called modular metrics, which provide a dissimilarity measurement between objects relative to a parameter. The new duality relationship allows us, on the one hand, to introduce a technique for generating the new indistinguishability operators from modular metrics and vice versa and, on the other hand, to derive, as a consequence, a technique for generating fuzzy metrics from modular metrics and vice versa. Furthermore, we yield examples that illustrate the new results. Full article
Open AccessArticle Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness
Received: 6 October 2017 / Revised: 23 October 2017 / Accepted: 26 October 2017 / Published: 28 October 2017
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Abstract
Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy
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Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space. Full article
Open AccessArticle Orness For Idempotent Aggregation Functions
Received: 23 August 2017 / Revised: 15 September 2017 / Accepted: 17 September 2017 / Published: 20 September 2017
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Abstract
Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the
[...] Read more.
Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the supremum of the given data, which is guaranteed only when the aggregation functions are idempotent. Ordered weighted averaging (OWA) operators are particular cases of this kind of function, with the particularity that the obtained global value depends on neither the source nor the expert that provides each datum, but only on the set of values. They have been classified by means of the orness—a measurement of the proximity of an OWA operator to the OR-operator. In this paper, the concept of orness is extended to the framework of idempotent aggregation functions defined both on the real unit interval and on a complete lattice with a local finiteness condition. Full article
Open AccessArticle New Order on Type 2 Fuzzy Numbers
Received: 5 June 2017 / Revised: 14 July 2017 / Accepted: 24 July 2017 / Published: 28 July 2017
Cited by 1 | PDF Full-text (1403 KB) | HTML Full-text | XML Full-text
Abstract
Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get
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Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers. Full article
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Open AccessArticle Assigning Numerical Scores to Linguistic Expressions
Received: 7 June 2017 / Revised: 29 June 2017 / Accepted: 30 June 2017 / Published: 6 July 2017
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Abstract
In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through
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In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed. Full article
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