axioms New Trends in Fuzzy Set Theory and Related Items Edited by Esteban Induráin, Humberto Bustince and Javier Fernandez Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms New Trends in Fuzzy Set Theory and Related Items New Trends in Fuzzy Set Theory and Related Items Special Issue Editors Esteban Indur´ain Humberto Bustince Javier Fernandez MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Esteban Indurain´ Humberto Bustince Universidad Publica´ de Navarra Universidad Publica´ de Navarra Spain Spain Javier Fernandez Universidad Publica´ de Navarra Spain Editorial Office MDPI St. Alban-Anlage 66 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) from 2017 to 2018 (available at: http://www.mdpi.com/journal/axioms/special issues/fuzzy set theory) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. 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Contents About the Special Issue Editors ..................................... vii Preface to ”New Trends in Fuzzy Set Theory and Related Items” .................. ix Humberto Bustince, Javier Fernandez and Esteban Indur´ain Introduction to Special Issue: New Trends in Fuzzy Set Theory andRelated Items Reprinted from: Axioms 2018, 7, 37, doi: 10.3390/axioms7020037 ................... 1 Mar´ıa Jesus ´ Campi´on, Edurne Falc´o, Jos´e Luis Garc´ıa-Lapresta and Esteban Indur´ain Assigning Numerical Scores to Linguistic Expressions Reprinted from: Axioms 2017, 6, 19, doi: 10.3390/axioms6030019 ................... 3 Pablo Hern´andez, Susana Cubillo, Carmen Torres-Blanc and Jos´e A. Guerrero New Order on Type 2 Fuzzy Numbers Reprinted from: Axioms 2017, 6, 22, doi: 10.3390/axioms6030022 ................... 20 Leire Legarreta, Inmaculada Lizasoain and Iraide Mardones-P´erez Orness For Idempotent Aggregation Functions Reprinted from: Axioms 2017, 6, 25, doi: 10.3390/axioms6030025 ................... 39 Paolo Bevilacqua, Gianni Bosi and Magal`ı Zuanon Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness Reprinted from: Axioms 2017, 6, 29, doi: 10.3390/axioms6040029 ................... 49 Juan-Jos´eMi˜nana and Oscar Valero On Indistinguishability Operators, Fuzzy Metrics and Modular Metrics Reprinted from: Axioms 2017, 6, 34, doi: 10.3390/axioms6040034 ................... 59 Teresa Gonz´alez-Artega, Rocio de Andr´es Calle and Luis Mart´ınez Managing Interacting Criteria: Application to Environmental Evaluation Practices Reprinted from: Axioms 2018, 7, 4, doi: 10.3390/axioms7010004 ................... 77 Young Bae Jun, Seok-Zun Song and Seon Jeong Kim Cubic Interval-Valued Intuitionistic Fuzzy Sets and TheirApplication in BCK/BCI-Algebras Reprinted from: Axioms 2018, 7, 7, doi: 10.3390/axioms7010007 ...................101 Christian Servin, Gerardo D. Muela and Vladik Kreinovich Fuzzy Analogues of Sets and Functions Can Be UniquelyDetermined from the Corresponding Ordered Category: A Theorem Reprinted from: Axioms 2018, 7, 8, doi: 10.3390/axioms7010008 ...................118 Krzysztof Piasecki Revision of the Kosinski’s´ Theory of Ordered Fuzzy Numbers Reprinted from: Axioms 2018, 7, 16, doi: 10.3390/axioms7010016 ...................125 Juan C. Candeal An Abstract Result on Projective Aggregation Functions Reprinted from: Axioms 2018, 7, 17, doi: 10.3390/axioms7010017 ...................141 Sundas Shahzadi and Muhammad Akram Graphs in an Intuitionistic Fuzzy Soft Environment Reprinted from: Axioms 2018, 7, 20, doi: 10.3390/axioms7020020 ...................149 v Marta Cardin Quantiles in Abstract Convex Structures Reprinted from: Axioms 2018, 7, 35, doi: 10.3390/axioms7020035 ...................165 vi About the Special Issue Editors Esteban Indur´ain, Dr., was born in Pamplona, Spain in 1960. He got his degree in Mathematical Sciences from the University of Zaragoza, Spain in 1982 and his doctorate (Ph.D.) in Mathematical Sciences, also from the University of Zaragoza, in 1985. He is currently full professor of Mathematical Analysis at the Public University of Navarre, Pamplona (Spain), since 2000. Dr. Indurain´ has published ca. 85 papers in well-ranked journals. His impact factor is h = 14 in “Web of Science” and ”Scopus”. He has been the main researcher in several projects. Dr. Indurain´ has co-advised seven Ph.D. theses to-date. One of them was awarded a prize for the best Ph.D. thesis in its ambit and period. Humberto Bustince, Dr., received his Bs.C. degree on Physics from the Salamanca University, Spain, in 1983 and his Ph.D. degree in Mathematics from the Public University of Navarra, Pamplona, Spain, in 1994. He has been a teacher at the Public University of Navarra since 1991, and he is currently a Full Professor with the Department of Automatics and Computation. He served as subdirector of the Technical School for Industrial Engineering and Telecommunications from 01/01/2003 to 30/10/2008, and he was involved in the implantation of Computer Science courses at the Public University of Navarra. He is currently involved in teaching artificial intelligence for students of computer sciences. Dr. Bustince has authored more than 120 journal papers (Web of Knowledge) and more than 100 contributions to international conferences. He has also been the co-author of four books on fuzzy theory and extensions of fuzzy sets. His main research interests are interval-valued fuzzy sets (interval type-2 fuzzy sets), Atanassov’s intuitionistic fuzzy sets, aggregation functions, implication operators, inclusion measures, image processing, decision making, and approximate reasoning. Javier Fernandez, Dr., graduated in Mathematics from the University of Zaragoza (1999) and got his doctorate (Ph.D.) in Mathematics from the University of the Basque Country in 2003. He has been a postdoctoral researcher at the CNRS (France) in 2003–2004 and 2005–2007, in the framework of the European Hyperbolic and Kynetic Equations (HYKE) research project. He has been a lecturer in the Public University of Navarre since 2008. He has co-advised two Ph.D. thesis. Dr. Fernandez is a member of the research Group on Artificial Intelligence and Approximate Reasoning (GIARA) of the Public University of Navarre, whose leader is Dr. Humberto Bustince. His main research interests are fuzzy theory, extensions of fuzzy theory, aggregation functions, computing, differential equations, and unique continuation, as well as image processing. He has authored more than 30 papers in JCR journals, most of them in Q1. Some of the most relevant ones are on linear orders for intervals. vii Preface to ”New Trends in Fuzzy Set Theory and Related Items” Fuzzy set theory is one of the main trend topics in Mathematics and Computer Sciences nowadays. Its wide, multidisciplinary applications, as well as the recent developments in its theoretical bases, provide an excellent set of frameworks for researchers interested in different scientific disciplines. Owing to these reasons, we have addressed the task of issuing this Special Issue, having in mind the possibility of collecting new contributions devoted to this topic, from many different points of view. In this sense, this book includes works authored by authors working from very different perspectives, but all of them are in the first line of research of their areas. The articles (twelve thereof) that have been included in this Special Issue cover a large set of the newest developments in this framework of research. Thus, always bearing in mind the notion of a fuzzy set, we can find ideas and new results on orderings, logic, aggregation of functions, functional equations, operators, metric, etc. We have also included some works devoted to applications in, e.g., social choice. Of course, the field of fuzzy sets theory is so broad nowadays that it is not possible to cover all of them, but we consider that the book provides a rather complete view of the state-of-the-art in the field. EstebanIndur´ain,HumbertoBustince,JavierFernandez SpecialIssueEditors ix axioms Editorial Introduction to Special Issue: New Trends in Fuzzy Set Theory and Related Items Humberto Bustince 1 ID , Javier Fernandez 1 ID and Esteban Induráin 2,* ID 1 Institute of Smart Cities (ISC), Departamento de Estadística, Matemáticas e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain; [email protected] (H.B.); [email protected] (J.F.) 2 Institute for Advanced Materials (InaMat), Departamento de Estadística, Matemáticas e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain * Correspondence: [email protected]; Tel.: +34-948-169-551 Received: 10 May 2018; Accepted: 1 June 2018; Published: 5 June 2018 Abstract: We focus on the articles recently published in the special issue of Axioms devoted