Fuzzy Mathematics Edited by Etienne E. Kerre and John Mordeson Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Fuzzy Mathematics Fuzzy Mathematics Special Issue Editors Etienne E. Kerre John Mordeson MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Etienne E. Kerre John Mordeson Ghent University Creighton University Belgium USA 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 Mathematics (ISSN 2227-7390) from 2017 to 2018 (available at: https://www.mdpi.com/journal/ mathematics/special issues/Fuzzy Mathematics) 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. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-322-5 (Pbk) ISBN 978-3-03897-323-2 (PDF) Articles in this volume are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book taken as a whole is c 2018 MDPI, Basel, Switzerland, distributed under the terms and conditions of the Creative Commons license CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents About the Special Issue Editors ..................................... vii Preface to ”Fuzzy Mathematics” ..................................... ix Erich Peter Klement and Radko Mesiar † L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New? Reprinted from: Mathematics 2018, 6, 146, doi: 10.3390/math6090146 ................ 1 Krassimir Atanassov On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets Reprinted from: Mathematics 2018, 6, 123, doi: 10.3390/math6070123 ................ 25 Young Bae Jun, Seok-Zun Song and Seon Jeong Kim N -Hyper Sets Reprinted from: Mathematics 2018, 6, 87, doi: 10.3390/math6060087 ................. 35 Muhammad Akram and Gulfam Shahzadi Hypergraphs in m-Polar Fuzzy Environment Reprinted from: Mathematics 2018, 6, 28, doi: 10.3390/math6020028 ................. 47 Noor Rehman, Choonkil Park, Syed Inayat Ali Shah and Abbas Ali On Generalized Roughness in LA-Semigroups Reprinted from: Mathematics 2018, 6, 112, doi: 10.3390/math6070112 ................ 65 Hsien-Chung Wu Fuzzy Semi-Metric Spaces Reprinted from: Mathematics 2018, 6, 106, doi: 10.3390/math6070106 ................ 73 E. Mohammadzadeh, R. A. Borzooei Nilpotent Fuzzy Subgroups Reprinted from: Mathematics 2018, 6, 27, doi: 10.3390/math6020027 ................. 92 Florentin Smarandache, Mehmet S¸ ahin and Abdullah Kargın Neutrosophic Triplet G-Module Reprinted from: Mathematics 2018, 6, 53, doi: 10.3390/math6040053 .................104 Pannawit Khamrot and Manoj Siripitukdet Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups Reprinted from: Mathematics 2017, 5, 71, doi: 10.3390/math5040071 .................113 Seok-Zun Song, Seon Jeong Kim and Young Bae Jun † Hyperfuzzy Ideals in BCK/BCI-Algebras Reprinted from: Mathematics 2017, 5, 81, doi: 10.3390/math5040081 .................127 Young Bae Jun, Seok-Zun Song and Seon Jeong Kim Length-Fuzzy Subalgebras in BCK/BCI-Algebras Reprinted from: Mathematics 2018, 6, 11, doi: 10.3390/math6010011 .................141 v Young Bae Jun, Florentin Smarandache,Seok-Zun Song and Hashem Bordbar Neutrosophic Permeable Values and Energetic Subsets withApplications in BCK/BCI-Algebras Reprinted from: Mathematics 2018, 6, 74, doi: ..............................153 Harish Garg and Jaspreet Kaur A Novel (R, S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making Reprinted from: Mathematics 2018, 6, 92, doi: 10.3390/math6060092 .................169 Dheeraj Kumar Joshi, Ismat Beg and Sanjay Kumar Hesitant Probabilistic Fuzzy Linguistic Sets with Applications in Multi-Criteria Group Decision Making Problems Reprinted from: Mathematics 2018, 6, 47, doi: 10.3390/math6040047 .................188 Irina Georgescu The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models Reprinted from: Mathematics 2018, 6, 133, doi: 10.3390/math6080133 ................208 Rudolf Seising The Emergence of Fuzzy Sets in the Decade of the Perceptron—Lotfi A. Zadeh’s and Frank Rosenblatt’s Research Work on Pattern Classification Reprinted from: Mathematics 2018, 6, 110, doi: 10.3390/math6070110 ................227 Mohamadtaghi Rahimi, Pranesh Kumar and Gholamhossein Yari Credibility Measure for Intuitionistic Fuzzy Variables Reprinted from: Mathematics 2018, 6, 50, doi: 10.3390/math6040050 .................247 Musavarah Sarwar and Muhammad Akram Certain Algorithms for Modeling Uncertain Data Using Fuzzy Tensor Product Be´zierSurfaces Reprinted from: Mathematics 2018, 6, 42, doi: 10.3390/math6030042 .................254 Lubna Inearat and Naji Qatanani Numerical Methods for Solving Fuzzy Linear Systems Reprinted from: Mathematics 2018, 6, 19, doi: 10.3390/math6020019 .................266 vi About the Special Issue Editors Etienne E. Kerre was born in Zele, Belgium on 8 May 1945. He obtained his M. Sc. degree in Mathematics in 1967 and his Ph.D. in Mathematics in 1970 from Ghent University. He has been a lector since 1984, and has been a full professor at Ghent University since 1991. In 2010, he became a retired professor. He is a referee for more than 80 international scientific journals, and also a member of the editorial boards of many international journals and conferences on fuzzy set theory. He has been an honorary chairman at various international conferences. In 1976, he founded the Fuzziness and Uncertainty Modeling Research Unit (FUM). Since then, his research has been focused on the modeling of fuzziness and uncertainty, and has resulted in a great number of contributions in fuzzy set theory and its various generalizations. Especially the theories of fuzzy relational calculus and of fuzzy mathematical structures owe a very great deal to him. Over the years he has also been a promotor of 30 Ph.D.s on fuzzy set theory. His current research interests include fuzzy and intuitionistic fuzzy relations, fuzzy topology, and fuzzy image processing. He has authored or co-authored 25 books and more than 500 papers appearing in international refereed journals and proceedings. John Mordeson is Professor Emeritus of Mathematics at Creighton University. He received his B.S., M.S., and Ph. D. from Iowa State University. He is a Member of Phi Kappa Phi. He is President of the Society for Mathematics of Uncertainty. He has published 15 books and over two hundred journal articles. He is on the editorial boards of numerous journals. He has served as an external examiner of Ph.D. candidates from India, South Africa, Bulgaria, and Pakistan. He has refereed for numerous journals and granting agencies. He is particularly interested in applying mathematics of uncertainty to combat the problems of human trafficking, modern slavery, and illegal immigration. vii Preface to ”Fuzzy Mathematics” This Special Issue on fuzzy mathematics is dedicated to Lotfi A. Zadeh. 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, the extension was 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 fuzztfications of the classical mathematical domains were launched by simply changing Cantor’s set-theoretic operations by Zadeh’s max-min extensions. The 1980s were characterized by an extension of the possible fuzzifications due to the discovery of triangular norms and conorms. Starting in the 1990s, more profound analysis was performed by studying the axiomatization of fuzzy structures and searching for links between the different models to represent imprecise and uncertain information. It was our aim to have this Special Issue comprise 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. This Special Issue starts with a corner-stone paper that should be read by all working in the field of fuzzy mathematics. Using lattice isomorphisms, it shows that the results of many of the variations and extensions of fuzzy set theory can be obtained immediately from the results of set theory itself. The paper is extremely valuable to reviewers in the field. This paper is followed by one which gives the definition of the most extended modal operator of the first type over interval-valued sets, and presents some of its basic properties. A new function called a negative-valued function is presented and applied to various structures. Results concerning N-hyper sets and hypergraphs in m-polar fuzzy setting are also presented. In the next paper, it is shown that the lower approximation of a subset of an LA-semigroup need not be an LA-subsemigroup/ideal of an LA-semigroup under