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Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures • Irina Cristea Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures

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Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures • Irina Cristea Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures Symmetry Classical Algebraic Fuzzy in and Hypercompositional Structures Symmetry in Classical • Irina Cristea and Fuzzy Algebraic Hypercompositional Structures Edited by Irina Cristea Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures Special Issue Editor Irina Cristea MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Special Issue Editor Irina Cristea Center for Information Technologies and Applied Mathematics, University of Nova Gorica Slovenia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Fuzzy Algebraic Hypercompositional Structures). 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 978-3-03928-708-6 (Pbk) ISBN 978-3-03928-709-3 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to “Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures” ... ix Dariush Heidari and Irina Cristea Breakable Semihypergroups Reprinted from: Symmetry 2019, 11, 100, doi:10.3390/sym11010100 ................. 1 Mario De Salvo, Dario Fasino, Domenico Freni and Giovanni Lo Faro On Hypergroups with a β-Class of Finite Height Reprinted from: Symmetry 2020, 12, 168, doi:10.3390/sym12010168 ................. 11 Stˇˇ ep´an Kˇrehl´ık and Jana Vyroubalov´a The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory Reprinted from: Symmetry 2020, 12, 54, doi:10.3390/sym12010054 .................. 25 Vahid Vahedi, Morteza Jafarpour and Irina Cristea Hyperhomographies on Krasner Hyperfields Reprinted from: Symmetry 2019, 11, 1442, doi:10.3390/sym11121442 ................. 41 Madeline Al Tahan, Sarkaˇ Hoˇskova-Mayerova and Bijan Davvaz Some Results on (Generalized) Fuzzy Multi-Hv-Ideals of Hv-Rings Reprinted from: Symmetry 2019, 11, 1376, doi:10.3390/sym11111376 ................. 61 Sarkaˇ Hoˇskova-Mayerova and Babatunde O. Onasanya Results on Functions on Dedekind Multisets Reprinted from: Symmetry 2019, 11, 1125, doi:10.3390/sym11091125 ................. 75 Hashem Bordbar, G. Muhiuddin and Abdulaziz M. Alanazi Primeness of Relative Annihilators in BCK-Algebra Reprinted from: Symmetry 2020, 12, 286, doi:10.3390/sym12020286 ................. 85 Xiao Long Xin, Rajab Ali Borzooei, Mahmood Bakhshi and Young Bae Jun Intuitionistic Fuzzy Soft Hyper BCK Algebras Reprinted from: Symmetry 2019, 11, 399, doi:10.3390/sym11030399 ................. 97 Jan Chvalina and Bedˇrich Smetana Series of Semihypergroups of Time-Varying Artificial Neurons and Related Hyperstructures Reprinted from: Symmetry 2019, 11, 927, doi:10.3390/sym11070927 .................109 Michal Nov´ak, Step´ˇ an Kˇrehl´ık and Kyriakos Ovaliadis Elements of Hyperstructure Theory in UWSN Design and Data Aggregation Reprinted from: Symmetry 2019, 11, 734, doi:10.3390/sym11060734 .................121 Anam Luqman, Muhammad Akram and Ali N.A. Koam An m-Polar Fuzzy Hypergraph Model of Granular Computing Reprinted from: Symmetry 2019, 11, 483, doi:10.3390/sym11040483 .................137 Muhammad Akram, Amna Habib and Ali N. A. Koam A Novel Description on Edge-Regular q-Rung Picture Fuzzy Graphs with Application Reprinted from: Symmetry 2019, 11, 489, doi:10.3390/sym11040489 .................159 v About the Special Issue Editor Irina Cristea received her Ph.D. degree in mathematics in 2007 from University Ovidius of Constanta, Romania. After a period of post doctorate studies at University of Udine, Italy, in 2012, she became Assistant Professor at the University of Nova Gorica, Slovenia, where she currently hold the position of Associate Professor and acting head of the Center for Information Technologies and Applied Mathematics. Her main research direction is theory of algebraic hypercompositional structures and their connections with fuzzy sets theory. In this area, she has published more than 60 reasearch articles in journals indexed by Scopus/Web of Science. She is also a co-author of the monograph “Fuzzy Algebraic Hyperstructures: An Introduction” published in 2015 by Springer. Prof. Irina Cristea is deeply involved in the editorial activities, being member of the Editorial Board of five international journals (among them Mathematics, published by MDPI) and acting as a reviewer for numerous journals. In 2019, she was awarded by Publons the “Top 1% Reviewers in Mathematics”. vii Preface to “Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures” Symmetry plays a fundamental role in our daily lives and in the study of the structure of different objects in physics, chemistry, biology, mathematics, architecture, arts, sociology, linguistics, etc. For example, the structure of molecules is well explained by their symmetry properties, described by symmetry elements and symmetry operations. A symmetry operation is a change, a transformation after which certain objects remain invariant, such as rotations, reflections, inversions, or permutation operations. Until now, the most efficient method to better describe symmetry was using mathematical tools offered by group theory. Naturally generalizing the concept of a group, by considering the result of the interaction between two elements of a non-empty set to be a set of elements (and not only one element, as for groups), Frederic Marty, in 1934, at only 23 years old, defined the concept of a hypergroup. The law characterizing such a structure is called multi-valued operation, hyperoperation, or hypercomposition, and the theory of the algebraic structures endowed with at least one multi-valued operation is known as hyperstructure theory or hypercompositional algebra. Marty’s motivation to introduce hypergroups was that the quotient of a group modulo any subgroup (not necessarily normal) is a hypergroup. The main aim of this Special Issue was to present recent advances in hypercompositional algebra, the crisp and fuzzy algebra, where symmetry plays a crucial role. Studies are related (but not limited) to equivalence relations, orderings, permutations, symmetrical groups, graphs and hypergraphs, lattices, fuzzy sets, representations, etc. Some applications of algebraic hypercompositional structures in engineering, information technologies, computer science, where symmetry, or the lack of symmetry, is clearly specified and laid out, are also underlined. Irina Cristea Special Issue Editor ix S S symmetry Article Breakable Semihypergroups Dariush Heidari 1 and Irina Cristea 2,* 1 Faculty of Science, Mahallat Institute of Higher Education, Mahallat 37811-51958, Iran; [email protected] 2 Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, Vipavska Cesta 13, 5000 Nova Gorica, Slovenia * Correspondence: [email protected]; Tel.: +386-0533-15-395 Received: 14 December 2018; Accepted: 12 January 2019; Published: 16 January 2019 Abstract: In this paper, we introduce and characterize the breakable semihypergroups, a natural generalization of breakable semigroups, defined by a simple property: every nonempty subset of them is a subsemihypergroup. Then, we present and discuss on an extended version of Rédei’s theorem for semi-symmetric breakable semihypergroups, proposing a different proof that improves also the theorem in the classical case of breakable semigroups. Keywords: breakable semigroup; semihypergroup; hyperideal; semi-symmetry 1. Introduction Breakable semigroups, introduced by Rédei [1] in 1967, have the property that every nonempty subset of them is a subsemigroup. It was proved that they are semigroups with empty Frattini-substructure [1]. For a structure S (i.e., a group, a semigroup, a module, a ring or a field), the set of those elements which may be omitted from each generating system (containing them) of S is a substructure of the same kind of S, called the Frattini-substructure of S. However, as mentioned in the book [1], there are some exceptions. The first one is when the Frattini-substructure is the empty set and this is the case of breakable semigroups, unit groups, zero modules or zero rings. The second one concerns the skew fields having the Frattini-substructure zero [1]. Based on the definition, it is easy to see that a semigroup S is breakable if and only if xy ∈{x, y} for any x, y ∈ S, i.e., the product of any two elements of the given semigroup is always one of the considered elements. Another characterization of these semigroups is given by Tamura and Shafer [2], using the associated power semigroup, ∗ i.e., a semigroup S is breakable if and only if its power semigroup P (S) is idempotent. An idempotent semigroup is a semigroup
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