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Symmetry
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Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures
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Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2020) | Viewed by 32376

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

Special Issue Editor

Dr. Irina Cristea

E-Mail Website
Guest Editor
Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia
Interests: theory of algebraic hypercompositional structures
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

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 way to better describe symmetry, is 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, only 23 years old, defined the concept of a hypergroup. The law characterizing such a structure is called multi-valued operation, or hyperoperation, or hypercomposition and the theory of the algebraic structures endowed with at least one multi-valued operation is known as the Hyperstructure Theory or Hypercompositional Algebra. Marty’s motivation to introduce hypergroups is that the quotient of a group modulo any subgroup (not necessarily normal) is a hypergroup.

The main aim of this Special Issue is to underline various aspects in Hypercompositional Algebra, the crisp and fuzzy one, where symmetry plays a crucial role. They are related (but not limited) to equivalence relations, orderings, permutations, symmetrical groups, graphs, lattices, fuzzy sets, representations, etc. Applications of algebraic hypercompositional structures in physics, chemistry, biology, social sciences, information technologies, computer science, etc, where symmetry, or the lack of symmetry, is clearly specified and laid out, are also welcome.

Prof. Dr. Irina Cristea
Guest Editor

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 submissions that pass pre-check are 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. Symmetry is an international peer-reviewed open access monthly 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 2400 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) hypercomposition
  • Symmetrical hypercomposition
  • Ordered hyperstructures
  • Symmetric group
  • Symmetry
  • Equivalence relation
  • Representation of hyperstructures
  • Graphs

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Published Papers (13 papers)

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Editorial

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4 pages, 206 KiB  
Editorial
Special Issue on Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures
by Irina Cristea
Symmetry 2022, 14(10), 2160; https://doi.org/10.3390/sym14102160 - 15 Oct 2022
Viewed by 960
Abstract
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 [...] Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)

Research

Jump to: Editorial

11 pages, 268 KiB  
Article
Primeness of Relative Annihilators in BCK-Algebra
by Hashem Bordbar, G. Muhiuddin and Abdulaziz M. Alanazi
Symmetry 2020, 12(2), 286; https://doi.org/10.3390/sym12020286 - 15 Feb 2020
Cited by 9 | Viewed by 2214
Abstract
Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal [...] Read more.
Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
14 pages, 306 KiB  
Article
On Hypergroups with a β-Class of Finite Height
by Mario De Salvo, Dario Fasino, Domenico Freni and Giovanni Lo Faro
Symmetry 2020, 12(1), 168; https://doi.org/10.3390/sym12010168 - 15 Jan 2020
Cited by 7 | Viewed by 1912
Abstract
In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height [...] Read more.
In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
16 pages, 737 KiB  
Article
The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory
by Štěpán Křehlík and Jana Vyroubalová
Symmetry 2020, 12(1), 54; https://doi.org/10.3390/sym12010054 - 26 Dec 2019
Cited by 4 | Viewed by 2290
Abstract
In the first part of our paper, we construct a cyclic hypergroup of matrices using the Ends Lemma. Its properties are then, in the second part of the paper, used to describe the symmetry of lower and upper approximations in certain rough sets [...] Read more.
In the first part of our paper, we construct a cyclic hypergroup of matrices using the Ends Lemma. Its properties are then, in the second part of the paper, used to describe the symmetry of lower and upper approximations in certain rough sets with respect to invertible subhypergroups of this cyclic hypergroup. Since our approach is widely used in autonomous robotic systems, we suggest an application of our results for the study of detection sensors, which are used especially in mobile robot mapping. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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20 pages, 356 KiB  
Article
Hyperhomographies on Krasner Hyperfields
by Vahid Vahedi, Morteza Jafarpour and Irina Cristea
Symmetry 2019, 11(12), 1442; https://doi.org/10.3390/sym11121442 - 23 Nov 2019
Cited by 9 | Viewed by 2336
Abstract
In this paper, we introduce generalized homographic transformations as hyperhomographies over Krasner hyperfields.These particular algebraic hyperstructues are quotient structures of classical fields modulo normal groups. Besides, we define some hyperoperations and investigate the properties of the derived hypergroups and H v -groups associated [...] Read more.
In this paper, we introduce generalized homographic transformations as hyperhomographies over Krasner hyperfields.These particular algebraic hyperstructues are quotient structures of classical fields modulo normal groups. Besides, we define some hyperoperations and investigate the properties of the derived hypergroups and H v -groups associated with the considered hyperhomographies. They are equipped hyperhomographies obtained as quotient sets of nondegenerate hyperhomographies modulo a special equivalence. Thus the symmetrical property of the equivalence relations plays a fundamental role in this constructions. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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14 pages, 310 KiB  
Article
Some Results on (Generalized) Fuzzy Multi-Hv-Ideals of Hv-Rings
by Madeline Al Tahan, Sarka Hoskova-Mayerova and Bijan Davvaz
Symmetry 2019, 11(11), 1376; https://doi.org/10.3390/sym11111376 - 6 Nov 2019
Cited by 15 | Viewed by 2123
Abstract
The concept of fuzzy multiset is well established in dealing with many real life problems. It is possible to find various applications of algebraic hypercompositional structures in natural, technical and social sciences, where symmetry, or the lack of symmetry, is clearly specified and [...] Read more.
The concept of fuzzy multiset is well established in dealing with many real life problems. It is possible to find various applications of algebraic hypercompositional structures in natural, technical and social sciences, where symmetry, or the lack of symmetry, is clearly specified and laid out. In this paper, we use fuzzy multisets to introduce the concept of fuzzy multi- H v -ideals as a generalization of fuzzy H v -ideals. Moreover, we introduce the concept of generalized fuzzy multi- H v -ideals as a generalization of generalized fuzzy H v -ideals. Finally, we investigate the properties of these new concepts and present different examples. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
9 pages, 265 KiB  
Article
Results on Functions on Dedekind Multisets
by Šárka Hošková-Mayerová and Babatunde Oluwaseun Onasanya
Symmetry 2019, 11(9), 1125; https://doi.org/10.3390/sym11091125 - 4 Sep 2019
Cited by 4 | Viewed by 2085
Abstract
Many real-life problems are well represented only by sets which allow repetition(s), such as the multiset. Although not limited to the following, such cases may arise in a database query, chemical structures and computer programming. The set of roots of a polynomial, say [...] Read more.
Many real-life problems are well represented only by sets which allow repetition(s), such as the multiset. Although not limited to the following, such cases may arise in a database query, chemical structures and computer programming. The set of roots of a polynomial, say f ( x ) , has been found to correspond to a multiset, say F. If f ( x ) and g ( x ) are polynomials whose sets of roots respectively correspond to the multisets F ( x ) and G ( x ) , the set of roots of their product, f ( x ) g ( x ) , corresponds to the multiset F ⊎ G , which is the sum of multisets F and G. In this paper, some properties of the algebraic sum of multisets ⊎ and some results on selection are established. Also, the count function of the image of any function on Dedekind multisets is defined and some of its properties are established. Some applications of these multisets are also given. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
12 pages, 296 KiB  
Article
Series of Semihypergroups of Time-Varying Artificial Neurons and Related Hyperstructures
by Jan Chvalina and Bedřich Smetana
Symmetry 2019, 11(7), 927; https://doi.org/10.3390/sym11070927 - 16 Jul 2019
Cited by 4 | Viewed by 2435
Abstract
Detailed analysis of the function of multilayer perceptron (MLP) and its neurons together with the use of time-varying neurons allowed the authors to find an analogy with the use of structures of linear differential operators. This procedure allowed the construction of a group [...] Read more.
Detailed analysis of the function of multilayer perceptron (MLP) and its neurons together with the use of time-varying neurons allowed the authors to find an analogy with the use of structures of linear differential operators. This procedure allowed the construction of a group and a hypergroup of artificial neurons. In this article, focusing on semihyperstructures and using the above described procedure, the authors bring new insights into structures and hyperstructures of artificial neurons and their possible symmetric relations. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
16 pages, 418 KiB  
Article
Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
by Michal Novák, Štepán Křehlík and Kyriakos Ovaliadis
Symmetry 2019, 11(6), 734; https://doi.org/10.3390/sym11060734 - 29 May 2019
Cited by 8 | Viewed by 3019
Abstract
In our paper we discuss how elements of algebraic hyperstructure theory can be used in the context of underwater wireless sensor networks (UWSN). We present a mathematical model which makes use of the fact that when deploying nodes or operating the network we, [...] Read more.
In our paper we discuss how elements of algebraic hyperstructure theory can be used in the context of underwater wireless sensor networks (UWSN). We present a mathematical model which makes use of the fact that when deploying nodes or operating the network we, from the mathematical point of view, regard an operation (or a hyperoperation) and a binary relation. In this part of the paper we relate our context to already existing topics of the algebraic hyperstructure theory such as quasi-order hypergroups, E L -hyperstructures, or ordered hyperstructures. Furthermore, we make use of the theory of quasi-automata (or rather, semiautomata) to relate the process of UWSN data aggregation to the existing algebraic theory of quasi-automata and their hyperstructure generalization. We show that the process of data aggregation can be seen as an automaton, or rather its hyperstructure generalization, with states representing stages of the data aggregation process of cluster protocols and describing available/used memory capacity of the network. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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37 pages, 1705 KiB  
Article
A Novel Description on Edge-Regular q-Rung Picture Fuzzy Graphs with Application
by Muhammad Akram, Amna Habib and Ali N. A. Koam
Symmetry 2019, 11(4), 489; https://doi.org/10.3390/sym11040489 - 4 Apr 2019
Cited by 26 | Viewed by 3295
Abstract
Picture fuzzy model is a generalized structure of intuitionistic fuzzy model in the sense that it not only assigns the membership and nonmembership values in the form of orthopair ( μ , ν ) to an element, but it assigns a triplet [...] Read more.
Picture fuzzy model is a generalized structure of intuitionistic fuzzy model in the sense that it not only assigns the membership and nonmembership values in the form of orthopair ( μ , ν ) to an element, but it assigns a triplet ( μ , η , ν ) , where η denotes the neutral degree and the difference π = 1 − ( μ + η + ν ) indicates the degree of refusal. The q-rung picture fuzzy set( q -RPFS) provides a wide formal mathematical sketch in which uncertain and vague conceptual phenomenon can be precisely and rigorously studied because of its distinctive quality of vast representation space of acceptable triplets. This paper discusses some properties including edge regularity, total edge regularity and perfect edge regularity of q-rung picture fuzzy graphs (q-RPFGs). The work introduces and investigates these properties for square q-RPFGs and q-RPF line graphs. Furthermore, this study characterizes how regularity and edge regularity of q-RPFGs structurally relate. In addition, it presents the concept of ego-networks to extract knowledge from large social networks under q-rung picture fuzzy environment with algorithm. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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22 pages, 1447 KiB  
Article
An m-Polar Fuzzy Hypergraph Model of Granular Computing
by Anam Luqman, Muhammad Akram and Ali N.A. Koam
Symmetry 2019, 11(4), 483; https://doi.org/10.3390/sym11040483 - 3 Apr 2019
Cited by 14 | Viewed by 2375
Abstract
An m-polar fuzzy model plays a vital role in modeling of real-world problems that involve multi-attribute, multi-polar information and uncertainty. The m-polar fuzzy models give increasing precision and flexibility to the system as compared to the fuzzy and bipolar fuzzy models. [...] Read more.
An m-polar fuzzy model plays a vital role in modeling of real-world problems that involve multi-attribute, multi-polar information and uncertainty. The m-polar fuzzy models give increasing precision and flexibility to the system as compared to the fuzzy and bipolar fuzzy models. An m-polar fuzzy set assigns the membership degree to an object belonging to [ 0 , 1 ] m describing the m distinct attributes of that element. Granular computing deals with representing and processing information in the form of information granules. These information granules are collections of elements combined together due to their similarity and functional/physical adjacency. In this paper, we illustrate the formation of granular structures using m-polar fuzzy hypergraphs and level hypergraphs. Further, we define m-polar fuzzy hierarchical quotient space structures. The mappings between the m-polar fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of a universal set is more efficient through m-polar fuzzy hypergraphs as compared to crisp hypergraphs. We also present some examples and a real-world problem to signify the validity of our proposed model. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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12 pages, 276 KiB  
Article
Intuitionistic Fuzzy Soft Hyper BCK Algebras
by Xiaolong Xin, Rajab Ali Borzooei, Mahmood Bakhshi and Young Bae Jun
Symmetry 2019, 11(3), 399; https://doi.org/10.3390/sym11030399 - 19 Mar 2019
Cited by 8 | Viewed by 2523
Abstract
Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision-making problem. Maji et al. also introduced the notion of intuitionistic fuzzy soft sets in [...] Read more.
Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision-making problem. Maji et al. also introduced the notion of intuitionistic fuzzy soft sets in the paper [P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 9 (2001), no. 3, 677–692]. The aim of this manuscript is to apply the notion of intuitionistic fuzzy soft set to hyper BCK algebras. The notions of intuitionistic fuzzy soft hyper BCK ideal, intuitionistic fuzzy soft weak hyper BCK ideal, intuitionistic fuzzy soft s-weak hyper BCK-ideal and intuitionistic fuzzy soft strong hyper BCK-ideal are introduced, and related properties and relations are investigated. Characterizations of intuitionistic fuzzy soft (weak) hyper BCK ideal are considered. Conditions for an intuitionistic fuzzy soft weak hyper BCK ideal to be an intuitionistic fuzzy soft s-weak hyper BCK ideal are provided. Conditions for an intuitionistic fuzzy soft set to be an intuitionistic fuzzy soft strong hyper BCK ideal are given. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
10 pages, 265 KiB  
Article
Breakable Semihypergroups
by Dariush Heidari and Irina Cristea
Symmetry 2019, 11(1), 100; https://doi.org/10.3390/sym11010100 - 16 Jan 2019
Cited by 11 | Viewed by 3598
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)
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