General Algebraic Structures

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 November 2019) | Viewed by 14244

Special Issue Editors

Department of Mathematics, Research Institute of Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea
Interests: BCK/BCI-algebras; fuzzy algebras; groupoid theory (= general algebraic structures); semirings; Fibonacci numbers
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Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
Interests: quasigroups; BCK-algebras; fuzzy algebras
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Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Interests: algebraic structures; algebraic logic; fuzzy sets and applications
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran
Interests: algebra; fuzzy logic
Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China
Interests: artificial intelligence; algebra; fuzzy logic; rough sets theory
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, Hubei University for Nationalities, Enshi 445000, Hubei, China
Interests: fuzzy (logical) algebra; rough set; soft set; granular computing; decision-making

Special Issue Information

Dear Colleagues,

R. H. Bruck's famous book, A Survey of Binary Systems, mainly discussed loops and semigroups. It is necessary to organize several groupoids dealing with various axioms. The area of “general algebraic structures” contains several groupoids, i.e., sets with a single binary operation satisfying some conditions. The well-known topics, e.g., groups, semigroups, monoids, BCK/BCI-algebra, etc., are not included in this area. It contains lots of generalized algebraic structures of these well-known mathematical structures simply by deleting/weakening/changing the axioms.

The notion of BCK/BCI-algebra was introduced by K. Iséki in 1965, alongside its generalizations, e.g., BCH-algebra, BH-algebra, BZ-algebra, BCC-algebra, pre-BCK-algebra and near-BCK-algebra. The notion of d-algebra was introduced by deleting two complicated axioms from BCK-algebra. After that, many algebraic structures appeared, e.g., B-, BE-, BF-, BG-, BM-, BN-, BO-, BP-, C-, CI-, Q, QS-algebra. Other important algebraic structures are implicative algebra, positive implication algebra, selective groupoids, pogroupoids, weak-zero groupoids, etc. These algebras have some inter-relationships with each other, and have more rooms for further research.

This Special Issue of Mathematics (MDPI) will provide an opportunity to construct an area of general algebraic structures, and will encourage researchers to publish their investigations in this area.

We will consider any paper in the area of general algebraic structures for possible publication. We will exclude papers on well-known algebras, e.g., groups, rings, fields, semigroups, lattices and posets, BCK/BCI-algebra, fuzzy algebraic theory, etc.

Prof. Dr. Hee Sik Kim
Prof. Dr. Wiesław A. Dudek
Prof. Dr. Arsham Borumand Saeid
Prof. Dr. Rajab Borzooei
Prof. Dr. Xiaohong Zhang
Prof. Dr. Jianming Zhan
Guest Editors

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Keywords

  • general algebraic structures
  • sets with a single binary operations (groupoids)
  • generalized groups
  • implicative algebra
  • generalized BCK/BCI-algebra

Published Papers (7 papers)

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Research

23 pages, 369 KiB  
Article
On Cocyclic Hadamard Matrices over Goethals-Seidel Loops
by Víctor Álvarez, José Andrés Armario, Raúl M. Falcón, María Dolores Frau, Félix Gudiel, María Belén Güemes and Amparo Osuna
Mathematics 2020, 8(1), 24; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010024 - 20 Dec 2019
Cited by 5 | Viewed by 1873
Abstract
About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended [...] Read more.
About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended to define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup, and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type (which is known not to be cocyclically developed over any group) is actually pseudococyclically developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices, the usual cocyclic Hadamard test is unexpectedly applicable. Full article
(This article belongs to the Special Issue General Algebraic Structures)
19 pages, 312 KiB  
Article
Foldness of Bipolar Fuzzy Sets and Its Application in BCK/BCI-Algebras
by Young Bae Jun and Seok-Zun Song
Mathematics 2019, 7(11), 1036; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111036 - 03 Nov 2019
Cited by 3 | Viewed by 1894
Abstract
Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for [...] Read more.
Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for example, already observed or experienced, indicates what is guaranteed to be possible, and negative information indicates that it is impossible, prohibited, or certainly false. The purpose of this paper is to apply the bipolar fuzzy set to BCK/BCI-algebras. The notion of (translated) k-fold bipolar fuzzy sets is introduced, and its application in BCK/BCI-algebras is discussed. The concepts of k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are introduced, and related properties are investigated. Characterizations of k-fold bipolar fuzzy subalgebra/ideal are considered, and relations between k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are displayed. Extension of k-fold bipolar fuzzy subalgebra is discussed. Full article
(This article belongs to the Special Issue General Algebraic Structures)
18 pages, 805 KiB  
Article
Extending Structures for Lie 2-Algebras
by Yan Tan and Zhixiang Wu
Mathematics 2019, 7(6), 556; https://0-doi-org.brum.beds.ac.uk/10.3390/math7060556 - 18 Jun 2019
Cited by 1 | Viewed by 2000
Abstract
The extending structures problem for strict Lie 2-algebras is studied. To provide the theoretical answer to this problem, this paper introduces the unified product of a given strict Lie 2-algebra g and 2-vector space V. The unified product includes some interesting products [...] Read more.
The extending structures problem for strict Lie 2-algebras is studied. To provide the theoretical answer to this problem, this paper introduces the unified product of a given strict Lie 2-algebra g and 2-vector space V. The unified product includes some interesting products such as semi-direct product, crossed product, and bicrossed product. The paper focuses on crossed and bicrossed products, which give the answer to the extension problem and factorization problem, respectively. Full article
(This article belongs to the Special Issue General Algebraic Structures)
14 pages, 274 KiB  
Article
Graphs Based on Hoop Algebras
by Mona Aaly Kologani, Rajab Ali Borzooei and Hee Sik Kim
Mathematics 2019, 7(4), 362; https://0-doi-org.brum.beds.ac.uk/10.3390/math7040362 - 21 Apr 2019
Cited by 1 | Viewed by 2432
Abstract
In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using [...] Read more.
In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and both complete and a tree under some conditions. Full article
(This article belongs to the Special Issue General Algebraic Structures)
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7 pages, 223 KiB  
Article
On Pre-Commutative Algebras
by Hee Sik Kim, Joseph Neggers and Sun Shin Ahn
Mathematics 2019, 7(4), 336; https://0-doi-org.brum.beds.ac.uk/10.3390/math7040336 - 08 Apr 2019
Cited by 4 | Viewed by 1777
Abstract
In this paper, we introduce the notions of generalized commutative laws in algebras, and investigate their relations by using Smarandache disjointness. Moreover, we show that every pre-commutative B C K -algebra is bounded. Full article
(This article belongs to the Special Issue General Algebraic Structures)
13 pages, 1204 KiB  
Article
The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops
by Xiaoying Wu and Xiaohong Zhang
Mathematics 2019, 7(3), 268; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030268 - 15 Mar 2019
Cited by 22 | Viewed by 2083
Abstract
In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every [...] Read more.
In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups. Full article
(This article belongs to the Special Issue General Algebraic Structures)
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7 pages, 236 KiB  
Article
Some Special Elements and Pseudo Inverse Functions in Groupoids
by Yong Lin Liu, Hee Sik Kim and Joseph Neggers
Mathematics 2019, 7(2), 173; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020173 - 14 Feb 2019
Cited by 4 | Viewed by 1548
Abstract
In this paper, we consider a theory of elements u of a groupoid ( X , ) that are associated with certain functions u ^ : X X , pseudo-inverse functions, which are generalizations of the inverses associated with units of [...] Read more.
In this paper, we consider a theory of elements u of a groupoid ( X , ) that are associated with certain functions u ^ : X X , pseudo-inverse functions, which are generalizations of the inverses associated with units of groupoids with identity elements. If classifying the elements u as special of one of twelve types, then it is possible to do a rather detailed analysis of certain cases, leftoids, rightoids and linear groupoids included, which demonstrates that it is possible to develop a successful theory and that a good deal of information has already been obtained with much more possible in the future. Full article
(This article belongs to the Special Issue General Algebraic Structures)
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