Neutrosophic Sets and Systems. Book Series, vol. 33, 2020

2020·13.5 MB·English

by  Smarandache, Florentin

Neutrosophic Sets and Systems
Book Series, Vol. 33, 2020

<A> <neutA> <antiA>
Editors
Florentin Smarandache . Mohamed Abdel-Basset
ISBN 978-1-59973-669-3
Neutrosophic Science
International Association (NSIA)
ISBN 978-1-59973-669-3
Neutrosophic
Sets
and
Systems
An International Book Series in Information Science and Engineering
University of New Mexico
ISBN 978-1-59973-669-3
University of New Mexico
Neutrosophic Sets and Systems
An International Book Series in Information Science and Engineering
Copyright Notice
Copyright @ Neutrosophics Sets and Systems
All rights reserved. The authors of the articles do hereby grant Neutrosophic Sets and Systems non-exclusive,
worldwide, royalty-free license to publish and distribute the articles in accordance with the Budapest Open Ini-
tiative: this means that electronic copying, distribution and printing of both full-size version of the journal and the
individual papers published therein for non-commercial, academic or individual use can be made by any user
without permission or charge. The authors of the articles published in Neutrosophic Sets and Systems retain their
rights to use this journal as a whole or any part of it in any other publications and in any way they see fit. Any
part of Neutrosophic Sets and Systems howsoever used in other publications must include an appropriate citation of
this journal.
Information for Authors and Subscribers
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set,
neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as
the neutrosophic structures developed in algebra, geometry, topology, etc.
The submitted papers should be professional, in good English, containing a brief review of a problem and obtained results.
Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their inter
actions with different ideational spectra.
This theory considers every notion or idea <A> together with its opposite or negation <antiA> and with their spectrum of
neutralities <neutA> in between them (i.e. notions or ideas supporting neither <A> nor <antiA>). The <neutA> and <antiA>
ideas together are referred to as <nonA>.
Neutrosophy is a generalization of Hegel's dialectics (the last one is based on <A> and <antiA> only).
According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of
equilibrium.
In a classical way <A>, <neutA>, <antiA> are disjoint two by two. But, since in many cases the borders between notions are
vague, imprecise, Sorites, it is possible that <A>, <neutA>, <antiA> (and <nonA> of course) have common parts two by two, or
even all three of them as well.
Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of
intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth (T), a
degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[.
Neutrosophic Probability is a generalization of the classical probability and imprecise probability.
Neutrosophic Statistics is a generalization of the classical statistics.
What distinguishes the neutrosophics from other fields is the <neutA>, which means neither <A> nor <antiA>. <neutA>,
which of course depends on <A>, can be indeterminacy, neutrality, tie game, unknown, contradiction, igno
rance, imprecision, etc.
All submissions should be designed in MS Word format using our template file:
http://fs.unm.edu/NSS/NSS-paper-template.doc.
A variety of scientific books in many languages can be downloaded freely from the Digital Library of Science:
http://fs.unm.edu/ScienceLibrary.htm.
To submit a paper, mail the file to the Editor-in-Chief. To order printed issues, contact the Editor-in-Chief. This journal is
non-commercial, academic edition. It is printed from private donations.
Information about the neutrosophics you get from the UNM website:
http://fs.unm.edu/neutrosophy.htm. The
home page of the journal is accessed on
http://fs.unm.edu/NSS.
The Educational Publisher Inc.
1313 Chesapeake Ave. Columbus, Ohio 43212, USA
Copyright © Neutrosophic Sets and Systems, 2020
Editorial Board ISBN 978-1-59973-669-3
University of New Mexico
Editors-in-Chief
Prof. Dr. Florentin Smarandache, Postdoc, Department of Mathematics, University of New Mexico, Gallup,
NM 87301, USA, Email: [email protected].
Dr. Mohamed Abdel-Basset, Faculty of Computers and Informatics, Zagazig University, Egypt,
Email: [email protected].
Associate Editors
Dr. Said Broumi, University of Hassan II, Casablanca, Morocco, Email: [email protected].
Prof. Dr. W. B. Vasantha Kandasamy, School of Computer Science and Engineering, VIT, Vellore 632014,
India, Email: [email protected].
Dr. Huda E. Khalid, University of Telafer, College of Basic Education, Telafer - Mosul, Iraq,
Email: [email protected].
Prof. Dr. Xiaohong Zhang, Department of Mathematics, Shaanxi University of Science &Technology, Xian
710021, China, Email: [email protected].
Editors
Yanhui Guo, University of Illinois at Springfield, One Mogadishu, Somalia, Email: [email protected].
University Plaza, Springfield, IL 62703, United States, NoohBany Muhammad, American University of Kuwait,
Email: [email protected]. Kuwait, Email: [email protected].
Le Hoang Son, VNU Univ. of Science, Vietnam National Univ. Soheyb Milles, Laboratory of Pure and Applied Mathematics,
Hanoi, Vietnam, Email: [email protected]. University of Msila, Algeria, Email: [email protected].
A. A. Salama, Faculty of Science, Port Said University, Egypt, Pattathal Vijayakumar Arun, College of Science and
Email: [email protected]. Technology, Phuentsholing, Bhutan,
Young Bae Jun, Gyeongsang National University, South Email: [email protected]
Korea, Email: [email protected]. Xindong Peng, School of Information Science and
Yo-Ping Huang, Department of Computer Science and Engineering, Shaoguan University, Shaoguan 512005, China,
Information, Engineering National Taipei University, New Email: [email protected].
Taipei City, Taiwan, Email: [email protected]. Xiao-Zhi Gao, School of Computing, University of Eastern
Vakkas Ulucay, Gaziantep University, Gaziantep, Turkey, Finland, FI-70211 Kuopio, Finland, [email protected].
Email: [email protected]. Madad Khan, Comsats Institute of Information Technology,
Peide Liu, Shandong University of Finance and Economics, Abbottabad, Pakistan, Email: [email protected].
China, Email: [email protected]. Dmitri Rabounski and Larissa Borissova, independent
Jun Ye, Department of Electrical and Information Engineering, researchers,
Shaoxing University, 508 Huancheng West Road, Shaoxing Emails: [email protected], [email protected].
312000, China; Email: [email protected]. Selcuk Topal, Mathematics Department, Bitlis Eren University,
Mehmet Şahin, Department of Mathematics, Gaziantep Turkey, Email: [email protected].
University, Gaziantep 27310, Turkey, Ibrahim El-henawy, Faculty of Computers and Informatics,
Email: [email protected]. Zagazig University, Egypt, Email: [email protected].
Muhammad Aslam & Mohammed Alshumrani, King A. A. A. Agboola, Federal University of Agriculture,
Abdulaziz Univ., Jeddah, Saudi Arabia, Abeokuta, Nigeria, Email: [email protected].
Emails [email protected], [email protected]. Abduallah Gamal, Faculty of Computers and Informatics,
Mutaz Mohammad, Department of Mathematics, Zayed Zagazig University, Egypt, Email: [email protected].
University, Abu Dhabi 144534, United Arab Emirates. Luu Quoc Dat, Univ. of Economics and Business, Vietnam
Email: [email protected]. National Univ., Hanoi, Vietnam, Email: [email protected].
Abdullahi Mohamud Sharif, Department of Computer Maikel Leyva-Vazquez, Universidad de Guayaquil, Ecuador,
Science, University of Somalia, Makka Al-mukarrama Road, Email: [email protected].
Copyright © Neutrosophic Sets and Systems, 2020
Editorial Board ISBN 978-1-59973-669-3
University of New Mexico
Tula Carola Sanchez Garcia, Facultad de Educacion de la Baath University, Homs, Syria, Email: riad-
Universidad Nacional Mayor de San Marcos, Lima, Peru, [email protected].
Email: [email protected]. Faruk Karaaslan, Çankırı Karatekin University, Çankırı,
Muhammad Akram, University of the Punjab, New Campus, Turkey, Email: [email protected].
Lahore, Pakistan, Email: [email protected]. Suriana Alias, Universiti Teknologi MARA (UiTM) Kelantan,
Irfan Deli, Muallim Rifat Faculty of Education, Kilis 7 Aralik Campus Machang, 18500 Machang, Kelantan, Malaysia,
University, Turkey, Email: [email protected]. Email: [email protected].
Ridvan Sahin, Department of Mathematics, Faculty of Science, Arsham Borumand Saeid, Dept. of Pure Mathematics, Faculty
Ataturk University, Erzurum 25240, Turkey, of Mathematics and Computer, Shahid Bahonar University of
Email: [email protected]. Kerman, Kerman, Iran, Email: [email protected].
Ibrahim M. Hezam, Department of computer, Faculty of Mohammad Hamidi, Department of Mathematics, Payame
Education, Ibb University, Ibb City, Yemen, Noor University (PNU), Tehran, Iran.
Email: [email protected]. Email: [email protected].
Aiyared Iampan, Department of Mathematics, School of Lemnaouar Zedam, Department of Mathematics, Faculty of
Science, University of Phayao, Phayao 56000, Thailand, Mathematics and Informatics, University Mohamed Boudiaf,
Email: [email protected]. M’sila, Algeria, Email: [email protected].
Ameirys Betancourt-Vázquez, 1 Instituto Superior Politécnico M. Al Tahan, Department of Mathematics, Lebanese
de Tecnologias e Ciências (ISPTEC), Luanda, International University, Bekaa, Lebanon,
Angola, Email: [email protected]. Email: [email protected].
Karina Pérez-Teruel, Universidad Abierta para Adultos R. A. Borzooei, Department of Mathematics, Shahid Beheshti
(UAPA), Santiago de los Caballeros, República Dominicana, University, Tehran, Iran, [email protected].
E-mail: [email protected]. Sudan Jha, Pokhara University, Kathmandu, Nepal,
Neilys González Benítez, Centro Meteorológico Pinar del Río, Email: [email protected].
Cuba, Email: [email protected]. Mujahid Abbas, Department of Mathematics and Applied
Jesus Estupinan Ricardo, Centro de Estudios para la Calidad Mathematics, University of Pretoria Hatfield 002, Pretoria,
Educativa y la Investigation Cinetifica, Toluca, Mexico, South Africa, Email: [email protected].
Email: [email protected]. Željko Stević, Faculty of Transport and Traffic
Victor Christianto, Malang Institute of Agriculture (IPM), Engineering Doboj, University of East Sarajevo, Lukavica, East
Malang, Indonesia, Email: [email protected]. Sarajevo, Bosnia and Herzegovina,
Wadei Al-Omeri, Department of Mathematics, Al-Balqa Email: [email protected].
Applied University, Salt 19117, Jordan, Michael Gr. Voskoglou, Mathematical Sciences School of
Email: [email protected]. Technological Applications, Graduate Technological
Ganeshsree Selvachandran, UCSI University, Jalan Menara Educational Institute of Western Greece, Patras, Greece,
Gading, Kuala Lumpur, Malaysia, Email: [email protected].
Email: [email protected]. Angelo de Oliveira, Ciencia da Computacao, Universidade
Ilanthenral Kandasamy, School of Computer Science Federal de Rondonia, Porto Velho - Rondonia, Brazil,
and Engineering (SCOPE), Vellore Institute of Technology Email: [email protected].
(VIT), Vellore 632014, Tamil Nadu, India, Valeri Kroumov, Okayama University of Science, Japan,
Email: [email protected] Email: [email protected].
Kul Hur, Wonkwang University, Iksan, Jeollabukdo, South Rafael Rojas, Universidad Industrial de Santander,
Korea, Email: [email protected]. Bucaramanga, Colombia,
Kemale Veliyeva & Sadi Bayramov, Department of Algebra Email: [email protected].
and Geometry, Baku State University, 23 Z. Khalilov Str., Walid Abdelfattah, Faculty of Law, Economics and
AZ1148, Baku, Azerbaijan, Email: [email protected], Management, Jendouba, Tunisia, Email:
Email: [email protected]. [email protected].
Irma Makharadze & Tariel Khvedelidze, Ivane Javakhishvili Galina Ilieva, Paisii Hilendarski, University of Plovdiv, 4000
Tbilisi State University, Faculty of Exact and Natural Sciences, Plovdiv, Bulgaria, Email: [email protected].
Tbilisi, Georgia. Paweł Pławiak, Institute of Teleinformatics, Cracow
Inayatur Rehman, College of Arts and Applied Sciences, University of Technology, Warszawska 24 st., F-5, 31-155
Dhofar University Salalah, Oman,Email: [email protected]. Krakow, Poland, Email: [email protected].
Riad K. Al-Hamido, Math Department, College of Science, Al- E. K. Zavadskas, Vilnius Gediminas Technical University,
Copyright © Neutrosophic Sets and Systems, 2020
Editorial Board ISBN 978-1-59973-669-3
University of New Mexico
Vilnius, Lithuania, Email: [email protected]. Email: [email protected].
Darjan Karabasevic, University Business Academy, Novi Sad, Tieta Putri, College of Engineering Department of Computer
Serbia, Email: [email protected]. Science and Software Engineering, University of Canterbury,
Dragisa Stanujkic, Technical Faculty in Bor, University of Christchurch, New Zeeland.
Belgrade, Bor, Serbia, Email: [email protected]. Mumtaz Ali, Deakin University, Victoria 3125, Australia,
Luige Vladareanu, Romanian Academy, Bucharest, Romania, Email: [email protected].
Email: [email protected]. Sergey Gorbachev, National Research Tomsk State
Hashem Bordbar, Center for Information Technologies and University, 634050 Tomsk, Russia, Email: [email protected].
Applied Mathematics, University of Nova Gorica, Slovenia, E- Willem K. M. Brauers, Faculty of Applied Economics,
mail: [email protected]. University of Antwerp, Antwerp, Belgium,
Mihaela Colhon, University of Craiova, Computer Science Email: [email protected].
Department, Craiova, Romania, M. Ganster, Graz University of Technology, Graz, Austria,
Emails: [email protected]. Email: [email protected].
Philippe Schweizer, Independant Researcher, Av. de Lonay Umberto Rivieccio, Department of Philosophy, University of
11, 1110 Morges, Switzerland, Email: [email protected]. Genoa, Italy, Email: [email protected].
Saeid Jafari, College of Vestsjaelland South, Slagelse, F. Gallego Lupiaňez, Universidad Complutense, Madrid,
Denmark, Email: [email protected]. Spain, Email: [email protected].
Fernando A. F. Ferreira, ISCTE Business School, BRU-IUL, Francisco Chiclana, School of Computer Science and
University Institute of Lisbon, Avenida das Forças Armadas, Informatics, De Montfort University, The Gateway, Leicester,
1649-026 Lisbon, Portugal, LE1 9BH, United Kingdom, Email: [email protected].
Email: [email protected]. Jean Dezert, ONERA, Chemin de la Huniere, 91120 Palaiseau,
Julio J. Valdés, National Research Council Canada, M-50, 1200 France, Email: [email protected].
Montreal Road, Ottawa, Ontario K1A 0R6, Canada,
Copyright © Neutrosophic Sets and Systems, 2020
ISBN 978-1-59973-669-3
University of New Mexico
Contents
Atena Tahmasbpour Meikola, Eight Kinds of Graphs of BCK-algebras Based on Ideal and Dual
Ideal…………………………………………………………………………………………………......1
Arif Mehmood, Wadood Ullah, Said Broumi, Muhammad Imran Khan, Humera Qureshi, Muhammad
Ibrar Abbas, Humaira Kalsoom and Fawad Nadeem, Neutrosophic Soft Structures…………………..23
T.Chalapathi and L. Madhavi, Neutrosophic Boolean Rings..………………………………………....59
R. Suresh and S. Palaniammal, Neutrosophic Weakly Generalized open and Closed Sets...………...67
Nivetha Martin and Florentin Smarandache, Concentric Plithogenic Hypergraph based on Plithogenic
Hypersoft sets – A Novel Outlook………………………………...…………………………………78
S. A. Edalatpanah, Systems of Neutrosophic Linear Equations ……………………………………..92
Rafif Alhabib and A. A. Salama, The Neutrosophic Time Series- Study Its Models (Linear -
Logarithmic) and test the Coefficients Significance of Its linear model................................................105
Memet Şahin, Abdullah Kargın and Murat Yücel, Neutrosophic Triplet Partial g - Metric Space…116
Dimacha Dwibrang Mwchahary and Bhimraj Basumatary, A Note on Neutrosophic Bitopological
Spaces………………………………………………………………………………………………...135
R. S. Alghamdi and N. O. Alshehri, Neutrosophic Fuzzy Soft BCK-submodules………………….146
Shigui Du, Rui Yong and Jun Ye, Subtraction Operational Aggregation Operators of Simplified
Neutrosophic Numbers and Their Multi-Attribute Decision Making Approach………………..158
Vasantha Kandasamy, Ilanthenral Kandasamy and Florentin Smarandache, Neutrosophic Quadruple
Algebraic Codes over Z2 and their properties………………………………………………………170
Muhammad Saeed, Muhammad Saqlain, Asad Mehmood, Khushbakht Naseer and Sonia Yaqoob, Multi-
Polar Neutrosophic Soft Sets with Application in Medical Diagnosis and Decision-Making…..184
Sudipta Gayen, Florentin Smarandache, Sripati Jha, Manoranjan Kumar Singh, Said Broumi and Ranjan
Kumar, Introduction to Plithogenic Hypersoft Subgroup………………………………………….209
Huseyin kamaci, Neutrosophic Cubic Hamacher Aggregation Operators and Their Applications in
Decision Making ……………………………………………………………………………………235
Shawkat Alkhazaleh, Plithogenic Soft Set …………………………………………………………...257
Anas Al-Masarwah and Abd Ghafur Ahmad, Structures on Doubt Neutrosophic Ideals of BCK/BCI-
Algebras under (S, T)-Norms…………………………………………………………………….....276
Copyright © Neutrosophic Sets and Systems, 2020
ISBN 978-1-59973-669-3
University of New Mexico
Florentin Smarandache, Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-
SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-)
HyperAlgebra……………………………………………………………………………….…….…291
Memet Şahin, Abdullah Kargın and Merve Sena Uz, Neutrosophic Triplet Partial Bipolar Metric
Spaces ……………………………………………………………………………………………..…298
Akbar Rezaei and Florentin Smarandache, The Neutrosophic Triplet of BI-algebras ……………..315
Mohamed Grida, Rehab Mohamed and Abdelnaser H. Zaid, A Novel Plithogenic MCDM Framework
for Evaluating the Performance of IoT Based Supply Chain……………………………………...323
Address:
“Neutrosophic Sets and Systems”
(An International Book Series in Information Science and Engineering )
Department of Mathematics and Science
University of New Mexico
705 Gurley Avenue Gallup, NM 87301, USA
E-mail: [email protected]
Home page: http://fs.unm.edu/NSS
Copyright © Neutrosophic Sets and Systems, 2020
Neutrosophic Sets and Systems, Vol. 33, 2020
University of New Mexico
Eight Kinds of Graphs of BCK-algebras Based on Ideal and Dual
Ideal
Atena Tahmasbpour Meikola 1,*
1 Department of mathematics. Islamic azad university, central Tehran branch, Iran;
*Correspondence: [email protected].
Abstract: In this paper, at first we introduce the concepts of ideal-annihilator, dual ideal- annihilator,
right- ideal- annihilator, left- ideal- annihilator, right- dual ideal- annihilator, left- dual ideal-
annihilator. Then by using of these concepts, we constructed six new types of graphs in a bounded
BCK-algebra based on ideal and dual ideal which are denoted by
, and , respectively. Then basic properties of graph
theory such as connectivity, regularity, and planarity on the structure of these graphs are
investigated. Finally, by utilizing of binary operations and , we construct graphs and
, respectively, some their interesting properties are presented.
Keywords: BCK- algebra; Diameter; Chromatic number; Euler graph.
1.Introduction
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra in
various combinatorial contexts and vice versa. Associating a graph to an algebraic structure is a
research subject in this area and has attracted considerable attention. In fact, the research in this
subject aims at exposing the relationship between algebra and graph theory and at advancing the
application of one to the other. The story goes back to a paper of Beck [4] in 1998, where he
introduced the idea of a zero-divisor graph of a commutative ring with identity. He defined
to be the graph whose vertices are elements of and in which two vertices and are adjacent if
and only if . Recently, Halas and Jukl in [7] introduced the zero divisor graphs of posets. The
study of the zero-divisor graphs of posets was then continued by Xue and Liu in [23], Maimani in
[12]. More recently, a different method of associating a zero-divisor graph to a poset was
proposed by Lu and Wu in [11]. In this paper, we deal with zero-divisor graphs of BCI/BCK-algebras
Atena Tahmasbpour Meikola, Eight Kinds of Graphs of BCK-algebras Based on Ideal and Dual Ideal
Neutrosophic Sets and Systems, Vol. 33, 2020 2

based on ideal and dual ideal. Imai and Iseki [8] in 1966 introduced the notion of BCK- algebra. In
the same year, Iseki [9] introduced BCI-algebra as a super class of the class of BCK- algebras. Jun and
Lee [10] defined the concept of associated graph of BCK- algebra and verified some properties of this
graph. Zahiri and Borzooei [24] associated a new graph to a BCI-algebra which is denoted by
, this definition is based on branches of , Tahmasbpour in [16, 17] studied chordality of graph
defined by Zahiri and Borzooei and introduced four types of graphs of BCK- algebras which are
constructed by equivalence classes determined by ideal and dual ideal . Also, Tahmasbpour in
[18, 21] introduced two new graphs of lattice implication algebras based on LI-ideal. Further,
Tahmasbpour in [19, 20] introduced two new graphs of BCK-algebras based on fuzzy ideal and
fuzzy dual ideal , two new graphs of lattice implication algebras based on fuzzy filter and
fuzzy LI- ideal . Futhermore, Tahmasbpour in [22] introduced twelve kinds of graphs of lattice
implication algebras based on filter and LI- ideal. This paper is divided into six parts.
In Section 2, we recall some concepts of graph theory such as connected graph, planar graph,
outerplanar graph, Eulerian graph, and chromatic number, among others.
Section 3, is an introduction to a general theory of BCK- algebras. We will first give the notions of
BCI/BCK- algebras, and investigate their elementary and fundamental properties , and then deal
with a number of basic concepts, such as ideal, and dual ideal, among others.
In Section 4, inspired by ideas from Behzadi et al. [5], we study the graphs of BCK-algebras which
are constructed from ideal-annihilator and dual ideal-annihilator, denoted by ,
respectively.
In Sction 5, inspired by ideas from Behzadi et al. [5], we study the graphs of BCK- algebras which are
constructed from right- ideal- annihilator, left- ideal- annihilator, right- dual ideal- annihilator, left-
dual ideal- annihilator, denoted by , , respectively.
In Section 6, inspired by ideas from Alizadeh et al. [3], we introduce the associated graphs
and which are constructed from binary operations and , respectively.

2. Preliminaries of graph theory
In this section, for convenience of the reader, we recall some definitions and notations concerning
graphs and posets for later use.

Atena Tahmasbpour Meikola, Eight Kinds of Graphs of BCK-algebras Based on Ideal and Dual Ideal