Table Of ContentNew Development
of Neutrosophic
Probability, Neutrosophic
Statistics, Neutrosophic
Algebraic Structures,
and Neutrosophic &
Plithogenic Optimizations
Edited by
Florentin Smarandache and Yanhui Guo
Printed Edition of the Special Issue Published in Symmetry
www.mdpi.com/journal/symmetry
New Development of Neutrosophic
Probability, Neutrosophic Statistics,
Neutrosophic Algebraic Structures,
and Neutrosophic Plithogenic
Optimizations
New Development of Neutrosophic
Probability, Neutrosophic Statistics,
Neutrosophic Algebraic Structures,
and Neutrosophic Plithogenic
Optimizations
Editors
Florentin Smarandache
Yanhui Guo
MDPI•Basel•Beijing•Wuhan•Barcelona•Belgrade•Manchester•Tokyo•Cluj•Tianjin
Editors
FlorentinSmarandache YanhuiGuo
Mathematics DepartmentofComputer
UniversityofNewMexico Science
Gallup UniversityofIllinois
UnitedStates Springfield
UnitedStates
EditorialOffice
MDPI
St. Alban-Anlage66
4052Basel,Switzerland
ThisisareprintofarticlesfromtheSpecialIssuepublishedonlineintheopenaccessjournalSymmetry
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Probability Statistics Algebraic Structures Plithogenic).
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Contents
AbouttheEditors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Preface to ”New Development of Neutrosophic Probability, Neutrosophic Statistics,
NeutrosophicAlgebraicStructures,andNeutrosophic PlithogenicOptimizations” . . . . . . ix
XinZhou,PingLi,FlorentinSmarandacheandAhmedMostafaKhalil
NewResultsonNeutrosophicExtendedTripletGroupsEquippedwithaPartialOrder
Reprintedfrom: Symmetry2019,11,1514,doi:10.3390/sym11121514 . . . . . . . . . . . . . . . . . 1
GuanshengYu,ShouzhenZengandChonghuiZhang
Single-Valued Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their
ApplicationinInvestmentMultipleAttributeGroupDecisionMaking
Reprintedfrom: Symmetry2020,12,207,doi:10.3390/sym12020207 . . . . . . . . . . . . . . . . . 15
AshrafAl-Quran,HazwaniHashimandLazimAbdullah
AHybridApproachofIntervalNeutrosophicVagueSetsandDEMATELwithNewLinguistic
Variable
Reprintedfrom: Symmetry2020,12,275,doi:10.3390/sym12020275 . . . . . . . . . . . . . . . . . 29
Vasantha W. B., Ilanthenral Kandasamy, Florentin Smarandache, Vinayak Devvrat and
ShivamGhildiyal
StudyofImaginativePlayinChildrenUsingSingle-ValuedRefinedNeutrosophicSets
Reprintedfrom: Symmetry2020,12,402,doi:10.3390/sym12030402 . . . . . . . . . . . . . . . . . 45
MajidKhan,MuhammadGulistan,MumtazAliandWathekChammam
The Generalized Neutrosophic Cubic Aggregation Operators and Their Application to
Multi-ExpertDecision-MakingMethod
Reprintedfrom: Symmetry2020,12,496,doi:10.3390/sym12040496 . . . . . . . . . . . . . . . . . 63
WeiYang,LuluCai,SeyedAhmadEdalatpanahandFlorentinSmarandache
Triangular Single Valued Neutrosophic Data Envelopment Analysis: Application to Hospital
PerformanceMeasurement
Reprintedfrom: Symmetry2020,12,588,doi:10.3390/sym12040588 . . . . . . . . . . . . . . . . . 79
NguyenThoThong,FlorentinSmarandache,NguyenDinhHoa,LeHoangSon,LuongThi
HongLanandCuNguyenGiapetal.
A Novel Dynamic Multi-Criteria Decision Making Method Based on Generalized Dynamic
Interval-ValuedNeutrosophicSet
Reprintedfrom: Symmetry2020,12,618,doi:10.3390/sym12040618 . . . . . . . . . . . . . . . . . 93
XiaohongZhang,WangtaoYuan,MingmingChenandFlorentinSmarandache
A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski
AssociativeNeutrosophicExtendedTripletGroupoids(TA-NET-Groupoids)
Reprintedfrom: Symmetry2020,12,714,doi:10.3390/sym12050714 . . . . . . . . . . . . . . . . . 113
VasanthaW.B.,IlanthenralKandasamyandFlorentinSmarandache
NeutrosophicComponentsSemigroupsandMultisetNeutrosophicComponentsSemigroups
Reprintedfrom: Symmetry2020,12,818,doi:10.3390/sym12050818 . . . . . . . . . . . . . . . . . 133
v
Abdul Alamin, Sankar Prasad Mondal, Shariful Alam, Ali Ahmadian, Soheil Salahshour
andMehdiSalimi
SolutionandInterpretationofNeutrosophicHomogeneousDifferenceEquation
Reprintedfrom: Symmetry2020,12,1091,doi:10.3390/sym12071091 . . . . . . . . . . . . . . . . . 145
CahitAslan,AbdullahKargınandMemetS¸ahin
NeutrosophicModelingofTalcottParsons’sActionandDecision-MakingApplicationsforIt
Reprintedfrom: Symmetry2020,12,1166,doi:10.3390/sym12071166 . . . . . . . . . . . . . . . . . 171
Ahmed Mostafa Khalil, Dunqian Cao, Abdelfatah Azzam, Florentin Smarandache and
WedadR.Alharbi
Combination of the Single-Valued Neutrosophic Fuzzy Set and the Soft Set with Applications
inDecision-Making
Reprintedfrom: Symmetry2020,12,1361,doi:10.3390/sym12081361 . . . . . . . . . . . . . . . . . 191
ShchurIryna,YuZhong,WenJiang,XinyangDengandJieGeng
Single-ValuedNeutrosophicSetCorrelationCoefficientandItsApplicationinFaultDiagnosis
Reprintedfrom: Symmetry2020,12,1371,doi:10.3390/sym12081371 . . . . . . . . . . . . . . . . . 209
YumingGong,ZeyuMa,MeijuanWang,XinyangDengandWenJiang
A New Multi-Sensor Fusion Target Recognition Method Based on Complementarity Analysis
andNeutrosophicSet
Reprintedfrom: Symmetry2020,12,1435,doi:10.3390/sym12091435 . . . . . . . . . . . . . . . . . 223
YaserSaber,FahadAlsharariandFlorentinSmarandache
OnSingle-ValuedNeutrosophicIdealsinSˇostakSense
Reprintedfrom: Symmetry2020,12,193,doi:10.3390/sym12020193 . . . . . . . . . . . . . . . . . 241
YaserSaber,FahadAlsharari,FlorentinSmarandacheandMohammedAbdel-Sattar
ConnectednessandStratificationofSingle-ValuedNeutrosophicTopologicalSpaces
Reprintedfrom: Symmetry2020,12,1464,doi:10.3390/sym12091464 . . . . . . . . . . . . . . . . . 261
Kritika Mishra, Ilanthenral Kandasamy, Vasantha Kandasamy W. B. and Florentin
Smarandache
ANovelFrameworkUsingNeutrosophyforIntegratedSpeechandTextSentimentAnalysis
Reprintedfrom: Symmetry2020,12,1715,doi:10.3390/sym12101715 . . . . . . . . . . . . . . . . . 281
MuhammadRayeesAhmad,MuhammadSaeed,UsmanAfzalandMiin-ShenYang
A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic
Environment
Reprintedfrom: Symmetry2020,12,1855,doi:10.3390/sym12111855 . . . . . . . . . . . . . . . . . 303
DongsikJo,S.Saleh,Jeong-GonLee,KulHurandChenXueyou
TopologicalStructuresviaInterval-ValuedNeutrosophicCrispSets
Reprintedfrom: Symmetry2020,12,2050,doi:10.3390/sym12122050 . . . . . . . . . . . . . . . . . 327
FahadAlsharari
£-SingleValuedExtremallyDisconnectedIdealNeutrosophicTopologicalSpaces
Reprintedfrom: Symmetry2020,13,53,doi:10.3390/sym13010053 . . . . . . . . . . . . . . . . . . 357
BhimrajBasumatary,NijwmWary,DimachaDwibrangMwchahary,AshokeKumarBrahma,
JwngsarMoshaharyandUshaRaniBasumataryetal.
AStudyonSomePropertiesofNeutrosophicMultiTopologicalGroup
Reprintedfrom: Symmetry2021,13,1689,doi:10.3390/sym13091689 . . . . . . . . . . . . . . . . . 375
vi
About the Editors
FlorentinSmarandache
FlorentinSmarandacheisaprofessorofmathematicsattheUniversityofNewMexico, United
States. He received his MSc in Mathematics and Computer Science from the University of Craiova,
Romania, PhD in Mathematics from the State University of Kishinev, and Postdoctoral in Applied
Mathematics from Okayama University of Sciences, Japan, and The Guangdong University of
Technology, Guangzhou, China. He has been the founder of neutrosophy (generalization of
dialectics), neutrosophicset, logic, probabilityandstatisticssince1995andhaspublishedhundreds
of papers and books on neutrosophic physics, superluminal and instantaneous physics, unmatter,
quantumparadoxes,absolutetheoryofrelativity,redshiftandblueshiftduetothemediumgradient
and refraction index in addition to the Doppler effect, paradoxism, outerart, neutrosophy as
a new branch of philosophy, Law of Included Multiple-Middle, multispace and multistructure,
hypersoft sets, degree of dependence and independence between neutrosophic components,
refined neutrosophic sets, neutrosophic over-under-off-sets, plithogenic sets / logic / probability /
statistics, neutrosophic triplet and duplet structures, quadruple neutrosophic structures, extension
of algebraic structures to NeutroAlgebras and AntiAlgebras, NeutroGeometry and AntiGeometry,
Dezert–Smarandache Theory, and so on to many peer-reviewed international journals and many
books and he has presented papers and plenary lectures at many international conferences around
theworld.
In addition, he has published many books of poetry, drama, children’s stories,
translations, essays, a novel, folklore collections, traveling memories, and art albums
(http://fs.unm.edu/FlorentinSmarandache.htm).
YanhuiGuo
YanhuiGuoreceivedhisB.S.degreeinAutomaticControlfromZhengzhouUniversity,China,
M.S. degree in Pattern Recognition and Intelligence System from Harbin Institute of Technology,
China, and Ph.D. degree in the Department of Computer Science, Utah State University, USA. He
wasaresearchfollowintheDepartmentofRadiologyattheUniversityofMichiganandanassistant
professorinSt. ThomasUniversity. Dr. GuoiscurrentlyanassistantprofessorintheDepartmentof
ComputerScienceattheUniversityofIllinoisatSpringfield. Dr. Guohaspublishedmorethan100
journal papers and 30 conference papers, completed more than 10 grant-funded research projects,
andworkedasanassociateeditorofdifferentinternationaljournals,areviewerfortopjournalsand
hasbeenapartofmanyconferences. Hisresearchareasincludecomputervision,machinelearning,
bigdataanalytics,andcomputer-aideddetection/diagnosis.
vii
Preface to ”New Development of Neutrosophic
Probability, Neutrosophic Statistics, Neutrosophic
Algebraic Structures, and Neutrosophic Plithogenic
Optimizations”
ThisSpecialIssuepresentsstate-of-the-artpapersonnewtopicsrelatedtoneutrosophictheories,
such as neutrosophic algebraic structures, neutrosophic triplet algebraic structures, neutrosophic
extended triplet algebraic structures, neutrosophic algebraic hyperstructures, neutrosophic
triplet algebraic hyperstructures, neutrosophic n-ary algebraic structures, neutrosophic n-ary
algebraic hyperstructures, refined neutrosophic algebraic structures, refined neutrosophic algebraic
hyperstructures, quadruple neutrosophic algebraic structures, refined quadruple neutrosophic
algebraicstructures,neutrosophicimageprocessing,neutrosophicimageclassification,neutrosophic
computer vision, neutrosophic machine learning, neutrosophic artificial intelligence, neutrosophic
dataanalytics,neutrosophicdeeplearning,andneutrosophicsymmetry,aswellastheirapplications
intherealworld.
The neutrosophic extended triplet group (NETG) is a novel algebra structure studied here
by Xin Zhou, Ping Li, Florentin Smarandache and Ahmed Mostafa Khalil in an article (Results
on Neutrosophic Extended Triplet Groups Equipped with a Partial Order) presenting the concept
of a partially ordered neutrosophic extended triplet group (po-NETG), considering the properties
and structure features of po-NETGs. The authors propose the concepts of the positive cone and
negative cone in a po-NETG, study the specificity of the positive cone in a partially ordered weak
commutative neutrosophic extended triplet group (po-WCNETG), and introduce the concept of a
po-NETGhomomorphismbetweentwopo-NETGs.
In the next selected paper (Single-Valued Neutrosophic Ideals in Sˇostak Sense), Yaser Saber,
Fahad Alsharari and Florentin Smarandache introduce the notion of single-valued neutrosophic
ideals sets in Sˇostak’s sense, and then the concept of a single-valued neutrosophic ideal open local
functionfor asingle-valuedneutrosophic topologicalspace, studying thebasicstructure, especially
a basis for such generated single-valued neutrosophic topologies and several relations between
different single-valued neutrosophic ideals and single-valued neutrosophic topologies. For the
purposeofsymmetry,theauthorsalsodefinethesingle-valuedneutrosophicrelations.
Guansheng Yu, Shouzhen Zeng and Chonghui Zhang study the single-valued neutrosophic
linguistic distance measures based on the induced aggregation method in their paper (-Valued
Neutrosophic Linguistic-Induced Aggregation Distance Measures and Their Application in
InvestmentMultipleAttributeGroupDecisionMaking), suggestinganewextensionoftheexisting
distance measures based on the induced aggregation view, namely the single-valued neutrosophic
linguistic-induced ordered weighted averaging distance (SVNLIOWAD) measure. Based on
SVNLIOWAD,inordertoeliminatethedefectsoftheexistingmethods,theauthorsdevelopanovel
induceddistanceforsingle-valuedneutrosophiclinguisticsets,calledthesingle-valuedneutrosophic
linguistic weighted induced ordered weighted averaging distance (SVNLWIOWAD). Then, the
relationship between the two proposed distance measures is explored, and a numerical example
concerning an investment selection problem is constructed to prove the efficiency of the proposed
methodunderasingle-valuedneutrosophiclinguisticenvironment.
AshrafAl-Quran,HazwaniHashimandLazimAbdullahextend,inthepaperHybridApproach
ix
