Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Second extended and improved edition

2017·2 MB·English

by  Florentin Smarandache

Florentin Smarandache
Neutrosophic Perspectives:
Triplets, Duplets, Multisets,
Hybrid Operators, Modal Logic,
Hedge Algebras.
And Applications
Peer-Reviewers:
Dr. Azeddine Elhassouny
Fulbright postdoctoral fellow, Department
of Mathematics and Science, University of
New Mexico, Gallup Campus, USA.

Prof. S. Jafari
College of Vestsjaelland South,
Herrestraede 11, 4200 Slagelse, Denmark.

Dr. Mohamed Abdel-Basset
Faculty of Computers and Informatics,
Zagazig University, Egypt.
Florentin Smarandache

Neutrosophic Perspectives:
Triplets, Duplets, Multisets,
Hybrid Operators, Modal Logic,
Hedge Algebras.
And Applications

Pons Publishing House
Brussels, 2017
Pons Publishing House / Pons asbl
Quai du Batelage, 5
1000 - Bruxelles
Belgium

DTP: George Lukacs

ISBN 978-1-59973-527-6
Florentin Smarandache
Neutrosophic Perspectives
TABLE OF CONTENTS

PREFACE
Neutrosophic Perspectives 9
I.1. Positively or Negatively Qualitative Neutrosophic
Components 12
I.2. The Average Positive Qualitative Neutrosophic Function
and The Average Negative Qualitative Neutrosophic
Function 13
II.1. Neutrosophic Overnumbers / Undernumbers /
Offnumbers 18
II.2. Spherical Neutrosophic Numbers 21
III.1. Neutrosophic Indeterminacy of Second Type 23
IV.1. n-Refined Neutrosophic Set and Logic and Its
Applications to Physics 26
V.1. Operations with Neutrosophic Numbers 45
V.2. Addition of Multiple Single-Valued Neutrosophic
Numbers 48
V.3. Interval-Valued Neutrosophic Number Operations 50
V.4. Operations with (t, i, f)-Neutrosophic Matrices 53
VI.1. Neutrosophic Hybrid Operators 62
VII.1. Neutrophic Perfect/Imperfect Triplets 67
VII.2. Neutrosophic Triplet Function 76
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Florentin Smarandache
Neutrosophic Perspectives
VII.3. Theorems on Neutrosophic Triplets 77
VII.4. Neutrosophic Triplet Group vs. Generalized Group. 91
VII.5. Neutrosophic Triplet Multiple Order 94
VIII.1. Neutrosophic Triplet Ring 95
VIII.2. Hybrid Neutrosophic Triplet Ring 97
VIII.3. Neutrosophic Triplet Field 99
VIII.4. Hybrid Neutrosophic Triplet Field 103
VIII.5. Neutrosophic Triplet Loop 104
VIII.6. Neutrosophic Triplet Structures 106
IX.1. Neutrosophic Duplets 111
IX.2. Neutrosophic Duplet Set and Neutrosophic Duplet
Structures 113
X.1. Neutrosophic Multiset 117
X.2. Neutrosophic Multiset Applied in Physical Processes 122
X.3. Neutrosophic Complex Multiset 124
XI.1. Classical Logic and Neutrosophic Logic. Answers to K.
Georgiev. 126
XI.2. Neutrosophic Tautological, Contradictional, and
Provability Tresholds, and Neutrosophic Completeness.
Answers to U. Rivieccio 147
XII.1. Neutrosophic Predicate Logic 178
XII.2. Neutrosophic Decidability System 182
XII.3. Neutrosophic Modal Logic 184
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Florentin Smarandache
Neutrosophic Perspectives
XII.4. Neutrosophic Hedge Algebras 212
XIII.1. Neutrosophic MCDM 228
XIII.2 Neutrosophic Psychology 229
XIII.3. Neutrosophic Function as The Equatorial Virtual Line
230
XIII.4. PCR5 and Neutrosophic Probability in Target
Identification 231
XIII.5. Easier to Break a Dynamic System from Inside than from
Outside 245
XIII.6. Neutrosophic Quantum Computer 281
XIII.7. Theory of Neutrosophic Evolution: Degrees of Evolution,
Indeterminacy, and Involution 297
XIII.8. Neutrosophic Triplet Structures in Practice 325
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Florentin Smarandache
Neutrosophic Perspectives
PREFACE
Neutrosophic Perspectives
This book is part of the book-series dedicated
to the advances of neutrosophic theories and their
applications, started by the author in 1998.
Its aim is to present the last developments in
the field. For the first time, we now introduce:
— Neutrosophic Duplets and the Neutrosophic
Duplet Structures;
— Neutrosophic Multisets (as an extension of
the classical multisets);
— Neutrosophic Spherical Numbers;
— Neutrosophic Overnumbers / Undernumbers
/ Offnumbers;
— Neutrosophic Indeterminacy of Second Type;
— Neutrosophic Hybrid Operators (where the
heterogeneous t-norms and t-conorms may be
used in designing neutrosophic aggregations);
— Neutrosophic Triplet Loop;
— Neutrosophic Triplet Function;
— Neutrosophic Modal Logic;
— and Neutrosophic Hedge Algebras.
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