Neutrosophic Sets

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures).
Let <A> be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to <A> and <antiA>, and one corresponding to neutral (indeterminate) <neutA> (also denoted <neutroA>) between the opposites}, which may or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space).
A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements).

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosoph-ic sets), which is a set whose elements are characterized by many attributes' values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life.

This paper applies the concepts and achievements of the neutrosophic logic (Smarandache 1995, 2002) to argumentation in natural languages. By this approach, each statement is considered to have a fraction of truth in a subset T, a fraction of indeterminacy in a subset I, and a fraction of falsity in a subset F, with T, I, F as defined in Smarandache 1995. Neutrosophy employs subsets of truth, indeterminacy, and falsity instead of just a number since in many cases the fractions of truth and of falsity cannot be always precisely determined. The reflection on the reasoning involved in the verbal exchange that is everyday argumentation is the starting point in this argumentation.
Argumentation is generally designed as the reasoning done in natural language, underpinned by the communicative logic of natural language. Pillars of the argumentation, arguments are topoi, paths that we must follow in order to reach a specific conclusion. In argumentation logic, we distinguish between arguments, counter-arguments and non-arguments. The limit between these categories of statements is not always clear, as their status is discovered through interpretative rules at a semantic and pragmatic level, allowing construction and deconstruction of speech. Another specificity of argumentation in natural languages relies on its relativity. Argumentation operates with categories such as possible, probable and plausible instead of true and false. Its validity is a matter of degree: the stronger arguments win over the less strong ones.

This is the third volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The introduction contains a short history of neutrosophics, together with links to the main papers and books.
Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

Contributors to current issue (listed in papers’ order):
Ibrahim Yasser, Abeer Twakol, A. A. Abd El-Khalek, A. A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Remya P. B, Francina Shalini, Masoud Ghods, Zahra Rostami, A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes, K.R. Vijayalakshmi, Prakasam Muralikrishna, Surya Manokaran, Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar, Rakhal Das, Binod Chandra Tripathy, Nidhi Singh, Avishek Chakraborty, Nilabhra Paul, Deepshikha Sarma, Akash Singh, Uttam Kumar Bera, Fatimah M. Mohammed, Sarah W. Raheem, Muhammad Riaz, Florentin Smarandache, Faruk Karaaslan, Masooma Raza Hashmi, Iqra Nawaz, Kousik Das, Sovan Samanta, Kajal De, Xavier Encarnacion, Nivetha Martin, I. Pradeepa, N. Ramila Gandhi, P. Pandiammal, Aiman Muzaffar, Md Tabrez Nafis, Shahab Saquib Sohail, Abhijit Saha, Jhulaneswar Baidya, Debjit Dutta, Irfan Deli, Said Broumi, Mohsin Khalid, Neha Andaleeb Khalid, Md. Hanif Page, Qays Hatem Imran, Shilpi Pal, S. Satham Hussain, Saeid Jafari, N. Durga, Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad, Ali Khatami Firouzabadi, Mohammad Rahmanimanesh, Mujahid Abbas, Ghulam Murtaza, K. Porselvi, B. Elavarasan, Y. B. Jun, Chinnadurai V, Sindhu M P, K.Radhika, K. Arun Prakash, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Assia Bakali,  Nivetha Martin, R. Dhavaseelan, Ali Hussein Mahmood Al-Obaidi, Suman Das, Surapati Pramanik, Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan, M. Sarwar Sindhu, Tabasam Rashid, Agha Kashif, Rajesh Kumar Saini, Atul Sangal, Manisha.

Papers in current issue (listed in papers’ order):
COVID-X: Novel Health-Fog Framework Based on Neutrosophic Classifier for Confrontation Covid-19; Introduction to Decision Making for Neutrosophic Environment “Study on the Suez Canal Port, Egypt”; Neutrosophic Vague Binary BCK/BCI-algebra; Introduction to Topological Indices in Neutrosophic Graphs; Assessment of MCDM problems by TODIM using aggregated weights; MBJ – Neutrosophic – Ideal of – Algebra; A Study of Social Media linked MCGDM Skill under Pentagonal Neutrosophic Environment in the Banking Industry; Neutrosophic Multiset Topologica Space; Impact of Social Media in Banking Sector under Triangular Neutrosophic Arena Using MCGDM Technique; A Generalized Neutrosophic Solid Transportation Model with Insufficient Supply; Generalized b Closed Sets and Generalized b Open Sets in Fuzzy Neutrosophic bi-Topological Spaces; Neutrosophic Soft Rough Topology and its Applications to Multi-Criteria Decision-Making; A Study on Discrete Mathematics: Sum Distance in Neutrosophic Graphs with Application; Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric Neutrosophic Hypergraphic Approach; Neutrosophy Logic and its Classification: An Overview; MADM Using m-Generalized q-Neutrosophic Sets; Hesitant Triangular Neutrosophic Numbers and Their Applications to MADM; Translative and Multiplicative Interpretation of Neutrosophic Cubic Set; Neutrosophic Generalized Homeomorphism; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item with Ramp Type Demand under Shortages and Time Discounting; Operations on Neutrosophic Vague Graphs; Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach; Basic operations on hypersoft sets and hypersoft point; Neutrosophic ℵ −bi-ideals in semigroups; A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using Neutrosophic Sets and An Output Validation in Real Life Application; Ranking of Pentagonal Neutrosophic Numbers and its Applications to solve Assignment Problem; A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids; Introduction to Combined Plithogenic Hypersoft Sets; On Neutrosophic Generalized Alpha Generalized Continuity; Generalized neutrosophic b-open sets in neutrosophic topological space; Neutrosophic Soft Fixed Points; Selection of Alternative under the Framework of Single-Valued Neutrosophic Sets; Application of Single Valued Trapezoidal Neutrosophic Numbers in Transportation Problem.

Papers in current issue (listed in papers' order): De-Neutrosophication Technique of Pentagonal Neutrosophic Number and Application in Minimal Spanning Tree; Cyclic Associative Groupoids (CA-Groupoids) and Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids); An Integrated Neutrosophic and TOPSIS for Evaluating Airline Service Quality; .NET Framework to deal with Neutrosophic b∗gα-Closed Sets in Neutrosophic Topological Spaces; An Introduction to Neutrosophic Bipolar Vague Topological Spaces; Neutrosophic Almost Contra α-Continuous; Neutrosophic Cognitive Maps for Situation Analysis; Neutrosophic gb-closed Sets and Neutrosophic gb-Continuity; On Parametric Divergence Measure of Neutrosophic Sets with its Application in Decision-making Model; Single-Valued Neutrosophic Hyperrings and Hyperideals; Technique for Reducing Dimensionality of Data in Decision Making Utilizing Neutrosophic Soft Matrices; Vague –Valued Possibility Neutrosophic Vague Soft Expert Set Theory and Its Applications; Neutrosophic complex connectedness in neutrosophic complex topological spaces; Unraveling Neutrosophic Transportation Problem Using Costs Mean and Complete Contingency Cost Table; Neutrosophic Shortest Path Problem (NSPP) in a Directed Multigraph; Direct and Semi-Direct Product of Neutrosophic Extended Triplet Group; Data Envelopment Analysis for Simplified Neutrosophic Sets; Neutrosophic Vague Binary Sets; Multi-level linear programming problem with neutrosophic numbers: A goal programming strategy.

“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophicset, 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.

“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.

Contributors to current issue (listed in papers' order): M. Mohseni Takallo, Hashem Bordbar, R.A. Borzooei, Young Bae Jun, J. Kavikumar, D. Nagarajan, Said Broumi, F. Smarandache, M. Lathamaheswari, Nur Ain Ebas, R.Dhavaseelan, R. Subash Moorthy, A. Bakali, M. Talea, Wadei F. Al-Omeri, Saeid Jafari, R. Narmada Devi, G. Jayaparthasarathy, V. F. Little Flower , M. Arockia Dasan, Chaoqun Li, Jun Ye, Wenhua Cui, Shigui Du, Misturah Adunni Alaran, Abdul Akeem Adesina Agboola, Adio Taofik Akinwale, Olusegun Folorunso, Majdoleen Abu Qamar, Nasruddin Hassan, Muhammad Saqlain, Muhammad Saeed, Muhammad Rayees Ahmad, M. Al-Tahan, B. Davvaz, Memet Şahin, Abdullah Kargın, V.Banu priya, S.Chandrasekar, Qays Hatem Imran, Muhammad Akram, Nabeela Ishfaq, S. Khademan, M. M. Zahedi, C. Barrionuevo de la Rosa, B. Cárdenas Bolaños, H. Cárdenas Echeverría, R. Cabezas Padilla, G. A. Sandoval Ruilova, J. A. Montalván Espinoza, P. Alburquerque Proaño, J. R. Medina Villavicencio, M. Alexander Villegas, Abdel Nasser H. Zaied, Mahmoud Ismail, Abduallah Gamal.

“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. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.

This Special Issue of Symmetry (no. 9, issue 10, October 2017 - Special Issue: Neutrosophic Theories Applied in Engineering) presents original research papers that report on state-of-the-art and recent advancements in neutrosophic sets and logic in soft computing, artificial intelligence, big and small data mining, decision making problems, and practical achievements, approaching topics as retinal vessels detection, shearlet transform, indeterminacy filtering, neural network, fundus image, joint roughness coefficient (JRC), scale effect, anisotropy, fuzzy time series, forecasting, fuzzy logical relationship, Jaccard similarity, image segmentation, graph cut, indeterminate filtering, multiple criteria decision-making (MCDM), weighted aggregated sum product assessment (WASPAS), integrated criteria weight information, solar-wind power station location selection, linguistic neutrosophic number, score function, accuracy function, linguistic neutrosophic number weighted arithmetic averaging (LNNWAA) operator, linguistic neutrosophic number weighted geometric averaging (LNNWGA) operator, multiple attribute group decision-making, single-valued neutrosophic sets, generalized single-valued neutrosophic approximation spaces, rough sets, single-valued neutrosophic relations, neutrosophic cubic set, similarity measure, cosine measure, merger and acquisition (M&A) target selection, group decision making, multigranulation rough sets over two universes, Atanassov’s intuitionistic fuzzy sets, extreme learning machine (ELM), neutrosophic c-means (NCM), imbalanced data set, metallic mine project, investment risks evaluation, maximum deviation, extended TOPSIS, refined simplified neutrosophic set, refined single-valued neutrosophic set, refined interval neutrosophic set, vector similarity measure, Fuzzy k-NN, data classification, and so on.

Contributors to current issue (listed in papers' order): T. Chalapathi, R. V. Kiran Kumar, A. A. Salama, Hewayda ElGhawalby, Mehmet Şahin, Abdullah Kargın, Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi, T. Gharibah, Suriana Alias, Daud Mohamad, Adibah Shuib, Fatimah M. Mohammed, Shaymaa F. Matar, Sahidul Islam, Partha Ray, Ramona Parraga Alava, José Muñoz Murillo, Roy Barre Zambrano, María Isabel Zambrano Vélez, Maikel Leyva Vázquez, Chaitali Kar, Bappa Mondal, T. K. Roy, Surapati Pramanik, Partha Pratim Dey, Milton Villegas Alava, Stella Paola Delgado Figueroa, Hilda Mercedes Blum Alcivar, Shyamal Dalapati, Nguyen Xuan Thao, Florentin Smarandache.

Papers in current issue (listed in papers' order): Neutrosophic Units of Neutrosophic Rings and Fields; Neutrosophic Approach to Grayscale Images Domain; Neutrosophic Triplet Normed Ring Space; On Neutrosophic Crisp Semi Alpha Closed Sets; Rough Neutrosophic Multisets Relation with Application in Marketing Strategy; Fuzzy Neutrosophic Alpham-Closed Sets in Fuzzy Neutrosophic Topological Spaces; Neutrosophic Crisp Bi-Topological Spaces; Multi-Objective Portfolio Selection Model with Diversification by Neutrosophic Optimization Technique; PEST Analysis Based on Neutrosophic Cognitive Maps: A Case Study for Food Industry; An Inventory Model under Space Constraint in Neutrosophic Environment: A Neutrosophic Geometric Programming Approach; Bi-level Linear Programming Problem with Neutrosophic Numbers; Single Valued Neutrosophic Numbers and Analytic Hierarchy Process for Project Selection; A Revisit to NC-VIKOR Based MAGDM Strategy in Neutrosophic Cubic Set Environment; Divergence measure of neutrosophic sets and applications; Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets – Revisited.

“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.

Contributors to current issue (listed in papers' order): Avishek Chakraborty, Shreyashree Mondal, Said Broumi, Xiaohong Zhang, Zhirou Ma, Wangtao Yuan, Abdel Nasser H. Zaied, Abduallah Gamal, Mahmoud Ismail, Saranya S , Vigneshwaran M, Mohana K, Princy R, Florentin Smarandache, R. Dhavaseelan , Md. Hanif PAGE, Aasmim Zafar, Mohd Anas, C.Maheswari , S. Chandrasekar, Abhishek Guleria, Saurabh Srivastava , Rakesh Kumar Bajaj, D. Preethi, S. Rajareega, J.Vimala, Ganeshsree Selvachandran, Abhishek Guleria, Anjan Mukherjee, M. Karthika, M. Parimala, Saeid Jafari, Mohammed Alshumrani, Cenap Ozel, R. Udhayakumar, S. Krishna Prabha, S.Vimala, Siddhartha Sankar Biswas, T. Nandhini , M. Vigneshwaran, Moges Mekonnen Shalla, Necati Olgun, S. A. Edalatpanah, Remya.P.B, Francina Shalini.A, Surapati Pramanik, Partha Pratim Dey.

The authors and co-authors, listed in the order of their published neutrosophic papers: Muhammad Akram, Muzzamal Sitara, A. A. A. Agboola, B. Davvaz, F. Smarandache, Ali Hassan, Muhammad Aslam Malik, Said Broumi, Assia Bakali, Mohamed Talea, K. Hur, P. K. Lim, J. G. Lee, J. Kim, Young Bae Jun, Maryam Nasir, and A. Borumand Saeid, would like to thank Prof. Kul Hur, the Editorin-Chief of the international journal Annals of Fuzzy Mathematics and Informatics (AFMI), for dedicating the whole Vol. 14, No.1, published on 25 July 2017, to the neutrosophic theories and applications. The papers included in this volume are especially referring to neutrosophic (single-valued and interval-valued) graphs and bipolar graphs, and their applications in multi-criteria decision making (MCDM), and to neutrosophic algebraic structures, such as: category of neutrosophic crisp sets, neutrosophic quadruple algebraic hyperstructures, and neutrosophic subalgebras of BCK/BCI-algebras. We would also like to bring our gratitude to many reviewers of the neutrosophic community, from around the world, community that has grew to over eight hundred peoples (students, faculty, and researchers).

Contributors to current issue (listed in papers' order): Arulpandy P., Trinita Pricilla M., A. Rohini, M. Venkatachalam, Said Broumi, F. Smarandache, Abdel Nasser H. Zaied, Mahmoud Ismail, Abduallah Gamal, Nabil Mostafa, Shazia Rana, Madiha Qayyum, Muhammad Saeed, Bakhtawar Ali Khan, A. Edward Samuel, R. Narmadhagnanam, Azeddine Elhassouny, Soufiane Idbrahim, S. Satham Hussain, R. Jahir Hussain, Young Bae Jun, Phattharaphon Rangsuk, Pattarin Huana, Aiyared Iampan, Abhijit Saha, Selçuk Topal, Assia Bakali, Mohamed Talea, Anitha S, Mohana K, Mary Margaret A, Shawkat Alkhazaleh, Kawther Fawzi Hamza Alhasan, Avishek Chakraborty, Prem Kumar Singh, Surya Manokaran, Muralikrishna Prakasam, Mohsin Khalid, Rakib Iqbal, Balasubramanian Elavarasan, Satya Kumar Das, Sahidul Islam.

“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. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.

Neutrosophic logic and set are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. A number of new neutrosophic theories have been proposed and have been applied in Multi-Criteria Decision Making, computational intelligence, multiple attribute decision making, image processing, medical diagnosis, fault diagnosis, optimization design, and so on. Neutrosophic logic, set, probability, statistics, etc., are, respectively, generalizations of fuzzy and intuitionistic fuzzy logic and set, classical and imprecise probability, and classical statistics and so on.
This Special Issue gathers 11 original research papers that report on state-of-the-art and recent advancements in Multi-Criteria Decision Making using neutrosophic environment to computing, artificial intelligence, big and small data mining, group decision making problems, pattern recognition, information processing, image processing, and many other practical achievements.

“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.

ISBN 978-3-03943-288-2 (Hbk); ISBN 978-3-03943-289-9 (PDF)
https://doi.org/10.3390/books978-3-03943-289-9 (registering DOI)
© 2020 by the authors; CC BY licence
New Challenges in Neutrosophic Theory and Applications
Stefan Vladutescu andMihaela Colhon (Eds.)
Pages: 348
Published: October 2020
(This book is a printed edition of the Special Issue New Challenges in Neutrosophic Theory and Applications that was published in Mathematics)

“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.

“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.

This book is an excellent exposition of the use of Data Envelopment Analysis (DEA) to generate data analytic insights to make evidence-based decisions, to improve productivity, and to manage cost-risk and benefit-opportunity in public and private sectors. The design and the content of the book make it an up-to-date and timely reference for professionals, academics, students, and employees, in particular those involved in strategic and operational decision-making processes to evaluate and prioritize alternatives to boost productivity growth, to optimize the efficiency of resource utilization, and to maximize the effectiveness of outputs and impacts to stakeholders. It is concerned with the alleviation of world changes, including changing demographics, accelerating globalization, rising environmental concerns, evolving societal relationships, growing ethical and governance concern, expanding the impact of technology; some of these changes have impacted negatively the economic growth of private firms, governments, communities, and the whole society. (John R. Edwards)

In the rising trends of information technology, the concepts of uncertainty have started gaining greater importance with time in solving operational research problems in supply chain model, project management, transportation problem, or inventory control problems. Moreover, day-by-day competition is becoming tougher in imprecise environments. For instance, customer demand is often being affected by several varying factors like production price, income level, and the like. In these cases, the demand either remains unfulfilled or is difficult to obtain with certainty in the real-world market. Fuzzy sets are not always able to directly depict such uncertainties because they exhibit numeric only membership functions, whereas neutrosophic sets are found to be more suitable to accommodate inherent uncertainties.

“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.

“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 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 Initiative: 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.

The papers included in this volume are especially referring to neutrosophic (single-valued and interval-valued) graphs and bipolar graphs, and their applications in multi-criteria decision making (MCDM), and to neutrosophic algebraic structures, such as: category of neutrosophic crisp sets, neutrosophic quadruple algebraic hyperstructures, and neutrosophic subalgebras of BCK/BCI-algebras.

“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.

In this paper, we define the notion of neutrosophic semiopen (resp. preopen and α-open) functions and investigate relation among them. We give a characterization of neutrosophic α-open set, and provide conditions for a neutrosophic set to be a neu-trosophic α-open set. We discuss characterizations of neutrosophic pre-continuous (resp. α-continuous) functions. We give a condition for a function of neutrosophic topological spaces to be a neutrosophic α-continuous function.

“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.

This volume is a collection of ten papers by contributors F. Smarandache, F. Yuhua, K. Mondal, S. Pramanik, S. Broumi, J. Ye, A. A. Salama,, N. Easa, S. A. Elhafez, M. M. Lotfy, L. Kong, Y. Wu, P. Biswas, B. C. Giri, A. Mukkerjee, and S. Sarkar, focusing on a new kind of algebraic structures called (T, I, F)- Neutrosophic Structures; Expanding Uncertainty Principle to Certainty-Uncertainty Principles with Neutrosophy and Quad-stage Methods; Rough Neutrosophic Multi-Attribute Decision-Making Based on Rough Accuracy Score Function; an Extended TOPSIS Method for Multiple Attribute Decision Making based on Interval Neutrosophic Uncertain Linguistic Variable; Review of Recommender Systems Algorithms Utilized in Social Networks based e-Learning Systems & Neutrosophic System; Fault Diagnosis Method of Gasoline Engines Using the Cosine Similarity Measure of Neutrosophic Numbers; Cosine Similarity Measure Based Multi-attribute Decision-making with Trapezoidal Fuzzy Neutrosophic Numbers; Thesis-Antithesis-Neutrothesis, and Neutrosynthesis; Negating Four Color Theorem with Neutrosophy and Quadstage Method; and A new method of measuring similarity between two neutrosophic soft sets and its application in pattern recognition problems.

This is the second volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation.
The authors are listed alphabetically, and represent the following countries: Angola, Argentina, P.R. China, Denmark, Dominican Republic, Ecuador, Egypt, India, Iraq, Iran, Jordan, South Korea, Morocco, Nigeria, Pakistan, Romania, Serbia, Syria, Turkey, S.R. Vietnam.
The introduction contains an updated history of neutrosophics, together with links to the most important papers and books.
Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy.
In the past years, the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete /  indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

“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 present book starts by proposing an approach for data mining with single valued neutrosophic information from large amounts of data and then progresses to topics in decision making in neutrosophic environment and neutrosophic hybrid environment, e-learning, graph
theory, medical diagnosis, neutrosophic models in sociology, topology, and some more.
The book collects thirty original research and application papers from different perspectives covering different areas of neutrosophic studies, such data mining, decision making, e-learning, graph theory, medical diagnosis, probability theory, topology, and some theoretical papers. This
book shows examples applications of neutrosophic set and neutrosophic hybrid set in multiple
criteria and multiple attribute decision making, medical diagnosis, etc.