Neutrosophic Sets and Systems
Neutrosophic Sets and Systems
Volume 2 Article 1
1-1-2014
Full Issue
Neutrosophic Sets and Systems
Follow this and additional works at: https://digitalrepository.unm.edu/nss_journal
Recommended Citation Systems, Neutrosophic Sets and. "Full Issue." Neutrosophic Sets and Systems 2, 1 (2014). https://digitalrepository.unm.edu/nss_journal/vol2/iss1/1
This Full Issue is brought to you for free and open access by UNM Digital Repository. It has been accepted for inclusion in Neutrosophic Sets and Systems by an authorized editor of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Neutrosophic Sets and Systems, Vol. 1, 2013 1 Vol. 2, 2014
Neutrosophic
Sets and
Systems
An International Journal in Information Science and Engineering
ISSN 2331-6055 (print) ISSN 2331-608X (online) ISSN 2331-6055 (print) ISSN 2331-608X (online)
Neutrosophic Sets and Systems
An International Journal in Information Science and Engineering
Editor-in-Chief: Associate Editors: Dmitri Rabounski and Larissa Borissova, independent researchers. Prof. Florentin Smarandache W. B. Vasantha Kandasamy, IIT, Chennai, Tamil Nadu, India. Department of Mathematics and Science Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. University of New Mexico A. A. Salama, Faculty of Science, Port Said University, Egypt. 705 Gurley Avenue Yanhui Guo, School of Science, St. Thomas University, Miami, USA. Gallup, NM 87301, USA Francisco Gallego Lupiañez, Universidad Complutense, Madrid, Spain. E-mail: [email protected] Peide Liu, Shandong Universituy of Finance and Economics, China. Home page: http://fs.gallup.unm.edu/NSS Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Jun Ye, Shaoxing University, China. Ştefan Vlăduţescu, University of Craiova, Romania.
Volume 2 2014 Contents
Shawkat Alkhazaleh and Emad Marei. Mappings on W. B. Vasantha Kandasamy, Florentin Smarandache. Measure and 42 Neutrosophic Soft Classes...... 3 Neutrosophic Lattices...... 8 Neutro...... 3 Said Broumi and Florentin Smarandache. On Jun Ye, Qiansheng Zhang. Single Valued Neutrosophic Neutrosophic Implications...... 9 Similarity Measures for Multiple Attribute Decision- Making...... 48 Rıdvan Şahin. Neutrosophic Hierarchical Clustering Algoritms...... 18 Mumtaz Ali, Florentin Smarandache, Muhammad A. A. Salama, Florentin Smarandache, Valeri Shabir, Munazza Naz. Soft Neutrosophic Bigroup and 55 Kroumov. Neutrosophic Crisp Sets & Neutrosophic Soft Neutrosophic N-Group ...... Crisp Topological Spaces...... 25 Surapati Pramanik and Tapan Kumar Roy. Karina Pérez-Teruel and Maikel Leyva-Vázquez. Neutrosophic Game Theoretic Approach to Indo-Pak Neutrosophic Logic for Mental Model Elicitation and Analysis...... Conflict over Jammu-Kashmir...... 31 82 A.A.A. Agboola, B. Davvaz. On Neutrosophic Pranab Biswas, Surapati Pramanik, and Bibhash C. Giri. Entropy Based Grey Relational Analysis Method Canonical Hypergroups and Neutrosophic Hyperrings.. 34 for Multi-Attribute Decision Making under Single 10 Valued Neutrosophic Assessments...... 2 Pingping Chi, and Peide Liu. An Extended TOPSIS Method for the Multiple Attribute Decision Making Problems Based on Interval Neutrosophic Set ………... 63
Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 2, 2014
pss -Header will be provided by the publisher
1 2 3 Neutrosophic Sets and Systems 4 An International Journal in Information Science and Engineering 5 6 7 Copyright Notice 8 9 10 Copyright @ Neutrosophics Sets and Systems academic or individual use can be made by any user 11 without permission or charge. The authors of the articles 12 All rights reserved. The authors of the articles do hereby published in Neutrosophic Sets and Systems retain their 13 grant Neutrosophic Sets and Systems non-exclusive, rights to use this journal as a whole or any part of it in any 14 worldwide, royalty-free license to publish and distribute other publications and in any way they see fit. Any part of 15 the articles in accordance with the Budapest Open Neutrosophic Sets and Systems howsoever used in other 16 Initiative: this means that electronic copying, distribution 17 publications must include an appropriate citation of this and printing of both full-size version of the journal and the 18 journal. 19 individual papers published therein for non-commercial, 20 21 22 23 Information for Authors and Subscribers 24 25 Neutrosophic Sets and Systems has been created for Neutrosophic Probability is a generalization of the classical 26 publications on advanced studies in neutrosophy, neutrosophic probability and imprecise probability. 27 set, neutrosophic logic, neutrosophic probability, neutrosophic Neutrosophic Statistics is a generalization of the classical 28 statistics that started in 1995 and their applications in any field, statistics. 29 such as the neutrosophic structures developed in algebra, What distinguishes the neutrosophics from other fields is the 30 geometry, topology, etc.
CopyrightCopyright © Neutrosophicline will be provided Sets and by theSystems publisher Neutrosophic Sets and Systems, Vol. 2, 2014 3
Mappings on Neutrosophic Soft Classes
Shawkat Alkhazaleh1 and Emad Marei2
1 Department of Mathematics, Faculty of Science and Art, Shaqra University, Saudi Arabia. E-mail: [email protected] 2 Department of Mathematics, Faculty of Science and Art, Shaqra University, Saudi Arabia. E-mail: [email protected]
Abstract. In 1995 Smarandache introduced the concept of classes where the neutrosophic soft classes are collections of neutrosophic set which is a mathematical tool for handling neutrosophic soft set. We also define and study the properties of problems involving imprecise, indeterminacy and inconsistent neutrosophic soft images and neutrosophic soft inverse images of data. In 2013 Maji introduced the concept of neutrosophic soft neutrosophic soft sets. set theory as a general mathematical tool for dealing with uncertainty. In this paper we define the notion of a mapping on
Keywords: soft set, fuzzy soft set, neutrosophic soft set, neutrosophic soft classes, mapping on neutrosophic soft classes, neutrosophic soft images, neutrosophic soft inverse images. .
1 Introduction properties and operations. They presented the applications of this concept to decision making problems. In 2012 Most of the problems in engineering, medical science, Salleh et al. [1] introduced the notion of economics, environments etc. have various uncertainties. multiparameterized soft set and studied its properties. In in 1995, Smarandache talked for the first time about 2010 Kharal and Ahmad [14] introduced the notion of neutrosophy and in 1999 and 2005 [15, 16] he initiated the mapping on soft classes where the soft classes are theory of neutrosophic set as a new mathematical tool for collections of soft sets. They also defined and studied the handling problems involving imprecise, indeterminacy, properties of soft images and soft inverse images of soft and inconsistent data. Molodtsov [8] initiated the concept sets and gave the application of this mapping in medical of soft set theory as a mathematical tool for dealing with diagnosis. They defined the notion of a mapping on classes uncertainties. Chen et al. [7] and Maji et al. [11, 9] studied of fuzzy soft sets. They also defined and studied the some different operations and application of soft sets. properties of fuzzy soft images and fuzzy soft inverse Furthermore Maji et al. [10] presented the definition of images of fuzzy soft sets (see [13]). In 2009 Bhowmik and fuzzy soft set and Roy et al. [12] presented the applications Pal [18] studied the concept of intuitionistic neutrosophic of this notion to decision making problems. Alkhazaleh et set, and Maji [17] introduced neutrosophic soft set, al. [4] generalized the concept of fuzzy soft set to established its application in decision making, and thus neutrosophic soft set and they gave some applications of opened a new direction, new path of thinking to engineers, this concept in decision making and medical diagnosis. mathematicians, computer scientists and many others in They also introduced the concept of fuzzy parameterized various tests. In 2013 Said and Smarandache [19] defined interval-valued fuzzy soft set [3], where the mapping is the concept of intuitionistic neutrosophic soft set and defined from the fuzzy set parameters to the interval- introduced some operations on intuitionistic neutrosophic valued fuzzy subsets of the universal set, and gave an soft set and some properties of this concept have been application of this concept in decision making. Alkhazaleh established. In this paper we define the notion of a and Salleh [2] introduced the concept of soft expert sets mapping on classes where the neutrosophic soft classes are where the user can know the opinion of all experts in one collections of neutrosophic soft set. We also define and model and gave an application of this concept in decision study the properties of neutrosophic soft images and making problem. As a generalization of Molodtsov’s soft neutrosophic soft inverse images of neutrosophic soft sets. set, Alkhazaleh et al. [5] presented the definition of a soft multiset and its basic operations such as complement, 2 Preliminaries union and intersection. In 2012 Alkhazaleh and Salleh [6] introduced the concept of fuzzy soft multiset as a In this section, we recall some basic notions in combination of soft multiset and fuzzy set and studied its neutrosophic set theory, soft set theory and neutrosophic
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes 4 Neutrosophic Sets and Systems, Vol. 2, 2014
soft set theory . Smarandache defined neutrosophic set in We denote it by (FAGB , ) ( , ). (,)FA is said to be the following way. neutrosophic soft super set of (,)GB if (,)GB is a
neutrosophic soft subset of (,)FA . We denote it by Definition 2.1 [15] A neutrosophic set A on the universe (FAGB , ) ( , ). of discourse X is defined as A={<, x T (), x I (), x F ()>, x x X } AAA T I FX: ] 0,1 [ 0T ( x ) I ( x ) F ( x ) 3 . where , , and AAA Definition 2.5 [17] Let (,)HA and (,)GB be two NSSs over
the common universe U . Then the union of (,)HA and (,)GB Smarandache explained his concept as follows: is denoted by `(HAGB , ) ( , )` and is defined by "for example, neutrosophic logic is a generalization of the (HAGBKC , ) ( , ) = ( , ) , where CAB= and the truth- fuzzy logic. In neutrosophic logic a proposition is T true , I indeterminate , and F false . For example, let’s analyze the membership, indeterminacy-membership and falsity- following proposition: "Pakistan will win against India in membership of (,)KC are defined as follows: the next soccer game". This proposition can be (0.6,0.3,0.1) TKH( e )( m ) = T ( e )( m ), if e A B , which means that there is possibility of 60% that Pakistan =T ( e )( m ), if e B A , wins, 30% that Pakistan has a tie game, and 10% that G Pakistan looses in the next game vs. India." =maxTemTem (HG ( )( ), ( )( )), if eA B . Molodtsov defined soft set in the following way. IKH( e )( m ) = I ( e )( m ), if e A B , Let U be a universe and E be a set of parameters. Let PU() denote the power set of U and AE . =IG ( e )( m ), if e B A ,
I( e )( m ) I ( e )( m )) Definition 2.2 [8] A pair FA, is called a soft set over U, =HG , ife A B . 2 where F is a mapping
FAPU:. FKH( e )( m ) = F ( e )( m ), if e A B , In other words, a soft set over U is a parameterized family of subsets of the universe U. For AF, may be =FG ( e )( m ), if e B A , considered as the set of -approximate elements of the =minFemFem (HG ( )( ), ( )( )), if eA B . soft set FA, . Definition 2.6 [17] Let (,)HA and (,)GB be two NSSs over Definition 2.3 [17] Let U be an initial universe set and E the common universe . Then the intersection of (,)HA be a set of parameters. Consider AE . Let PU() denotes and (,)GB is denoted by `(HAGB , ) ( , )` and is defined by the set of all neutrosophic sets of U . The collection (,)FA is (HAGBKC , ) ( , ) = ( , ) , where CAB= and the truth- termed to be the neutrosophic soft set (NSS in short) over membership, indeterminacy-membership and falsity- U , where F is a mapping given by FAPU:() . membership of (,)KC are as follows:
Example 2.1 Suppose that U= c1 , c 2 , c 3 is the set of color TKHG( em )( ) = minT ( ( emT )( ), ( em )( )) , cloths under consideration, A= e1 , e 2 , e 3 is the set of I( e )( m ) I ( e )( m )) parameters, where e1 stands for the parameter I( e )( m ) = HG, K 2 ‘color‘ which consists of red, green and blue, e stands for 2 the parameter ‘ingredient‘ which is made from wool, FemKHG( )( ) = maxFemFem ( ( )( ), ( )( )), eC . cotton and acrylic, and e stands for the parameter 3 ‘price‘ which can be various: high, medium and low. We Definition 2.7 [17] Let (,)HA and (,)GB be two NSSs over define neutrosophic soft set as follows: the common universe U . Then the ‘AND‘ operation on F e1 = < c 1 ,0.4,0.2,0.3 >,< c 2 ,0.7,0.3,0.4 >,< c 3 ,0.5,0.2,0.2 > , them is denoted by `(HAGB , ) ( , )` and is defined by F e2 = < c 1 ,0.6,0.2,0.6 >,< c 2 ,0.9,0.4,0.1>,< c 3 ,0.4,0.3,0.3 > , (HAGBKAB , ) ( , ) = ( , ) , where the truth-membership, F e3 = < c 1 ,0.3,0.3,0.7 >,< c 2 ,0.7,0.2,0.4 >,< c 3 ,0.5,0.6,0.4 > . indeterminacy-membership and falsity-membership of Definition 2.4 [17] Let (,)FA and (,)GB be two (,)KAB are as follows: neutrosophic soft sets over the common universe U . (,FA) TKHG(,)()= m min ( T ()(), m T ()()) m , is said to be neutrosophic soft subset of (,GB) if AB , and I( )( m ) I ( )( m )) T( e )( x ) T ( e )( x ) , I( e )( x ) I ( e )( x ) , F( e )( x ) F ( e )( x ) , e A,. x U Im( , )( ) = HG and FG FG FG K 2
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes Neutrosophic Sets and Systems, Vol. 1, 2013 5
F(,)()= mmaxF ( ()(), mF ()()), m A , B . KHG Definition 3.3 Let XE, and YE, ' be neutrosophic soft Definition 2.8 [17] Let (,)HA and (,)GB be two NSSs over classes. Let r: X Y and s: E E' be mappings. Then a U the common universe . Then the ‘OR‘ operation on them mapping f1 :,, Y E' X E is defined as follows: is denoted by `(HAGB , ) ( , )` and is defined by (HAGBOAB , ) ( , ) = ( , ) , where the truth-membership, ' 1 For a neutrosophic soft set GB, in YE, , f G, B is a indeterminacy-membership and falsity-membership of (,)OAB are as follows: neutrosophic soft set in XE, obtained as follows: G s r x,, if s B T(,)()= m max ( T ()(), m T ()()) m 1 OHG , f G, B x = 0,0,0 otherwise . IHG( )( m ) I ( )( m )) Im( , )( ) = 1 1 O 2 For s B E and xX . f G, B is called a and neutrosophic soft inverse image of the neutrosophic soft
FOHG(,)()= mminF ( ()(), mF ()()), m A , B . set GB, .
Example 3.1 Let X={ x1 , x 2 , x 3 } , Y={ y1 , y 2 , y 3 } and let 3 Mapping on Neutrosophic Soft Classes E= e , e , e and E''''= e , e , e . Suppose that XE, and YE, ' In this section, we introduce the notion of 1 2 3 1 2 3 mapping on neutrosophic soft classes. Neutrosophic soft are neutrosophic soft classes. Define r: X Y and s: E E' classes are collections of neutrosophic soft sets. We also as follows: define and study the properties of neutrosophic soft images r( x11 ) = y , r( x23 ) = y , r( x33 ) = y , and neutrosophic soft inverse images of neutrosophic soft ' ' ' s( e11 ) = e , s( e23 ) = e , s( e32 ) = e . sets, and support them with example and theorems.
Let (,)FA and (,)GB be two neutrosophic soft sets Definition 3.1 Let X be a universe and E be a set of over X and Y respectively such that parameters. Then the collection of all neutrosophic soft sets over X with parameters from E is called a F, A ={ e1 , < x 1 ,0.4,0.2,0.3 >,< x 2 ,0.7,0.3,0.4 >,< x 3 ,0.5,0.2,0.2 > , neutrosophic soft class and is denoted as XE, . e2, < x 1 ,0.2,0.2,0.7 >,< x 2 ,0.3,0.1,0.8 >,< x 3 ,0.2,0.3,0.6 > ,
e3, < x 1 ,0.8,0.2,0.1>,< x 2 ,0.9,0.1,0.1>,< x 3 ,0.1,0.4,0.5 > } , Definition 3.2 Let XE, and YE, ' be neutrosophic soft ' classes. Let r: X Y and s: E E' be mappings. Then a G, B ={ e1 , < y 1 ,0.2,0.4,0.5 >,< y 2 ,0.1,0.2,0.6 >,< y 3 ,0.2,0.5,0.3 > , ' e2, < y 1 ,0.8,0.1,0.1>,< y 2 ,0.5,0.5,0.5 >,< y 3 ,0.3,0.4,0.4 > , mapping f:,, X E Y E' is defined as follows: ' e3, < y 1 ,0.7,0.3,0.3 >,< y 2 ,0.9,0.2,0.1>,< y 3 ,0.8,0.2,0.1> } ,
For a neutrosophic soft set FA, in XE, , f F, A is a Then we define a mapping f:,, X E Y E' as follows: neutrosophic soft set in YE, ' obtained as follows: For a neutrosophic soft set FA, in XE, , f F, A is a ' F , neutrosophic soft set in YE, and is obtained as follows: x r1 y 11 f F,, A y if r y and s A 0,0,0 otherwise . ' f F,= A e11 y F x r1 y 1 For s E E' , yY and sA1 .
f F, A is called a neutrosophic soft image of the neutrosophic soft set FA, .
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes 6 Neutrosophic Sets and Systems, Vol. 2, 2014
e2, < x 1 ,0.7,0.3,0.3 >,< x 2 ,0.8,0.2,0.1>,< x 3 ,0.8,0.2,0.1> , = F e, < x ,0.8,0.1,0.1>,< x ,0.3,0.4,0.4 >,< x ,0.3,0.4,0.4 > }. x{}{} x e 3 1 2 3 11
= (
= (
defined as follows: f F, A , B ={ e' , < y ,0.4,0.2,0.3 >,< y ,0,0,0 >,< y ,0.7,0.25,0.2 > , 1 1 2 3 ' e2, < y 1 ,0.8,0.2,0.1>,< y 2 ,0,0,0 >,< y 3 ,0.9,0.25,0.1> , 1 1 1 1 fFA , fGB , xfFA = , x fGB , x . ' e3, < y 1 ,0.2,0.2,0.7 >,< y 2 ,0,0,0 >,< y 3 ,0.3,0.2,0.6 > }.
Next for the neutrosophic soft inverse images, the 1 1 1 1 fFA , fGB , xfFA = , x fGB , x . mapping f1 :,, Y E' X E is defined as follows:
' ' 1 Theorem 3.1 Let f:,, X E Y E be a mapping. Then for For a neutrosophic soft set GB, in YE, , f G,, B A is a neutrosophic soft sets FA, and GB, in the neutrosophic neutrosophic soft set in XE, obtained as follows: 1 soft class XE, , [a.] f G,= B e1 x 1 G s e 1 r x 1 a. f = . ' = G e11 y b. f X Y . = 0.2,0.4,0.5 , 1 c. fFA , GB , = fFA , fGB , . f G,= B e1 x 2 G s e 1 r x 2 ' = G e13 y d. fFA ,,,, GB fFA fGB . = 0.2,0.5,0.3 , 1 f G,= B e1 x 3 G s e 1 r x 3 e. If FAGB,, , then f F,, A f G B . ' = G e13 y Proof. For (a), (b) and (e) the proof is trivial, so we just = 0.2,0.5,0.3 . give the proof of (c) and (d). By similar calculations, consequently, we get c. For E' and yY , we want to prove that 1 f G, B , A ={ e1 , < x 1 ,0.2,0.4,0.5 >,< x 2 ,0.2,0.5,0.3 >,< x 3 ,0.2,0.5,0.3 > ,
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes Neutrosophic Sets and Systems, Vol. 1, 2013 7
fFA, GB , yfFA = , y fGB , y FG 1 1 1 1 x r y s A x r y s B
=f F , A y G , B y For left hand side, consider =f F , A f G , B y . fFA , GB , yfHAB = , y . Then
H , This gives (d). 1 x r y f H,= A B y 11 (1) if r y and s A B , 0,0,0otherwise . f:,, X E Y E' Theorem 3.2 Let be mapping. Then for neutrosophic soft sets FA, , GB, in the neutrosophic soft where HFG =, . class XE, ' , we have: Considering only the non-trivial case, then Equation 1 1 becomes: 1. f = . 2. f1 Y = X .
f H, A B y = F , G (2) 1 1 1 1 f FA, GB , = fFA , fGB , x r y 3. . For right hand side and by using Definition 3.4, 1 1 1 4. f FA, GB , = fFA , fGB , . we have 5. If FAGB,, , then f11 F,, A f G B . fFA , fGB , yfFA = , y fGB , y = FG Proof. We use the same method as in the previous proof. 1 1 1 1 x r y s A x r y s B 4 Conclusion = FG 11 In this paper we have defined the notion of a x r y s A B mapping on classes where the neutrosophic soft classes are
collections of neutrosophic soft set. The properties of =, FG (3) neutrosophic soft images and neutrosophic soft inverse 1 x r y images of neutrosophic soft sets have been defined and studied.
From Equations 2 and 3, we get (c). ACKNOWLEDGEMENT d. For E' and yY , and using Definition 3.4, we have We would like to acknowledge the financial support fFA , GB , yfHAB = , y received from Shaqra University. With our sincere thanks and appreciation to Professor Smarandache for his support and his comments. = H 11 x r y s A B References = FG 11 [1] A. R. Salleh, S. Alkhazaleh, N. Hassan and A. G. Ahmad, x r y s A B Multiparameterized soft set, Journal of Mathematics and Sta- tistics, 8(1) (2012), 92–97. = FG 11 x r y s A B [2] S. Alkhazaleh and A. R. Salleh, Soft expert sets, Advances in Decision Sciences, 2011(2011), 15 pages.
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes 8 Neutrosophic Sets and Systems, Vol. 2, 2014
[3] S. Alkhazaleh, A. R. Salleh and N. Hassan, Fuzzy paramete- rized interval-valued fuzzy soft set, Applied Mathematical Sciences, 5(67) (2011), 3335-3346. Received: March 1st., 2014. Accepted: February 29, 2014
[4] S. Alkhazaleh, A. R. Salleh and N. Hassan, Neutrosophic soft set, Advances in Decision Sciences, 2011(2011), 18 pages.
[5] S. Alkhazaleh, A. R. Salleh and N. Hassan, Soft multisets theory, Applied Mathematical Sciences, 5(72) (2011), 3561– 3573.
[6] S. Alkhazaleh and A. R. Salleh, Fuzzy soft multiset, Abstract and Applied Analysis, 2012 (2012), 16 pages.
[7] D. Chen, E. C. C. Tsang, D. S. Yeung and X. Wang, The pa- rameterization reduction of soft sets and its application, Computers & Mathematics with Applications, 49(2005), 757–763.
[8] D. Molodtsov, Soft set theory–first results, Computers & Ma- thematics with Applications 37(2)(1999) 19–31.
[9] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Computers & Mathemat- ics with Applications, 44 (8–9) (2002), 1077–1083.
[10] P. K. Maji, A. R. Roy and R. Biswas, Fuzzy soft sets, Jour- nal of Fuzzy Mathematics, 9 (3) (2001), 589–602.
[11] P. K. Maji, A. R. Roy and R. Biswas, Soft set theory, Com- puters & Mathematics with Applications, 54 (4–5) (2003), 555–562.
[12] R. Roy and P. K. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics, 203 (2) (2007), 412–418.
[13] A. Kharal and B. Ahmad, Mappings on fuzzy soft classes, Advances in Fuzzy Systems 2009 (2009) 6 pages.
[14] A. Kharal and B. Ahmad, Mapping on soft classes. New Math. & Natural Comput, 7(3) (2011) , 471–481.
[15] F. Smarandache, Neutrosophic set, a generalisation of the in- tuitionistic fuzzy sets, Inter. J. Pure Appl. Math., 24 (2005), 287-297.
[16] F. Smarandache, Neutrosophy. / Neutrosophic Probabili- ty, Set, and Logic, Amer. Res. Press, Rehoboth, USA, 105 p., 1998.
[17] P. K. Maji, Neutrosophic soft set, Annals of Fuzzy Mathe- matics and Informatics, 5 (2013), 157–168.
[18] M. Bhowmik and M.Pal, Intuitionistic neutrosophic set, Journal of Information and Computing Science, 4 2. (2009), 142–152.
[19] Broumi Said and F. Smarandache, Intuitionistic neutrosophic soft set, Journal of Information and Computing Science, 8 2. (2013), 130–140.
Shawkat Alkhazaleh and Emad Marei, Mappings on Neutrosophic Soft Classes Neutrosophic Sets and Systems, Vol. 2, 2014 9
On Neutrosophic Implications
Said Broumi1 , Florentin Smarandache2
1 Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca B.P. 7951, Hassan II University Mohammedia- Casablanca, Morocco .E-mail: [email protected] 2Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA .E-mail: [email protected]
Abstract: In this paper, we firstly review the neutrosophic set, Furthermore, some of their basic properties and some and then construct two new concepts called neutrosophic results associated with the two neutrosophic implication of type 1 and of type 2 for neutrosophic sets. implications are proven.
Keywords: Neutrosophic Implication, Neutrosophic Set, N-norm, N-conorm.
1 Introduction indeterminacy, and the degree of non-membership Neutrosophic set (NS) was introduced by Florentin (or Falsehood) of the element x X to the set A Smarandache in 1995 [1], as a generalization of the fuzzy with the condition. set proposed by Zadeh [2], interval-valued fuzzy set [3], −0 ≤ + + ≤ 3+. (1) intuitionistic fuzzy set [4], interval-valued intuitionistic fuzzy set [5], and so on. This concept represents From philosophical point of view, the uncertain, imprecise, incomplete and inconsistent neutrosophic set takes the value from real information existing in the real world. A NS is a set standard or non-standard subsets of ]−0, 1+[. So where each element of the universe has a degree of truth, instead of ]−0, 1+[, we need to take the interval [0, indeterminacy and falsity respectively and with lies in] 0- , 1] for technical applications, because ]−0, 1+[will 1+ [, the non-standard unit interval. be difficult to apply in the real applications such NS has been studied and applied in different fields as in scientific and engineering problems. including decision making problems [6, 7, 8], Databases [10], Medical diagnosis problem [11], topology [12], Definition 2 (Single-valued Neutrosophic sets) [20] control theory [13], image processing [14, 15, 16] and so Let X be an universe of discourse with generic on. elements in X denoted by x. An SVNS A in X is In this paper, motivated by fuzzy implication [17] and characterized by a truth-membership function intutionistic fuzzy implication [18], we will introduce the , an indeterminacy-membership function definitions of two new concepts called neutrosophic , and a falsity-membership function , implication for neutrosophic set. for each point x in X, , , , [0, This paper is organized as follow: In section 2 some basic 1]. definitions of neutrosophic sets are presented. In section 3, When X is continuous, an SVNS A can be written we propose some sets operations on neutrosophic sets. as Then, two kind of neutrosophic implication are proposed. A= (2) Finally, we conclude the paper. When X is discrete, an SVNS A can be written as 2 Preliminaries A= (3) This section gives a brief overview of concepts of Definition 3 (Neutrosophic norm, n-norm) [19] neutrosophic sets, single valued neutrosophic sets, Mapping : (]-0,1+[ × ]-0,1+[ × ]-0,1+[)2→ ]- neutrosophic norm and neutrosophic conorm which will 0,1+[ × ]-0,1+[ × ]-0,1+[ be utilized in the rest of the paper. (x( , , ), y( , , ) ) = ( T(x,y), I(x,y), F(x,y), where Definition 1 (Neutrosophic set) [1] T(.,.), I(.,.), F(.,.) Let X be a universe of discourse then, the neutrosophic set are the truth/membership, indeterminacy, and A is an object having the form: respectively falsehood/ nonmembership A = {< x: , , >,x X}, where the components. functions T, I, F : X→ ]−0, 1+[ define respectively the degree of membership (or Truth), the degree of
Said Broumi and9 Florentin Smarandache, On Neutrosophic Implication
10 Neutrosophic Sets and Systems, Vol. 2, 2014
have to satisfy, for any x, y, z in the neutrosophic where the “ ” operator is a N-norm (verifying the logic/set M of the universe of discourse X, the following above N-conorms axioms); while the “ ” axioms operator, is a N-norm. a) Boundary Conditions: (x, 0) = 0, (x, 1) = x. For example, can be the Algebraic Product T- b) Commutativity: (x, y) = (y, x). norm/N-norm, so T1 T2= T1·T2 and can be the c) Monotonicity: If x ≤y, then (x, z) ≤ (y, z). Algebraic Product T-conorm/N-conorm, so d) Associativity: ( (x, y), z) = (x, (y, z)). T1 T2= T1+T2-T1·T2. represents the intersection operator in neutrosophic set Or can be any T-norm/N-norm, and any T- theory. conorm/N-conorm from the above. Let J {T, I, F} be a component. In 2013, A. Salama [21] introduced beside the Most known N-norms, as in fuzzy logic and set the T- intersection and union operations between two norms, are: neutrosophic set A and B, another operations • The Algebraic Product N-norm: J(x, y) = x · y defined as follows: • The Bounded N-Norm: J(x, y) = max{0, x + y −1} Definition 5 • The Default (min) N-norm: (x, y) = min{x, y}. Let A, B two neutrosophic sets A general example of N-norm would be this. A = min ( , ) ,max ( , ) , max( , ) Let x( , , ) and y ( , , ) be in the neutrosophic A B = (max ( , ) , max ( , ) ,min( , )) A B={ min ( , ), min ( , ), max ( , )} set M. Then: A B = (max ( , ) , min ( , ) ,min( , )) (x, y) = ( , , ) (4) = ( , , ). where the “ ” operator is a N-norm (verifying the above N-norms axioms); while the “ ” operator, is a N-conorm. Remark For example, can be the Algebraic Product T-norm/N- For the sake of simplicity we have denoted: norm, so = · and can be the Algebraic = min min max, = max min min Product T-conorm/N-conorm, so = + - · = min max max, = max max min. Or can be any T-norm/N-norm, and any T-conorm/N- Where , represent the intersection set and conorm from the above. the union set proposed by Florentin Smarandache and , represent the intersection set and the Definition 4 (Neutrosophic conorm, N-conorm) [19] union set proposed by A.Salama. Mapping : ( ]-0,1+[ × ]-0,1+[ × ]-0,1+[ )2→]-0,1+[ × ]- 0,1+[ × ]-0,1+[ 3 Neutrosophic Implications (x( , , ), y( , , )) = ( T(x,y), I(x,y), In this subsection, we introduce the set operations F(x,y)), on neutrosophic set, which we will work with. where T(.,.), I(.,.), F(.,.) are the truth/membership, Then, two neutrosophic implication are indeterminacy, and respectively falsehood/non mem- constructed on the basis of single valued bership components. neutrosophic set .The two neutrosophic have to satisfy, for any x, y, z in the neutrosophic implications are denoted by and . Also, logic/set M of universe of discourse X, the following important properties of and are axioms: a) Boundary Conditions: (x, 1) = 1, (x, 0) = x. demonstrated and proved. b) Commutativity: (x, y) = (y, x). c) Monotonicity: if x ≤y, then (x, z) ≤ (y, z). Definition 6 (Set Operations on Neutrosophic sets) d) Associativity: ( (x, y), z) = (x, (y, z)) Let and two neutrosophic sets , we propose represents respectively the union operator in the following operations on NSs as follows: neutrosophic set theory. @ = ( , , ) where Let J {T, I, F} be a component. Most known N- < , , ,< , , conorms, as in fuzzy logic and set the T-conorms, are: = ( , , ) ,where • The Algebraic Product N-conorm: J(x, y) = < , , ,< , , x + y −x · y • The Bounded N-conorm: J(x, y) = min{1, x # = ( , , ) , where + y} • The Default (max) N-conorm: J(x, y) = max{x, < , , ,< , , y}. B=( + - , , ) ,where A general example of N-conorm would be this. < , , ,< , , Let x( , , ) and y( , , ) be in the neutrosophic B= ( , + - , + - ), where set/logic M. Then: < , , ,< , , (x, y) = (T1 T2, I1 I2, F1 F2) (5)
Said Broumi and Florentin Smarandache, On Neutrosophic10 Implication
Neutrosophic Sets and Systems, Vol. 2, 2014 11 Obviously, for every two and , ( @ ), ( ), Max(min ( , ), min ( , ) >| x X} ( ) , B and B are also NSs. (11) Based on definition of standard implication denoted by “A Comparing the result of (10) and (11), we get B”, which is equivalent to “non A or B”. We extended Max( , min( , ))= Min( max( , ), it for neutrosophic set as follows: max( , )) Min( ,max ( , ) )= Max (min ( , ), min Definition 7 ( , )) Let A(x) ={
(ii) From definition in (5), we have Theorem 2 For A, B and C NS(X), A B ={Max( , min( , )) , Min( ,max ( , ) ) , Min( , max ( , ) >| x X} (10) i. A B C =( A C ) ( B and ( A B ) ( A C ) = { Said Broumi and11 Florentin Smarandache, On Neutrosophic Implication 12 Neutrosophic Sets and Systems, Vol. 2, 2014 ii. A B =( A B ) ( A C ) hence, A C = ( A C ) ( B C ) iii. A C = ( A C ) ( B C ) (iv) From definition in (5) ,we have iv. A B =( A B ) ( A C ) A B ={ Said Broumi and Florentin Smarandache, On Neutrosophic12 Implication Neutrosophic Sets and Systems, Vol. 2, 2014 13 From (24) and (25), we get A = = { ( , ) > | x (31) From (30) and (31), we get = = , , ) = (34) and Theorem 6 = For A, B NS(X), = , , ) ( , + - i. = , + - ) = ={< x, Max( , , Min ( , + - ii. = ), Min ( , + - ) | x } = = , , ) = (35) From (34) and (35 ), we get the result ( ii) iii. = (iii) From definition in (6) ,we have = =( iv. = , , ) = ( , , ) v. = = { Said Broumi and13 Florentin Smarandache, On Neutrosophic Implication 14 Neutrosophic Sets and Systems, Vol. 2, 2014 = ( , , )= (37) From (42) and (43), we get the result (vi). The following theorem is not valid. From ( 36) and (37), we get the result (iii). (iv) From definition in (6), we have Theorem 7 = ( , , , + - ) For A, B NS(X), ( , , ) i. = = { ,), Min( , ) > | x = = ( , , ) = (38) ii. = and = ( , , ) ( = + - , , ) iii. = ={< x, Max ( , + - ) ,Min( , ), Min( , ) > | x = ( , , ) = = (39) iv. = From (38) and (39), we get the result (iv). (v) From definition in (6), we have = = ( , v. = , ) ( , , ) = ={ Said Broumi and Florentin Smarandache, On Neutrosophic14 Implication Neutrosophic Sets and Systems, Vol. 2, 2014 15 (i) From definition in (6), we have = = ( + - , , ) ( , , ) ={ = = = = = (49) From (48) and (49), we get the result (iii). = (iv) From definition in (6), we have =( , , ) = = (45) = From ( 44) and (45), we get the result (i). (ii) From definition in (6) ,we have = = = = (50) and = = =( , , ) = = (46) and ={ Said Broumi and15 Florentin Smarandache, On Neutrosophic Implication 16 Neutrosophic Sets and Systems, Vol. 2, 2014 = 5- = = = 6- = = = = , , ) 8- = = (53) From (52) and (53), we get the result (v). = (vi) From definition in (2), we have 9- = = = Example We prove only the (i) 1- = = = ( , = (54) , ) and ={ < < A B A B V(A Remark , > , > We remark that if the indeterminacy values are < 0 ,1> < 0 ,1> < 1 ,0> < 1 ,0> < 1 ,0> restricted to 0, and the membership /non- < 0 ,1> < 1 ,0> < 1 ,0> < 1 ,0> < 1 ,0> membership are restricted to 0 and 1. The results < 1 ,0> < 0 ,1> < 0 ,1> < 0 ,1> < 0 ,1> of the two neutrosophic implications and < 1 ,0> < 1 ,0> < 1 ,0> < 1 ,0> < 1 ,0> collapse to the fuzzy /intuitionistic fuzzy Theorem 9 implications defined (V(A ) in [17] 1- = = Table 2- = Comparison of three kind of implications From the table, we conclude that fuzzy = /intuitionistic fuzzy implications are special case of neutrosophic implication. 3- = Conclusion In this paper, the neutrosophic implication is studied. The basic knowledge of the neutrosophic = set is firstly reviewed, a two kind of neutrosophic implications are constructed, and its properties. 4- = These implications may be the subject of further research, both in terms of their properties or comparison with other neutrosophic implication, = and possible applications. Said Broumi and Florentin Smarandache, On Neutrosophic16 Implication Neutrosophic Sets and Systems, Vol. 2, 2014 17 ACKNOWLEDGEMENTS [11] Ansari, Biswas, Aggarwal,”Proposal for The authors are highly grateful to the referees for their Applicability of Neutrosophic Set Theory in valuable comments and suggestions for improving the Medical AI”, International Journal of Computer Applications (0975 – 8887),Vo 27– No.5, (2011), paper. pp.5-11. [12] F.G Lupiáñez, "On neutrosophic topology", References Kybernetes, Vol. 37 Iss: 6,(2008), pp.797 - 800 Doi:10.1108/03684920810876990. [1] F. Smarandache, “An Unifying Field in Logics. [13] S. Aggarwal, R. Biswas, A.Q. Ansari, Neutrosophy: Neutrosophic Probability, Set and Logic”. “Neutrosophic Modeling and Control”,978-1- Rehoboth: American Research Press, (1999). 4244-9034-/10 IEEE,( 2010), pp.718-723. [2] L.A., Zadeh “Fuzzy Sets”, Information and Control, 8, [14] H. D. Cheng,& Y Guo. “A new neutrosophic (1965), pp.338-353. approach to image thresholding”. New [3] Atanassov K. T., Intuitionistic fuzzy sets. Fuzzy Sets and Mathematics and Natural Computation, 4(3), Systems 20(1), (1986), pp.87-96. (2008), pp. 291–308. [4] Atanassov K T , Intuitionistic fuzzy sets. Springer Physica- [15] Y.Guo,&, H. D. Cheng “New neutrosophic Verlag, Heidelberg,1999. approach to image segmentation”.Pattern Recognition, 42, (2009), pp.587–595. [5] K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems,Vol. 31, [16] M.Zhang, L.Zhang, and H.D.Cheng. “A Issue 3, (1989), pp. 343 – 349. neutrosophic approach to image segmentation based on watershed method”. Signal Processing [6] S.Broumi and F. Smarandache, “Intuitionistic 5, 90 , (2010), pp.1510-1517. Neutrosophic Soft Set”, Journal of Information and Computing Science, England, UK, ISSN 1746-7659,Vol. [17] A. G. Hatzimichailidis, B. K. Papadopoulos, 8, No. 2, (2013), pp.130-140. V.G.Kaburlasos. “An implication in Fuzzy Sets”, IEEE International Conference on Fuzzy [7] S.Broumi, “Generalized Neutrosophic Soft Set”, Systems, (2006) pp.1 – 6. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), ISSN: 2231-3605, [18] A. G. Hatzimichailidis, B. K. Papadopoulos., “A E-ISSN : 2231-3117, Vol.3, No.2, (2013), pp.17-30. new implication in the Intuitionistic fuzzy sets”, Notes on IFS, Vol. 8 (2002), No4 p. 79-104. [8] J. Ye,”Similarity measures between interval neutrosophic sets and their multicriteria decision-making method “ [19] F. Smarandache,” N-norm and N-conorm in Neutrosophic Logic and Set, and the Journal of Intelligent & Fuzzy Systems, DOI: 10.3233/IFS- Neutrosophic Topologies”, in Critical Review, 120724 ,(2013),pp. Creighton University, USA, Vol. III, 73-83, 2009. [9] M.Arora, R.Biswas,U.S.Pandy, “Neutrosophic Relational Database Decomposition”, International Journal of [20] H. Wang, F. Smarandache Y.Q. Zhang, et al., Advanced Computer Science and Applications, Vol. 2, No. Single valued neutrosophic sets,Multispace and 8, (2011), pp.121-125. Multistructure4(2010), 503410–413 [10] M. Arora and R. Biswas,” Deployment of Neutrosophic [21] A.A. Salama and S.A. Al-Blowi; “Neutrosophic technology to retrieve answers for queries posed in natural Set And Neutrosophic Topological Spaces”, language”, in 3rdInternational Conference on Computer IOSR Journal of Mathematics (IOSR-JM) ISSN: Science and Information Technology ICCSIT, IEEE 2278-5728. Volume 3, Issue 4 (Sep-Oct. 2012), catalog Number CFP1057E-art,Vol.3, ISBN: 978-1-4244- pp.31-35. 5540-9,(2010), pp. 435-439. Received: January 9th, 2014. Accepted: January 23th, 2014. Said Broumi and17 Florentin Smarandache, On Neutrosophic Implication Neutrosophic Sets and Systems, Vol. 2, 2014 Neutrosophic Hierarchical Clustering Algoritms Rıdvan Şahin Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey. [email protected] Abstract. Interval neutrosophic set (INS) is a generaliza- hierarchical clustering procedure, the single valued neu- tion of interval valued intuitionistic fuzzy set (IVIFS), trosophic aggregation operator, and the basic distance whose the membership and non-membership values of el- measures between SVNSs, we define a single valued neu- ements consist of fuzzy range, while single valued neutro- trosophic hierarchical clustering algorithm for clustering sophic set (SVNS) is regarded as extension of intuition- SVNSs. Then we extend the algorithm to classify an inter- istic fuzzy set (IFS). In this paper, we extend the hierar- val neutrosophic data. Finally, we present some numerical chical clustering techniques proposed for IFSs and IVIFSs examples in order to show the effectiveness and availabil- to SVNSs and INSs respectively. Based on the traditional ity of the developed clustering algorithms. Keywords: Neutrosophic set, interval neutrosophic set, single valued neutrosophic set, hierarchical clustering, neutrosophic aggre- gation operator, distance measure. 1 Introduction Clustering is an important process in data mining, pat- 2.1 Neutrosophic sets tern recognition, machine learning and microbiology analy- Definition 1. [3] Let 푋 be a space of points (objects) and sis [1, 2, 14, 15, 21]. Therefore, there are various types of 푥 ∈ 푋. A neutrosophic set 푁 in 푋 is characterized by a techniques for clustering data information such as numerical truth-membership function 푇 , an indeterminacy-member- information, interval-valued information, linguistic infor- 푁 ship function 퐼 and a falsity-membership function 퐹 , mation, and so on. Several of them are clustering algorithms 푁 푁 where 푇 (푥), 퐼 (푥) and 퐹 (푥) are real standard or non- such as partitional, hierarchical, density-based, graph-based, 푁 푁 푁 standard subsets of ]0−, 1+[. That is, 푇 ∶ 푈 →]0−, 1+[, 퐼 ∶ model-based. To handle uncertainty, imprecise, incomplete, 푁 푁 푈 →]0−, 1+[ and 퐹 ∶ 푈 →]0−, 1+[. and inconsistent information which exist in real world, 푁 There is no restriction on the sum of 푇 (푥), 퐼 (푥) and Smarandache [3, 4] proposed the concept of neutrosophic 푁 푁 퐹 (푥), so set (NS) from philosophical point of view. A neutrosophic 푁 0− ≤ sup 푇 (푥) + sup 퐼 (푥) + sup 퐹 (푥) ≤ 3+. set [3] is a generalization of the classic set, fuzzy set [13], 푁 푁 푁 intuitionistic fuzzy set [11] and interval valued intuitionistic Neutrosophic sets is difficult to apply in real scientific fuzzy set [12]. It has three basic components independently and engineering applications [5]. So Wang et al. [5] pro- of one another, which are truht-membership, indeterminacy- posed the concept of SVNS, which is an instance of neutro- membership, and falsity-membership. However, the neutro- sophic set. sophic sets is be diffucult to use in real scientiffic or engi- neering applications. So Wang et al. [5, 6] defined the con- 2.2 Single valued neutrosophic sets cepts of single valued neutrosophic set (SVNS) and interval neutrosophic set (INS) which is an instance of a neutro- A single valued neutrosophic set has been defined in [5] sophic set. At present, studies on the SVNSs and INSs is as follows: progressing rapidly in many different aspects [7, 8, 9, 10, 16, 18]. Yet, until now there has been little study on clustering Definition 2. Let 푋 be a universe of discourse. A single val- the data represented by neutrosophic information [9]. There- ued neutrosophic set 퐴 over 푋 is an object having the form: fore, the existing clustering algorithms cannot cluster the 퐴 = {〈푥, 푢퐴(푥), 푤퐴(푥), 푣퐴(푥)〉 ∶ 푥 ∈ 푋}, neutrosophic data, so we need to develop some new tech- where 푢퐴: 푋 → [0,1] , 푤퐴: 푋 → [0,1] and 푣퐴: 푋 → [0,1] niques for clustering SVNSs and INSs. with the condition 2 Preliminaries 0 ≤ 푢퐴(푥) + 푤퐴(푥) + 푣퐴(푥) ≤ 3, ∀푥 ∈ 푋. The numbers 푢퐴(푥), 푤퐴(푥) and 푣퐴(푥) denote the de- In this section we recall some definitions, operations and gree of truth-membership, indeterminacy membership and properties regarding NSs, SVNSs and INSs, which will be falsity-membership of 푥 to 푋, respectively. used in the rest of the paper. Definition 3. Let 퐴 and 퐵 be two single valued neutro- sophic sets, Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 퐴 = {〈푥, 푢퐴(푥), 푤퐴(푥), 푣퐴(푥)〉 ∶ 푥 ∈ 푋} Int[0,1] with the condition 퐵 = {〈푥, 푢퐵 (푥), 푤퐵 (푥), 푣퐵 (푥)〉 ∶ 푥 ∈ 푋} 0 ≤ sup 푢퐴̃(푥) + sup 푤퐴̃(푥) + sup 푣퐴̃(푥) ≤ 3, Then we can give two basic operations of 퐴 and 퐵 as fol- for all 푥 ∈ 푋. The intervals 푢 (푥), 푤 (푥) and 푣 (푥) denote the truth- lows: 퐴̃ 퐴̃ 퐴̃ membership degree, the indeterminacy membership degree 1. 퐴 + 퐵 = {< 푥, 푢퐴(푥) + 푢퐵(푥) − 푢퐴(푥). 푢퐵(푥), and the falsity-membership degree of 푥 to 퐴̃, respectively. 푤퐴(푥). 푤퐵(푥), 푣퐴(푥). 푣퐵(푥) >∶ 푥 ∈ 푋}; For convenience, if let 휆 휆 휆 2. 휆퐴 = {< 푥, 1 − (1 − 푢 (푥)) , (푤 (푥)) , (푣 (푥)) >: + − 퐴 퐴 퐴 푢퐴̃(푥) = [푢퐴̃ (푥), 푢퐴̃ (푥)] 푥 ∈ 푋 and 휆 > 0} + − 푤퐴̃(푥) = [푤퐴̃ (푥), 푤퐴̃ (푥)] + − Definition 4. Let 푋 = {푥1, 푥2, … , 푥푛} be a universe of dis- 푣퐴̃(푥) = [푣퐴̃ (푥), 푣퐴̃ (푥)] course. Consider that the elements 푥 (i = 1,2,...,n) in the 푖 then universe 푋 may have different importance, let 휔 = 푇 ̃ − + − + − + (휔1, 휔2, . . . , 휔푛) be the weight vector of 푥푖 (i = 1,2,...,n), 퐴 = {〈푥, [푢퐴̃ (푥), 푢퐴̃ (푥)], [푤퐴̃ (푥), 푤퐴̃ (푥)], [푣퐴̃ (푥), 푣퐴̃ (푥)]〉}: 푥 ∈ 푋} with 휔 ≥ 0, i = 1,2,...,n, ∑푛 휔푖 = 1. Assume that 푖 푖=1 with the condition 퐴 = {〈푥, 푢퐴(푥), 푤퐴(푥), 푣퐴(푥)〉: 푥 ∈ 푋} and + + + 0 ≤ sup 푢퐴̃ (푥) + sup 푤퐴̃ (푥) + sup 푣퐴̃ (푥) ≤ 3, 퐵 = {〈푥, 푢퐵(푥), 푤퐵(푥), 푣퐵(푥)〉: 푥 ∈ 푋} for all 푥 ∈ 푋 . If 푤 (푥) = [0,0] and sup 푢+(푥) + be two SVNSs. Then we give the following distance 퐴̃ 퐴̃ sup 푣+ ≤ 1 then 퐴̃ reduces to an interval valued intuition- measures: 퐴̃ istic fuzzy set. The weighted Hamming distance and normalized Hamming distance [9] Definition 6. [20] Let 퐴̃ and 퐵̃ be two interval neutrosophic sets, 푒휔(퐴, 퐵) = (1 ∑푛 휔 (|푢 (푥 ) − 푢 (푥 )| + |푤 (푥 ) − 푤 (푥 )| + 1 3 푖=1 푖 퐴 푖 퐵 푖 퐴 푖 퐵 푖 ̃ − + − + − + 퐴 = {〈푥, [푢퐴̃ (푥), 푢퐴̃ (푥)], [푤퐴̃ (푥), 푤퐴̃ (푥)], [푣퐴̃ (푥), 푣퐴̃ (푥)]〉: 푥 ∈ 푋}, |푣 (푥 ) − 푣 (푥 )|)). (1) 퐴 푖 퐵 푖 ̃ − + − + − + 퐵 = {〈푥, [푢퐵̃ (푥), 푢퐵̃ (푥)], [푤퐵̃ (푥), 푤퐵̃ (푥)], [푣퐵̃ (푥), 푣퐵̃ (푥)]〉: 푥 ∈ 푋}. ̃ ̃ Assume that 휔 = (1⁄푛 , 1⁄푛 , . . . , 1⁄푛)푇, then Eq. (1) is re- Then two basic operations of 퐴 and 퐵 are given as follows: ̃ ̃ − − + − + duced to the normalized Hamming distance 1. 퐴 + 퐵 = {< 푥, [푢퐴̃ (푥) + 푢퐵̃ (푥)−, 푢퐴̃ (푥) ⋅ 푢퐵̃ (푥), 푢퐴̃ (푥) + 푛 1 푛 + + + − − + + 푒 (퐴, 퐵) = ( ∑ (|푢 (푥 ) − 푢 (푥 )| + |푤 (푥 ) − 푤 (푥 )| + 푢퐵̃ (푥) − 푢퐴̃ (푥) ⋅ 푢퐵̃ (푥)], [푢퐴̃ (푥) ⋅ 푤퐵̃ (푥), 푤퐴̃ (푥) ⋅ 푤퐵̃ (푥)], 2 3푛 푖=1 퐴 푖 퐵 푖 퐴 푖 퐵 푖 − − + + [푣퐴̃ (푥) ⋅ 푣퐵̃ (푥), 푣퐴̃ (푥) ⋅ 푣퐵̃ (푥)]: 푥 ∈ 푋} |푣퐴(푥푖) − 푣퐵(푥푖)|)) (2) 휆 휆 ̃ −( ) +( ) The weighted Euclidean distance and normalized Euclidean 2. 휆퐴 = {< 푥, [, 1 − (1 − 푢퐴̃ 푥 ) , 1 − (1 − 푢퐴̃ 푥 ) ] , 휆 휆 휆 휆 distance [7] − + − + [(푤퐴̃ (푥)) , (푤퐴̃ (푥)) ] , [(푣퐴̃ (푥)) , (푣퐴̃ (푥)) ] >: 푥 ∈ 푋 and 휆 > 0}. 푒휔(퐴, 퐵) = (1 ∑푛 휔 (|푢 (푥 ) − 푢 (푥 )|)2 + (|푤 (푥 ) − 푤 (푥 )|)2 + 3 3 푖=1 푖 퐴 푖 퐵 푖 퐴 푖 퐵 푖 1 Definition 7. Let 푋 = {푥1, 푥2, . . . , 푥푛} be a universe of dis- 2 2 (|푣퐴(푥푖) − 푣퐵(푥푖)|) ) (3) course. Consider that the elements 푥푖 (i = 1,2,...,n) in the uni- verse 푋 may have different importance, let 휔 = T 푇 Assume that ω = (1/n,1/n,...,1/n) , then Eq. (3) is reduced to (휔1, 휔2, . . . , 휔푛) be the weight vector of 푥푖 (i = 1,2,...,n), 푛 ̃ the normalized Euclidean distance with 휔푖 ≥ 0, i = 1,2,...,n, ∑푖=1 휔푖 = 1. Suppose that 퐴 and ̃ 1 퐵 are two interval neutrosophic sets. Ye [6] has defined the 푒푛(퐴, 퐵) = ( ∑푛 (|푢 (푥 ) − 푢 (푥 )|)2 + (|푤 (푥 ) − 4 3푛 푖=1 퐴 푖 퐵 푖 퐴 푖 distance measures for INSs as follows: 1 The weighted Hamming distance and normalized Hamming 2 2 2 푤퐵(푥푖)|) + (|푣퐴(푥푖) − 푣퐵(푥푖)|) ) (4) distance: 휔 1 푛 − − + + 푑 (퐴̃, 퐵̃) = ( ∑ 휔 (|푢 ̃ (푥) − 푢 (푥)| + |푢 ̃ (푥) − 푢 (푥)| 1 6 푖=1 푖 퐴 퐵 퐴 퐵 2.3 Interval neutrosophic sets + + − − + + +|푤퐴̃ (푥) − 푤퐵 (푥)| + |푣퐴̃ (푥) − 푢퐵 (푥)| + |푣퐴̃ (푥) − 푢퐵 (푥)|) (5) Definition 5. [3] Let 푋 be a set and Int[0,1] be the set of all 푇 closed subsets of [0,1]. An INS 퐴̃ in 푋 is defined with the Assume that 휔 = (1⁄푛 , 1⁄푛 , . . . , 1⁄푛) , then Eq. (5) is re- form duced to the normalized Hamming distance 휔 ̃ 1 푛 − − 푑2 (퐴, 퐵̃) = ( ∑ (|푢 ̃ (푥) − 푢 ̃ (푥)| + 퐴̃ = {〈푥, 푢퐴̃(푥), 푤퐴̃(푥), 푣퐴̃(푥)〉 ∶ 푥 ∈ 푋} 6푛 푖=1 퐴 퐵 + + + + − − |푢퐴̃ (푥) − 푢퐵̃ (푥)| + |푤퐴̃ (푥) − 푤퐵̃ (푥)| + |푣퐴̃ (푥) − 푢퐵̃ (푥)| + where 푢퐴̃: 푋 → Int[0,1] , 푤퐴̃: 푋 → Int[0,1] and 푣퐴̃: 푋 → Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 + + |푣퐴̃ (푥) − 푢퐵̃ (푥)|) (6) (푘 = 1,2, . . . , 푛) be a collection single valued neutro- sophic sets. A mapping 퐹휔 ∶ 푆푉푁푆푛 → 푆푉푁푆 is called a The weighted Euclidean distance and normalized Hamming single valued neutrosophic weighted averaging operator of distance dimension 푛 if it is satisfies 휔 1 푛 − − 2 + + 2 푑 (퐴̃, 퐵̃) = ( ∑ 휔 (|푢 ̃ (푥) − 푢 (푥)| + |푢 ̃ (푥) − 푢 (푥)| + 푛 3 6 푖=1 푖 퐴 퐵̃ 퐴 퐵̃ 퐹휔 (퐴1, 퐴2, . . . , 퐴푛) = ∑푘=1 휔푘퐴푘 − − 2 + + 2 − − 2 푇 |푤퐴̃ (푥) − 푤퐵̃ (푥)| + |푤퐴̃ (푥) − 푤퐵̃ (푥)| + |푣퐴̃ (푥) − 푢퐵̃ (푥)| + where 휔 = (휔1, 휔2, … , 휔푛) is the weight vector of 푛 1 퐴푘(푘 = 1,2, . . . , 푛), 휔푘 ∈ [0,1] and ∑푘=1 휔푘 = 1. + + 2 2 |푣퐴 (푥) − 푢퐵 (푥)| )) (7) Theorem 2. Suppose that Assume that 휔 = (1/푛, 1/푛, . . . ,1/푛)푇, then Eq. (7) is re- 퐴 = 〈푢 , 푤 , 푣 〉 duced to the normalized Hamming distance 푘 퐴푘 퐴푘 퐴푘 푛 1 푛 − − 2 + + 2 (푘 = 1,2, . . . , 푛) are single valued neutrosophic sets. Then 푑 (퐴̃, 퐵̃) = ( ∑ (|푢 ̃ (푥) − 푢 (푥)| + |푢 ̃ (푥) − 푢 (푥)| + 4 6푛 푖=1 퐴 퐵̃ 퐴 퐵̃ the aggregation result through using the single valued neu- 2 2 2 − − + + − − trosophic weighted averaging operator 퐹휔 is single neutro- |푤퐴̃ (푥) − 푤퐵̃ (푥)| + |푤퐴̃ (푥) − 푤퐵̃ (푥)| + |푣퐴̃ (푥) − 푢퐵̃ (푥)| + 1 sophic set and + + 2 2 |푣퐴̃ (푥) − 푢퐵̃ (푥)| )) (8) 퐹휔 (퐴1, 퐴2, . . . , 퐴푛) = 퐴푘 휔푘 ∏푛 ( ) Definition 8. [20] Let =< 1 − 푘=1 (1 − 푢퐴푘 푥 ) , 퐴̃ = 〈[푢−(푥), 푢+(푥)], [푤−(푥), 푤+(푥)], [푣−(푥), 푣+(푥)]〉 휔푘 휔푘 푘 퐴̃ 퐴̃ 퐴̃ 퐴̃ 퐴̃ 퐴̃ ∏푛 ( ) ∏푛 ( ) 푘=1 (푤퐴푘 푥 ) , 푘=1 (푣퐴푘 푥 ) > (10) 푘 = 1,2, … , . 푛) be a collection of interval neutrosophic sets. 푇 ̃ 푛 where 휔 = (휔1, 휔2, … , 휔푛) is the weight vector of A mapping 퐹휔 ∶ 퐼푁푆 → 퐼푁푆 is called an interval neutro- 푛 sophic weighted averaging operator of dimension 푛 if it is 퐴푘(푘 = 1,2, . . . , 푛), 휔푘 ∈ [0,1] and ∑푘=1 휔푘 = 1. 푇 satisfies Suppose that 휔 = (1/푛, 1/푛, … ,1/푛) , then the 퐹휔 is ̃ ̃ ̃ ̃ 푛 ̃ 퐹휔(퐴1, 퐴2, … , 퐴푛) = ∑푘=1 휔푘퐴푘 called an arithmetic average operator for SVNSs. 푇 where 휔 = (휔1, 휔2, . . . , 휔푛) is the weight vector of 퐴̃푘 (푘 = 3 Neutrosophic hierarchical algorithms 푛 1,2, . . . , 푛), 휔푘 ∈ [0,1] and ∑푘=1 휔푘 = 1. The traditional hierarchical clustering algorithm [17, Theorem 1. [20] Suppose that 19] is generally used for clustering numerical information. 퐴̃ = 〈[푢−(푥), 푢+(푥)], [푤−(푥), 푤+(푥)], [푣−(푥), 푣+(푥)]〉 By extending the traditional hierarchical clustering algo- 푘 퐴 퐴 퐴 퐴 퐴 퐴 rithm, Xu [22] introduced an intuitionistic fuzzy hierar- 푘 = 1,2, … , . 푛) are interval neutrosophic sets. Then the ag- chical clustering algorithm for clustering IFSs and extended gregation result through using the interval neutrosophic it to IVIFSs. However, they fail to deal with the data infor- weighted averaging operator Fω is an interval neutrosophic mation expressed in neutrosophic environment. Based on set and extending the intuitionistic fuzzy hierarchical clustering al- ̃ ̃ ̃ ̃ gorithm and its extended form, we propose the neutrosophic 퐹̃휔(퐴1, 퐴2, … , 퐴푛) = 퐴푘 휔 휔 푛 − 푘 푛 + 푘 hierarchical algorithms which are called the single valued =< [1 − ∏ (1 − 푢 ̃ (푥)) , 1 − ∏ (1 − 푢 ̃ (푥)) ], 푘=1 퐴푘 푘=1 퐴푘 neutrosophic hierarchical clustering algorithm and interval 휔푘 휔푘 [∏푛 (푤− (푥)) , ∏푛 (푤+ (푥)) ], neutrosophic hierarchical clustering algorithm. 푘=1 퐴̃푘 푘=1 퐴̃푘 휔 휔 푛 − 푘 푛 + 푘 Algorithm 1. Let us consider a collection of n SVNSs [∏푘=1 (푣퐴̃ (푥)) , ∏푘=1 (푣퐴̃ (푥)) ] > (9) 푘 푘 퐴푘(푘 = 1,2, . . . , 푛). In the first stage, the algorithm starts by assigning each of the n SVNSs to a single cluster. Based 푇 ̃ where 휔 = (휔1, 휔2, . . . , 휔푛) is the weight vector of 퐴푘 on the weighted Hamming distance (1) or the weighted Eu- 푛 (푘 = 1,2, . . . , 푛), 휔 ∈ [0,1] and ∑ 휔 = 1. 푘 푘=1 푘 clidean distance (3), the SVNSs 퐴푘(푘 = 1,2, . . . , 푛) are Suppose that 휔 = (1/푛, 1/푛, . . . ,1/푛)푇 then the 퐹̃휔 is then compared among themselves and are merged them into called an arithmetic average operator for INSs. a single cluster according to the closest (with smaller dis- tance) pair of clusters. The process are continued until all Since INS is a generalization of SVNS, according to Defi- the SVNSs 퐴푘 are merged into one cluster i.e., clustered into nition 8 and Theorem 1, the single valued neutrosophic a single cluster of size n. In each stage, only two clusters can weighted averaging operator can be easily obtained as fol- be merged and they cannot be separated after they are lows. merged, and the center of each cluster is recalculated by us- Definition 9. Let ing the arithmetic average (from Eq. (10)) of the SVNSs 〈 〉 퐴푘 = 푢퐴푘, 푤퐴푘, 푣퐴푘 proposed to the cluster. The distance between the centers of Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 each cluster is considered as the distance between two clus- 퐴5 = {(푥1, 0.90,0.20,0.00), (푥2, 0.90,0.40,0.10), ters. (푥3, 0.80,0.05,0.10), (푥4, 0.70,0.45,0.20), (푥 , 0.50,0.25,0.15), (푥 , 0.30,0.30,0.65), However, the clustering algorithm given above cannot 5 6 (푥 , 0.15,0.10,0.75), (푥 , 0.65,0.50,0.80)} cluster the interval neutrosophic data. Therefore, we need 7 8 another clustering algorithm to deal with the data repre- Now we utilize Algorithm 1 to classify the building ma- sented by INSs. terials 퐴푘(푘 = 1,2, . . . ,5): Algorithm 2. Let us consider a collection of n INSs ̃ Step1 In the first stage, each of the SVNSs 퐴푘(푘 = 퐴푘(푘 = 1,2, . . . , 푛). In the first stage, the algorithm starts 1,2, . . . ,5) is considered as a unique cluster by assigning each of the n INSs to a single cluster. Based on {퐴1}, {퐴2}, {퐴3}, {퐴4}, {퐴5}. the weighted Hamming distance (5) or the weighted Euclid- ean distance (7), the INSs 퐴̃푘(푘 = 1,2, . . . , 푛) are then Step2 Compare each SVNS 퐴푘 with all the other four compared among themselves and are merged them into a SVNSs by using Eq. (1): single cluster according to the closest (with smaller dis- tance) pair of clusters. The process are continued until all 휔 푒1 (퐴1, 퐴2) = 푑1 (퐴2, 퐴1) = 0.6403 the INSs 퐴̃푘 are merged into one cluster i.e., clustered into a 휔 푒1 (퐴1, 퐴3) = 푑1 (퐴3, 퐴1) = 0.5191 single cluster of size n. In each stage, only two clusters can 휔 푒1 (퐴1, 퐴4) = 푑1 (퐴4, 퐴1) = 0.7120 be merged and they cannot be separated after they are 휔 푒1 (퐴1, 퐴5) = 푑1 (퐴5, 퐴1) = 0.5435 merged, and the center of each cluster is recalculated by us- 휔 푒1 (퐴2, 퐴3) = 푑1 (퐴3, 퐴2) = 0.5488 ing the arithmetic average (from Eq. (9)) of the INSs pro- 휔 푒1 (퐴2, 퐴4) = 푑1 (퐴4, 퐴2) = 0.4546 posed to the cluster. The distance between the centers of 휔 푒1 (퐴2, 퐴5) = 푑1 (퐴5, 퐴2) = 0.6775 each cluster is considered as the distance between two clus- 휔 푒1 (퐴3, 퐴4) = 푑1 (퐴4, 퐴3) = 0.3558 ters. 휔 푒1 (퐴3, 퐴5) = 푑1 (퐴5, 퐴3) = 0.2830 휔 푒1 (퐴4, 퐴5) = 푑1 (퐴5, 퐴4) = 0.3117 3.1 Numerical examples. and hence 휔 Let us consider the clustering problem adapted from 푒1 (퐴1, 퐴3) = 휔 휔 휔 휔 [21]. min{푒1 (퐴1, 퐴2), 푒1 (퐴1, 퐴3) , 푒1 (퐴1, 퐴4), 푒1 (퐴1, 퐴5)} = 0.5191, Example 1. Assume that five building materials: sealant, 휔 floor varnish, wall paint, carpet, and polyvinyl chloride 푒1 (퐴2, 퐴4) = 휔( ) 휔( ) 휔( ) 휔( ) flooring, which are represented by the SVNSs 퐴푘(푘 = min{푒1 퐴2, 퐴1 , 푒1 퐴2, 퐴3 , 푒1 퐴2, 퐴4 , 푒1 퐴2, 퐴5 } 1,2, . . . ,5) in the feature space 푋 = {푥1, 푥2, . . . , 푥8}. = 0.4546, 휔 = (0.15,0.10,0.12,0.15,0.10,0.13,0.14,0.11) is the 휔 푒1 (퐴3, 퐴5) = weight vector of 푥푖(푖 = 1,2, . . . ,8), and the given data are 휔 휔 휔 휔 min{푒1 (퐴3, 퐴1), 푒1 (퐴3, 퐴2) , 푒1 (퐴3, 퐴4), 푒1 (퐴3, 퐴5)} listed as follows: = 0.2830. 퐴1 = {(푥1 ,0.20,0.05,0.50), (푥2, 0.10,0.15,0.80), (푥3 ,0.50,0.05,0.30), (푥4, 0.90,0.55,0.00), Then since only two clusters can be merged in each stage, (푥5 ,0.40,0.40,0.35), (푥6, 0.10,0.40,0.90), the SVNSs 퐴푘(푘 = 1,2, . . . ,5) can be clustered into the fol- (푥7 ,0.30,0.15,0.50), (푥8, 1.00,0.60,0.00), } lowing three clusters at the second stage {퐴1}, {퐴2, 퐴4}, {퐴3, 퐴5 }. 퐴2 = {(푥1, 0.50,0.60,0.40), (푥2, 0.60,0.30,0.15)}, (푥3, 1.00,0.60,0.00), (푥4, 0.15,0.05,0.65), Step3 Calculate the center of each cluster by using Eq. (10) (푥5, 0.00,0.25,0.80), (푥6, 0.70,0.65,0.15), { } (푥7, 0.50,0.50,0.30), (푥8, 0.65,0.05,0.20)} 푐 퐴1 = 퐴1 푐{퐴2, 퐴4} = 퐹휔(퐴2, 퐴4) 퐴3 = {(푥1, 0.45,0.05,0.35), (푥2, 0.60,0.50,0.30)}, = {(푥1, 1.00,0.62,0.00), (푥2, 1.00,0.27,0.00), (푥3, 0.90,0.05,0.00), (푥4, 0.10,0.60,0.80), (푥3, 1.00,0.62,0.00), (푥4, 0.17,0.05,0.72), (푥5, 0.20,0.35,0.70), (푥6, 0.60,0.40,0.20), (푥5, 0.07,0.27,0.82), (푥6, 0.48,0.62,0.32), (푥7, 0.15,0.05,0.80), (푥8, 0.20,0.60,0.65)} (푥 , 0.40,0.54,0.45), (푥 , 0.58,0.13,0.37)} 7 8 퐴4 = {(푥1, 1.00,0.65,0.00), (푥2, 1.00,0.25,0.00)}, 푐{퐴3, 퐴5} = 퐹휔(퐴3, 퐴5) (푥3, 0.85,0.65,0.10), (푥4, 0.20,0.05,0.80), = {(푥1, 0.76,0.10,0.00), (푥2, 0.80,0.44,0.17), (푥5, 0.15,0.30,0.85), (푥6, 0.10,0.60,0.70), (푥3, 0.85,0.05,0.00), (푥4, 0.48,0.51,0.40), (푥7, 0.30,0.60,0.70), (푥8, 0.50,0.35,0.70)} (푥5, 0.36,0.29,0.32), (푥6, 0.47,0.34,0.36), (푥7, 0.15,0.07,0.77), (푥8, 0.47,0.54,0.72)}. Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 and then compare each cluster with the other two clus- (푥5, [0.70,0.75], [0.15,0.55], [0.10,0.20]), ters by using Eq. (1): (푥6, [0.90,1.00], [0.30,0.35], [0.00,0.00])}. 퐴̃ = (푥 , [0.20,0.30], [0.85,0.60], [0.40,0.45), 푒휔(푐{퐴 }, 푐{퐴 , 퐴 }) = 푒휔(푐{퐴 , 퐴 }, 푐{퐴 }) = 0.7101, 3 1 1 1 2 4 1 2 4 1 (푥 , [0.80,0.90], [0.10,0.25], [0.00,0.10]), 푒휔(푐{퐴 }, 푐{퐴 , 퐴 }) = 푒휔(푐{퐴 , 퐴 }, 푐{퐴 }) = 5266, 2 1 1 3 5 1 3 5 1 (푥 , [0.10,0.20], [0.00,0.05], [0.70,0.80]), 푒휔(푐{퐴 , 퐴 }, 푐{퐴 , 퐴 }) = 푒휔(푐{퐴 , 퐴 }, 푐{퐴 , 퐴 }) 3 1 2 4 3 5 1 3 5 2 4 (푥 , [0.15,0.20], [0.25,0.45], [0.70,0.75]), = 0.4879. 4 (푥5, [0.00,0.10], [0.25,0.35], [0.80,0.90]), Subsequently, the SVNSs 퐴푘(푘 = 1,2, . . . ,5) can be (푥6, [0.60,0.70], [0.15,0.25], [0.20,0.30])}. clustered into the following two clusters at the third 퐴̃4 = (푥1, [0.60,0.65], [0.05,0.10], [0.30,0.35]), stage {퐴1}, {퐴2, 퐴3, 퐴4, 퐴5 }. (푥2, [0.45,0.50], [0.45,0.55], [0.30,0.40]), Finally, the above two clusters can be further clustered (푥3, [0.20,0.25], [0.05,0.25], [0.65,0.70]), into a unique cluster {퐴1, 퐴2, 퐴3, 퐴4, 퐴5 }. (푥4, [0.20,0.30], [0.35,0.45], [0.50,0.60]), (푥 , [0.00,0.10], [0.35,0.75], [0.75,0.80]), All the above processes can be presented as in Fig. 1. 5 (푥6, [0.50,0.60], [0.00,0.05], [0.20,0.25])}. Here Algorithm 2 can be used to classify the enter- {A1,A2,A3,A4,A5} } prises퐴̃푘(푘 = 1,2,3,4): ̃ Step 1 In the first stage, each of the INSs 퐴푘(푘 = {A2,A3,A4,A5} 1,2,3,4) is considered as a unique cluster } {퐴̃1}, {퐴̃2}, {퐴̃3}, {퐴̃4} {A2,A4} {A3,A5} Step 2 Compare each INS 퐴̃ with all the other three } } 푘 {A1} {A2} {A4} {A3} {A5} INSs by using Eq. (5) FIGURE 1: Classification of the building materi- 휔 ̃ ̃ 휔 ̃ ̃ 푑1 (퐴1, 퐴2) = 푑1 (퐴2, 퐴1) = 0.3337, als 퐴 (푘 = 1,2, . . . ,5) 휔 ̃ ̃ 휔 ̃ ̃ 푘 푑1 (퐴1, 퐴3) = 푑1 (퐴3, 퐴1) = 0.2937, 휔 ̃ ̃ 휔 ̃ ̃ Example 2. Consider four enterprises, represented by 푑1 (퐴1, 퐴4) = 푑1 (퐴4, 퐴1) = 0.2041, 휔 ̃ ̃ 휔 ̃ ̃ the INSs 퐴̃푘(푘 = 1,2,3,4) in the attribute set 푋 = 푑1 (퐴2, 퐴3) = 푑1 (퐴3, 퐴2) = 0.3508, 휔 ̃ ̃ 휔 ̃ ̃ {푥1, 푥2, . . . , 푥6}, where (1) 푥1 −the ability of sale; (2) 푑1 (퐴2, 퐴4) = 푑1 (퐴4, 퐴2) = 0.2970, 휔 ̃ ̃ 휔 ̃ ̃ 푥2 −the ability of management; (3) 푥3 −the ability of 푑1 (퐴3, 퐴4) = 푑1 (퐴4, 퐴3) = 0.2487, production; (4) 푥 −the ability of technology; (5) 4 then the INSs 퐴̃ (푘 = 1,2,3,4) can be clustered into 푥 −the ability of financing; (6) 푥 −the ability of risk 푘 5 6 the following three clusters at the second stage bearing (the weight vector of 푥 (푖 = 1,2, . . . ,6) is 휔 = 푖 {퐴̃ , 퐴̃ }, {퐴̃ }, {퐴̃ }. (0.25,0.20,15,0.10,0.15,0.15) . The given data are 1 4 2 3 listed as follows. Step 3 Calculate the center of each cluster by using Eq. (9) 퐴̃1 = {(푥1, [0.70,0.75], [0.25,0.45], [0.10,0.15]), (푥2, [0.00,0.10], [0.15,0.15], [0.80,0.90]), 푐{퐴̃2} = {퐴̃2}, 푐{퐴̃3} = {퐴̃3}, (푥3, [0.15,0.20], [0.05,0.35], [0.60,0.65]), ̃ ̃ ̃ ̃ (푥4, [0.50,0.55], [0.45,0.55], [0.30,0.35]), 푐{퐴1, 퐴4} = 퐹휔(퐴1, 퐴4) = (푥5, [0.10,0.15], [0.40,0.60], [0.50,0.60]), (푥1, [0.60,0.70], [0.11,0.21], [0.17,0.22]), (푥6, [0.70,0.75], [0.20,0.25], [0.10,0.15])} (푥2, [0.25,0.32], [0.25,0.28], [0.48,0.60]), (푥3, [0.17,0.22], [0.05,0.29], [0.62,0.67]), 퐴̃2 = (푥1, [0.40,0.45], [0.00,0.15], [0.30,0.35]), (푥4, [0.36,0.43], [0.39,0.49], [0.38,0.45]), (푥2, [0.60,0.65], [0.10,0.25], [0.20,0.30]), (푥5, [0.05,0.12], [0.37,0.67], [0.61,0.69]), (푥3, [0.80,1.00], [0.05,0.75], [0.00,0.00]), (푥6, [0.61,0.68], [0.00,0.011], [0.14,0.19])}. (푥4, [0.70,0.90], [0.35,0.65], [0.00,1.00]), and then compare each cluster with the other two clus- ters by using Eq. (5) Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 휔 ̃ ̃ 휔 ̃ ̃ 푑1 (푐{퐴2}, 푐{퐴3}) = 푑1 (푐{퐴3}, 푐{퐴2}) = 0.3508 which is based on the traditional hierarchical clustering pro- 휔 ̃ ̃ ̃ 휔 ̃ ̃ ̃ cedure, the single valued neutrosophic aggregation operator, 푑 (푐{퐴2}, 푐{퐴1, 퐴4}) = 푑 (푐{퐴4, 퐴 }, 푐{퐴2}) = 0.3003 1 1 1 and the basic distance measures between SVNSs. Then, we 푑휔(푐{퐴̃ }, 푐{퐴̃ , 퐴̃ }) = 푑휔(푐{퐴̃ , 퐴̃ }, 푐{퐴̃ }) = 0.2487. 1 3 1 4 1 4 1 3 have extented the algorithm to INSs for clustering interval neutrosophic data. Finally, an illustrative example is pre- then the INSs 퐴̃푘(푘 = 1,2,3,4) can be clustered into sented to demonstrate the application and effectiveness of the following two clusters in the third stage the developed clustering algorithms. Since the NSs are a ̃ ̃ ̃ ̃ {퐴2}, {퐴1, 퐴3, 퐴4}. more general platform to deal with uncertainties, the pro- posed neutrosophic hierarchical algorithms are more prior- In the final stage, the above two clusters can be further ity than the other ones. In the future we will focus our atten- ̃ ̃ ̃ ̃ clustered into a unique cluster {퐴1, 퐴2, 퐴3, 퐴4}. tion on the another clustering methods of neutrosophic in- formation. Note that the clustering results obtained in Example 1 and 2 are different from ones in [21]. References All the above processes can be presented as in Fig. 2. [1] A.K. Jain, M.N. Murty and P.J. Flynn, Data clustering: A view. ACM Computing Surveys, 31 (1999) 264–323. [2] B. S. Everitt, S. Landau, M. Leese. Cluster analysis. New {A1,A2,A3,A4} York: Oxford University Press, (2001). } [3] F. Smarandache, A unifying field in logic neutrosophy: Neu- trosophic probability, set and logic, American Research Pres, Rehoboth, 1999. {A1,A3,A4} [4] F. Smarandache, A Unifying Field in Logics Neutrosophic } Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Proba- bility, American Research Press, 2003. {A1,A4} [5] H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, } Single valued neutrosophic sets, Multispace and Multistruc- {A1} {A4} {A3} {A2} ture 4 (2010) 410-413. [6] H. Wang, F. Smarandache, Y.Q. Zhang, and R. Sunderraman, FIGURE 2: Classification of the enterprises 퐴̃푘(푘 = Interval neutrosophic sets and logic: Theory and applications 1,2,3,4) in computing, Hexis, Phoenix, AZ, (2005). [7] H. D. Cheng and Y. Guo, A new neutrosophic approach to Interval neutrosophic information is a generalization of image thresholding. New Math. Nat. Comput. 4(3) (2008) 291-308. interval valued intuitionistic fuzzy information while the [8] J. Ye, Similarity measures between interval neutrosophic sets single valued neutrosophic information extends the intui- and their applications in multicriteria decisionmaking, Journal tionistic fuzzy information. In other words, The components of Intelligent and Fuzzy Systems (2013) doi: 10.3233/IFS- of IFS and IVIFS are defined with respect to 푇 and 퐹, i.e., 120724. membership and nonmembership only, so they can only [9] J. Ye, Single-Valued Neutrosophic Minimum Spanning Tree handle incomplete information but not the indetermine in- and Its Clustering Method, Journal of Intelligent Systems, formation. Hence INS and SVNS, whose components are (2014) DOI: 10.1515/jisys-2013-0075. the truth membership, indeterminacy-membership and fal- [10] J. Ye, Single valued neutrosophic cross-entropy for mul- sity membership functions, are more general than others that ticriteria decision making problems, Applied Mathematical do not include the indeterminacy-membership. Therefore, it Modelling (2013), DOI:10.1016/j.apm.2013.07.020. is a natural outcome that the neutrosophic hierarchical clus- [11] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Sys- tering algorithms developed here is the extension of both the tems 20 (1986), 87-96. intuitionistic hierarchical clustering algorithm and its extend [12] K. Atanassov and G. Gargov, Interval valued intuitionistic form. The above expression clearly indicates that clustering fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343-349. analysis under neutrossophic environment is more general [13] L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353. and more practical than existing hierarchical clustering al- [14] M. Eissen, P. Spellman, P. Brown and D. Botstein, Cluster gorithms. analysis and display of genome- wide expression patterns. Proceeding of National Academy of Sciences of USA, (1998) 4 Conclusion 95: 14863–14868. To cluster the data represented by neutrosophic infor- [15] M. R. Anderberg, Cluster analysis for applications. New York: mation, we have discussed on the clustering problems of Academic Press, (1972). [16] P. Majumdar, S. K. Samant, On similarity and entropy of neu- SVNSs and INSs. Firstly, we have proposed a single valued trosophic sets, Journal of Intelligent and fuzzy Systems, (2013) neutrosophic hierarchical algorithm for clustering SVNSs, Doi:10.3233/IFS-130810. [17] S.A. Mingoti, J.O. Lima, Comparing SOM neural network Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 with fuzzy c-means k-means and traditional hierarchical clus- Making Problems, The Scientific World Journal, to appear tering algorithms. European Journal of Operational Research, (2013). (2006) 174: 1742–1759. [21] Z. M. Wang, Yeng Chai Soh, Qing Song, Kang Sim, Adaptive [18] Y. Guo and H. D. Cheng, New neutrosophic approach to im- spatial information-theoretic clustering for image segmenta- age segmentation. Pattern Recognition, 42 (2009) 587-595. tion, Pattern Recognition, 42 (2009) 2029-2044. [19] Y. H. Dong, Y. T. Zhuang, K. Chen and X. Y. Tai (2006). A [22] Z. S. Xu, Intuitionistic fuzzy hierarchical clustering algo- hierarchical clustering algorithm based on fuzzy graph con- rithms, Journal of Systems Engineering and Electronics 20 nectedness. Fuzzy Sets and Systems, 157, 1760–1774. (2009) 90–97. [20] Z. Hongyu, J. Qiang Wang, and X. Chen, Interval Neutro- sophic Sets and its Application in Multi-criteria Decision Received: January 13th, 2014. Accepted: February 3th, 2014 Rıdvan Şahin, Neutrosophic Hierarchical Clustering Algoritms Neutrosophic Sets and Systems, Vol. 2, 2014 25 Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces A. A. Salama1, Florentin Smarandache2 and Valeri Kroumov3 1 Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt [email protected] 2Department of Mathematics, University of New Mexico Gallup, NM, USA [email protected] 3Valeri Kroumov, Okayama University of Science, Okayama, Japan [email protected] Abstract. In this paper, we generalize the crisp topological compact spaces. Finally, some characterizations spaces to the notion of neutrosophic crisp topological space, and concerning neutrosophic crisp compact spaces are we construct the basic concepts of the neutrosophic crisp presented and one obtains several properties. Possible topology. In addition to these, we introduce the definitions of application to GIS topology rules are touched upon. neutrosophic crisp continuous function and neutrosophic crisp Keywords: Neutrosophic Crisp Set; Neutrosophic Topology; Neutrosophic Crisp Topology. 1 Introduction crisp set and its operations. We now improve some results by the following. Neutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both 3 Neutrosophic Crisp Sets their crisp and fuzzy counterparts, the most used one being the neutrosophic set theory [6, 7, 8]. After the 3.1 Definition introduction of the neutrosophic set concepts in [1, 2, 3, 4, 5, 9, 10, 11, 12] and after haven given the fundamental Let X be a non-empty fixed set. A definitions of neutrosophic set operations, we generalize neutrosophic crisp set (NCS for short) is an the crisp topological space to the notion of neutrosophic object having the form A A1,A2,A3 where crisp set. Finally, we introduce the definitions of A , A and A are subsets of X satisfying neutrosophic crisp continuous function and neutrosophic 1 2 3 crisp compact space, and we obtain several properties and A1 A2 , A1 A3 and A2 A3 . some characterizations concerning the neutrosophic crisp compact space. 3.1 Remark A neutrosophic crisp set A A1, A2 , A3 2 Terminology can be identified as an ordered triple A1, A2 , A3 , We recollect some relevant basic preliminaries, and where A , A , A are subsets on , and one can in particular, the work of Smarandache in [6, 7, 8, 12], 1 2 3 X and Salama et al. [1, 2, 3, 4, 5, 9, 10, 11, 12]. define several relations and operations between Smarandache introduced the neutrosophic components T, NCSs. I, F which represent the membership, indeterminacy, and Since our purpose is to construct the tools for developing neutrosophic crisp sets, we must non-membership values respectively, where -0 ,1 is introduce the types of NCSs N , X N in X as non-standard unit interval. follows: Hanafy and Salama et al. [10, 12] considered some 1) may be defined in many ways as a possible definitions for basic concepts of the neutrosophic N NCS, as follows: A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic25 Crisp Sets & Neutrosophic Crisp Topological Spaces 26 Neutrosophic Sets and Systems, Vol. 2, 2014 ii) i) N X ,,, or ,, BABABABA 332211 ii) XX ,,, or c N 3) [ ,,] AAAA 121 . iii) N X ,,, or c 4) ,, AAAA 323 . iv) N ,, 3.2 Proposition 2) X N may also be defined in many ways as a For all two neutrosophic crisp sets A and B NCS: on X, then the followings are true: i) N XX ,,, 1) 2) ii) XXX ,,, N We can easily generalize the operations of iii) intersection and union in definition 3.2 to arbitrary family of neutrosophic crisp subsets as Every crisp set A formed by three disjoint subsets of a non-empty set is obviously a NCS having the form follows: ,, AAAA . 321 3.3 Proposition 3.2 Definition Let be arbitrary family of neutrosophic crisp subsets in X, then Let a NCS on , then the 1) A j may be defined as the following c complement of the set , ( A for short may be types : defined in three different ways: i) Aj Aj1 ,, AA jj ,or c ccc 32 C ,, AAAA , 1 321 ii) A Aj ,, AA . j 1 jj 32 c C ,, AAAA 2 123 2) A j may be defined as the following c c types : C 3 ,, AAAA 123 i) A Aj ,, AA or One can define several relations and operations j 1 jj 32 between NCSs as follows: ii) A Aj ,, AA . j 1 jj 32 3.3 Definition 3.5 Definition Let be a non-empty set, and the NCSs and B The product of two neutrosophic crisp sets X A and B is a neutrosophic crisp set BA in the form , , then we may ,, AAAA 321 ,, BBBB 321 given by consider two possible definitions for subsets AB ,, BABABABA 332211 . may be defined in two ways: 4 Neutrosophic Crisp Topological Spaces 1) , and BABABABA 332211 or Here we extend the concepts of topological space and intuitionistic topological space to the 2) , and BABABABA 332211 case of neutrosophic crisp sets. 3.1 Proposition For any neutrosophic crisp set the following hold: 4.1 Definition i) N A, NN . A neutrosophic crisp topology (NCT for ii) , XXXA NNN . short) on a non-empty set is a family of neutrosophic crisp subsets in satisfying 3.4 Definition the following axioms Let X is a non-empty set, and the NCSs and i) , X NN . in the form ,, AAAA 321 , ,, BBBB 321 . Then: AB 1) may be defined in two ways: ii) AA 21 for any A1 and A2 . i) ,, BABABABA 332211 or ii) ,, BABABABA 332211 iii) Aj . 2) AB may also be defined in two ways: i) ,, BABABABA 332211 or A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic26 Crisp Sets & Neutrosophic Crisp Topological Spaces Neutrosophic Sets and Systems, Vol. 2, 2014 27 In this case the pair X , is called a neutrosophic Furthermore, is the coarsest NCT on crisp topological space for short) in X . The (NCTS containing all topologies. elements in are called neutrosophic crisp open sets (NCOSs for short) in . A neutrosophic crisp set F is C Proof closed if and only if its complement F is an open Obvious. Now, we define the neutrosophic crisp neutrosophic crisp set. closure and neutrosophic crisp interior operations on neutrosophic crisp topological spaces: 4.1 Remark 4.3 Definition Neutrosophic crisp topological spaces are very natural generalizations of topological spaces and Let be NCTS and ,, AAAA 321 be a intuitionistic topological spaces, and they allow more NCS in . Then the neutrosophic crisp closure general functions to be members of topology. of A (NCCl(A) for short) and neutrosophic TS ITS NCTS interior crisp (NCInt (A ) for short) of A are defined by 4.1 Example NCCl :)( KKA is NCSan Xin and KA Let ,,, dcbaX , , X NN be any types of the universal and empty subsets, and A, B two neutrosophic :)( GGANCInt is NCOSan Xin and AG , crisp subsets on X defined by ,,, cdbaA , where NCS is a neutrosophic crisp set, and NCOS ,, cbaB , then the family ,,, BAX is NN is a neutrosophic crisp open set. a neutrosophic crisp topology on X. It can be also shown that NCCl A)( is a NCCS 4.2 Example (neutrosophic crisp closed set) and ANCInt )( is a Let X, be a topological space such that CNOS in is not indiscrete. Suppose i : JiG be a family and i :, JiGX . Then we can construct the a) A is in if and only if NCCl )( AA . following topologies as follows i) Two intuitionistic topologies b) A is a NCCS in if and only if )( AANCInt . a) 1 iII ,,, JiGX . 4.2 Proposition b) c ,,, JiGX 2 II i For any neutrosophic crisp set in X , we have ii) Four neutrosophic crisp topologies c c c (a) NCCl A (()( ANCInt )) , a) 1 X NN i ,,,, JiG (b) ANCInt c NCCl(()( A)) c . b) 2 , iNN ,,, JiGX Proof a) Let and suppose that the c c) 3 , iiNN ,,, JiGGX , family of neutrosophic crisp subsets contained in are indexed by the family if c NCSs contained in are indexed by the d) 4 , iNN ,,, JiGX family :,, JiAAAA . Then 21 jjj 3 we see that we have two types of 4.2 Definition ,,)( AAAANCInt or 1 2 jjj 3 Let XX ,,, 21 be two neutrosophic crisp ,,)( AAAANCInt hence 1 2 jjj 3 topological spaces on . Then is said be contained in 1 c (( ANCInt )) ,, AAA jjj or 2 (in symbols 21 ) if G 2 for each G 1. In 1 2 3 this case, we also say that is coarser than . c 1 2 (( ANCInt )) ,, AAA . 1 2 jjj 3 4.1 Proposition Hence NCCl Ac (()( ANCInt )) c , which Let j : Jj be a family of NCTs on . Then is analogous to (a). j is a neutrosophic crisp topology on . A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic27 Crisp Sets & Neutrosophic Crisp Topological Spaces 28 Neutrosophic Sets and Systems, Vol. 2, 2014 4.3 Proposition Let X , be a NCTS and AB, be two Proof neutrosophic crisp sets in X . Then the following Obvious. properties hold: 5.2 Definition (a) AANCInt ,)( Let X, 1 and Y, 2 be two NCTSs, and (b) A NCCl(A), let : YXf be a function. Then is said to (c) ()( BNCIntANCIntBA ), be continuous iff the preimage of each NCS in (d) BA NCCl A NCCl()( B), 2 is a NCS in 1 . (e) ()()( BNCIntANCIntBANCInt ), 5.3 Definition (f) NCCl BA NCCl A NCCl()()( B), Let X, 1 and be two NCTSs and let : YXf be a function. Then is said to (g) XXNCInt ,)( NN be open iff the image of each NCS in is a (h) NCCl )( NN NCS in 2 . Proof. (a), (b) and (e) are obvious; (c) follows from (a) 5.1 Example and from definitions. Let X, o and Y, o be two NCTSs 5 Neutrosophic Crisp Continuity (a) If is continuous in the usual Here come the basic definitions first sense, then in this case, is continuous in the sense of Definition 5.1 too. Here we consider 5.1 Definition the NCTs on X and Y, respectively, as follows : c (a) If ,, BBBB 321 is a NCS in Y, then the 1 :,, GGG o and 1 c preimage of B under f , denoted by (Bf ), is a 2 :,, HHH o, NCS in X defined by ,, HH c 1 1 1 1 In this case we have, for each 2 , ()( 1), ( 2), BfBfBfBf 3 .)( (b) If ,, AAAA 321 is a NCS in X, then the image H o , of A under f , denoted by (Af ), is the a NCS in c 1 c 111 c Y defined by ()( ), ( ), AfAfAfAf 321 .))( ,, ( ), (), HffHfHHf )( Here we introduce the properties of images and preimages 1 c some of which we shall frequently use in the following (, ), (( HffHf )) 1 . sections . (b) If is open in the usual sense, 5.1 Corollary then in this case, is open in the sense Let A, i : JiA , be NCSs in X, and of Definition 3.2. Now we obtain some B, j : KjB NCS in Y, and : YXf a characterizations of continuity: function. Then 5.1 Proposition (a) ()( AfAfAA 2121 ), Let 1 YXf 2 ),(),(: . 1 1 21 1 ()( BfBfBB 2), f is continuous if the preimage of each (b) 1 (( AffA )) and if f is injective, then CNCS (crisp neutrosophic closed set) in is a 1 AffA ))(( . CNCS in . 1 (c) (( )) BBff and if is surjective, then 5.2 Proposition 1 BBff ,))(( . The following are equivalent to each other: (d) 1 1())( BfBf ), 1 1())( BfBf ), i i i i (a) is continuous. (e) and if f is 1 1 i ()( AfAf i ); i ()( AfAf i ); (b) (()(( BfCNIntBCNIntf )) injective, then i ()( AfAf i ); for each CNS B in Y. (c) 1 1 (f) 1 XYf ,)( f 1 )( . CNCl (( )) fBf CNCl(( B)) NN NN for each CNC B in Y. (g) if is subjective. f NN ,)( YXf NN ,)( A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic28 Crisp Sets & Neutrosophic Crisp Topological Spaces Neutrosophic Sets and Systems, Vol. 2, 2014 29 5.2 Example In this case Let Y, 2 be a NCTS and : YXf be a function. is an NCT on X, which is not a neutrosophic 1 crisp compact. In this case 1 :)( HHf 2 is a NCT on X. Indeed, it is the coarsest NCT on X which makes the 6.1 Corollary function continuous. One may call it the initial A NCTS X , is a neutrosophic crisp neutrosophic crisp topology with respect to f . compact iff every family 6 Neutrosophic Crisp Compact Space (NCCS) :,,, JiGGGX of NCCSs in X having iii 321 First we present the basic concepts: the FIP has nonempty intersection. 6.1 Definition 6.2 Corollary Let X , be an NCTS. Let X,1 , Y,2 be NCTSs and (a) If a family :,, JiGGG of NCOSs in iii 321 : YXf be a continuous surjection. If X, 1 X satisfies the condition is a neutrosophic crisp compact, then so is :,,, XJiGGGX , then it is called iii 321 N an neutrosophic open cover of X. (b) A finite subfamily of an open cover 6.3 Definition on X, which is also a (a) If a family of neutrosophic open cover of X , is called a NCCSs in X satisfies the condition neutrosophic finite subcover :,, JiGGGA , then it is :,, JiGGG . iii 321 iii 321 (c) A family :,, JiKKK of NCCSs in X called a neutrosophic crisp open cover iii 321 satisfies the finite intersection property (FIP for of A. short) iff every finite subfamily (b) Let’s consider a finite subfamily of a 2,1:,, ,...,niKKK of the family neutrosophic crisp open subcover of iii 321 satisfies the condition . . iii :,, JiKKK N 321 A neutrosophic crisp set ,, AAAA 321 in a 6.2 Definition NCTS X , is called neutrosophic crisp A NCTS is called neutrosophic crisp compact compact iff every neutrosophic crisp open cover of A has a finite neutrosophic crisp open iff each crisp neutrosophic open cover of X has a finite subcover. subcover. 6.1 Example 6.3 Corollary a) Let X and let’s consider the NCSs Let X, 1 , be NCTSs and (neutrosophic crisp sets) given below: : YXf be a continuous surjection. If A is a neutrosophic crisp compact in , then so is A1 4,3,2 ,... ,,, A2 4,3 ,... ,1,, Af )( in . A3 6,5,4 ,... ,2,1,, … 7 Conclusion nnnA 3,2,1 ,... 3,2,1,, ,...n 1 n . In this paper we introduce both the neutrosophic crisp topology and the neutrosophic crisp compact Then , AX nNN : 5,4,3 ,... is a NCT on X and space, and we present properties related to them. is a neutrosophic crisp compact. b) Let X 1,0 and let’s take the NCSs XA 1 n1 ,0,,,, 1 n n nn , n 5,4,3 ,...in X. A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces 29 30 Neutrosophic Sets and Systems, Vol. 2, 2014 References [28] Florentin Smarandache, An introduction to the Neutrosophy probability applied in Quantum [22] A.A. Salama and S.A. Alblowi, "Generalized Neutrosophic Physics, International Conference on Set and Generalized Neutrousophic Topological Spaces ", Neutrosophic Physics, Neutrosophic Logic, Set, Journal computer Sci. Engineering, Vol. 2, No. 7, (2012), Probability, and Statistics University of New pp. 29-32. Mexico, Gallup, NM 87301, USA, 2-4 December (2011) . [23] A.A. Salama and S.A. Alblowi, Neutrosophic set and neutrosophic topological space, ISORJ. Mathematics, Vol. [29] F. Smarandache. A Unifying Field in Logics: 3, Issue 4, (2012), pp. 31-35. Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American [24] A.A. Salama and S.A. Alblowi, Intuitionistic Fuzzy Ideals Research Press, Rehoboth, NM, (1999). Topological Spaces, Advances in Fuzzy Mathematics, Vol. 7, Number 1, (2012), pp. 51- 60. [30] I. Hanafy, A.A. Salama and K. Mahfouz, Correlation of neutrosophic Data, International [25] A.A. Salama, and H. Elagamy, "Neutrosophic Filters," Refereed Journal of Engineering and Science International Journal of Computer Science Engineering (IRJES) , Vol.(1), Issue 2 , (2012), pp. 39-43. and Information Technology Research (IJCSEITR), Vol.3, Issue (1), (2013), pp. 307-312. [31] I.M. Hanafy, A.A. Salama and K.M. Mahfouz,” Neutrosophic Crisp Events and Its Probability" [26] S. A. Alblowi, A. A. Salama & Mohmed Eisa, New International Journal of Mathematics and Concepts of Neutrosophic Sets, International Journal of Computer Applications Research (IJMCAR) Vol. Mathematics and Computer Applications Research 3, Issue 1, (2013), pp. 171-178. (IJMCAR),Vol.3, Issue 4, Oct 2013, (2013), pp. 95-102. [32] A. A. Salama,"Neutrosophic Crisp Points & [27] Florentin Smarandache , Neutrosophy and Neutrosophic Neutrosophic Crisp Ideals", Neutrosophic Sets Logic, First International Conference on Neutrosophy, and Systems, Vol.1, No. 1, (2013) pp. 50-54. Neutrosophic Logic, Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA [33] A. A. Salama and F. Smarandache, "Filters via (2002) . Neutrosophic Crisp Sets", Neutrosophic Sets and Systems, Vol.1, No. 1, (2013) pp. 34-38. Received: January 28th, 2014. Accepted: February 13th, 2014 A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic30 Crisp Sets & Neutrosophic Crisp Topological Spaces Neutrosophic Sets and Systems, Vol. 2, 2014 31 Neutrosophic Logic for Mental Model Elicitation and Analysis Karina Pérez-Teruel1, Maikel Leyva-Vázquez2 1 Universidad de las Ciencias Informáticas, La Habana, Cuba. E-mail: [email protected] 2 Universidad de las Ciencias Informáticas, La Habana, Cuba. E-mail: [email protected] Abstract. Mental models are personal, internal representations mental models elicitation and analysis based on of external reality that people use to interact with the world neutrosophic Logic is presented. An illustrative around them. They are useful in multiple situations such as example is provided to show the applicability of the muticriteria decision making, knowledge management, complex proposal. The paper ends with conclusion future system learning and analysis. In this paper a framework for research directions. Keywords: mental model, neutrosophic Logic, neutrosophic cognitive maps, static analysis. 1 Introduction 3 Proposed Framework Mental models are useful in multiple situations such as muticriteria decision making [1], knowledge management, The following steps will be used to establish a complex system learning and analysis [2]. In this paper, framework for mental model elicitation and we propose the use of an innovative technique for analysis with NCM (Fig. 1). processing uncertainty and indeterminacy in mental models. Figure 1: Mental model. The outline of this paper is as follows: Section 2 is dedicated to mental models and neutrosophic logic and • Mental model development. neutrosophic cognitive maps. The proposed framework is presented in Section 3. An illustrative example is discussed Mental model develoment in Section 4. The paper closes with concluding remarks, and discussion of future work in Section 5. • Nodes determination • Causal relationships determination. • Weights and signs determination. 2 Mental Models and neutrosophic Logic Mental models are personal, internal representations of external reality that people use to interact with the world Mental Model analysis around them [3]. The development of more effective end- • Degree centrality determination user mental modelling tools is an active area of research • De-neutrosophication process [4]. A cognitive map is form of structured knowledge representation introduced by Axelrod [5]. Mental models This Activity begins with determination have been studied using cognitive mapping [6]. of nodes. Finally causal relationships, its Another approach is based in fuzzy cognitive maps [7]. weights and signs are elicited [10]. FCM utilizes fuzzy logic in the creation of a directed cognitive map. FCM are a further extension of Axelrod‟s • Mental model analysis definition of cognitive maps [7] . Neutrosophic logic is a generalization of fuzzy logic based Static analysis is develop to define the on neutrosophy [8]. If indeterminacy is introduced in importance of each node based on the cognitive mapping it is called Neutrosophic Cognitive degree centrality measure [11]. A de- Map (NCM) [9]. neutrosophication process gives an NCM are based on neutrosophic logic to represent interval number for centrality. Finally uncertainty and indeterminacy in cognitive maps [8]. A the nodes are ordered. NCM is a directed graph in which at least one edge is an indeterminacy denoted by dotted lines [6]. Karina Pérez-Teruel, Maikel Leyva-Vázquez, Neutrosophic31 Logic for Mental Model Elicitation and Analysis 32 Neutrosophic Sets and Systems, Vol. 2, 2014 4 Illustrative example The next step is the de-neutrosophication process as proposes by Salmeron and Smarandache [13]. In this section, we present an illustrative example in order I ∈[0,1] is repalaced by both maximum and to show the applicability of the proposed model. We minimum values. selected a group of concepts related to people factor in agile software develoment projects sucess (Table 1) [12]. A 1.75 Table I. FCM nodes B [0.75,1.75] Node Description C [0.25,1.25] A Competence and expertise of team members D 0.75 B Motivation of tem members E 0.75 C Managers knowledge of F [0.75,2.75] agile development D Team training E Customer relationship Finally we work with extreme values [14] for F Customer giving a total order: commitment and presence The FCM is developed integrating knowledge from one Competence and expertise of team members, expert. The FCM with weighs is represented in Fig. 4. Customer commitment and presence are the more important factors in his mental model. A 0.75 5 Conclusions 0.75 In this paper, we propose a new framework for 0.25 D processing uncertainty and indeterminacy in mental models. Future research will focus on B conducting further real life experiments and the development of a tool to automate the process. C The use of the computing with words (CWW) is another area of research. E References 0.75 F 1. Montibeller, G. and V. Belton, Causal maps and the evaluation of decision options—a review. Journal of the Operational Research Figure 2: Mental model. Society, 2006. 57(7): p. 779-791. The neutrosophic score of each node based on the centrality measure is as follows: 2. Ross, J., Assessing Understanding of A 1.75 Complex Causal Networks Using an Interactive Game. 2013, University of B 0.75+I California: Irvine. C 0.25+I 3. Jones, N.A., et al., Mental models: an interdisciplinary synthesis of theory and D 0.75 methods. Ecology and Society, 2011. 16(1): p. 46. E 0.75 4. Durham, J.T., Mental Model Elicitation F 0.75+2I Device (MMED) Methods and Apparatus. 2012, US Patent 20,120,330,869. Karina Pérez-Teruel, Maikel Leyva-Vázquez, Neutrosophic32 Logic for Mental Model Elicitation and Analysis Neutrosophic Sets and Systems, Vol. 2, 2014 33 5. Axelrod, R.M., I.o.p.p. studies, and I.o.i. studies, 11. Samarasinghea, S. and G. Strickert, A New Structure of decision: The cognitive maps of political Method for Identifying the Central Nodes in elites. Vol. 1. 1976: Princeton university press Fuzzy Cognitive Maps using Consensus Princeton. Centrality Measure, in 19th International Congress on Modelling and Simulation. 6. Salmeron, J.L. and F. Smarandache, Processing 2011: Perth, Australia. Uncertainty and Indeterminacy in Information Systems projects success mapping, in Computational Modeling 12. Chow, T. and D.-B. Cao, A survey study of in Applied Problems: collected papers on critical success factors in agile software econometrics, operations research, game theory and projects. Journal of Systems and Software, simulation. 2006, Hexis. p. 94. 2008. 81(6): p. 961-971. 7. Gray, S., E. Zanre, and S. Gray, Fuzzy Cognitive Maps 13. Salmerona, J.L. and F. Smarandacheb, as Representations of Mental Models and Group Redesigning Decision Matrix Method with an Beliefs, in Fuzzy Cognitive Maps for Applied Sciences indeterminacy-based inference process. and Engineering. 2014, Springer. p. 29--48. Multispace and Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of 8. Smarandache, F., A unifying field in logics: Sciences), 2010. 4: p. 151. neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability and statistics. 2005: 14. Merigó, J., New extensions to the OWA American Research Press. operators and its application in decision making, in Department of Business 9. Kandasamy, W.V. and F. Smarandache, Analysis of Administration, University of Barcelona. social aspects of migrant labourers living with 2008. HIV/AIDS using Fuzzy Theory and Neutrosophic Cognitive Maps. 2004: American Research Press. 10. Leyva-Vázquez, M.Y., et al., Modelo para el análisis de escenarios basado en mapas cognitivos difusos. Ingeniería y Universidad 2013. 17(2). Received: December 23th, 2013. Accepted: January 5th, 2014 33 Karina Pérez-Teruel, Maikel Leyva-Vázquez, Neutrosophic Logic for Mental Model Elicitation and Analysis Neutrosophic Sets and Systems, Vol. 2, 2014 On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings A.A.A. Agboola1 and B. Davvaz2 1 Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria. E-mail: [email protected] 2 Department of Mathematics, Yazd University, Yazd, Iran. E-mail: [email protected] Abstract. A neutrosophic hyperstructure is an algebraic tion are hyperoperations in a neutrosophic environment. structure generated by a given hyperstructure H and an Some basic properties of neutrosophic canonical hyper- indeterminacy factor I under the hyperoperation(s) of H. groups and neutrosophic hyperrings are presented. Quo- The objective of this paper is to study canonical hyper- tient neutrosophic canonical hypergroups and neutro- groups and hyperrings in which addition and multiplica- sophic hyperrings are presented. Keywords: neutrosophic canonical hypergroup, neutrosophic subcanonical hypergroup, neutrosophic hyperring, neutrosophic subhyperring, neutrosophic hyperideal. 1 Introduction 2 A Review of Well Known Definitions Given any hyperstructure H, a new hyperstructure H(I) may be generated by H and I under the hyperoperation(s) In this section, we provide basic definitions, notations of H. Such new hyperstructures H(I) are called neutrosoph- and results that will be used in the sequel. ic hyperstructures where I is an indeterminate or a neutro- sophic element. Generally speaking, H(I) is an extension of Definition 2.1. Let (G, ) be any group and let H but some properties of H may not hold in H(I). However, )( IGIG . The couple (( IG ),) is called a neu- H(I) may share some properties with H and at times may trosophic group generated by G and I under the binary op- possess certain algebraic properties not present in H. eration . The indeterminacy factor I is such that Neutrosophic theory was introduced by F. III . If is ordinary multiplication, then Smarandache in 1995 and some known algebraic structures ... n IIIIII and if is ordinary addition, in the literature include neutrosophic groups, neutrosophic then ... IIII nI for Nn . semigroups, neutrosophic loops, neutrosophic rings, neu- G(I) is said to be commutative if abba for all trosophic fields, neutrosophic vector spaces, neutrosophic (, IGba ). modules etc. Further introduction to neutrosophy and neu- trosophic algebraic structures can be found in Theorem 2.2. [26] Let G(I) be a neutrosophic group. [1,2,3,4,16,26,27]. (1) G(I) in general is not a group; In 1934, Marty [18] introduced the theory of hyper- (2) G(I) always contain a group. structures at the 8th Congress of Scandinavian Mathemati- cians. In 1972, Mittas [21] introduced the theory of canon- Definition 2.3. Let G(I) be a neutrosophic group. ical hypergroups. A class of hyperrings (R, +, .), where + (1) A proper subset A(I) of G(I) is said to be a neutro- and . are hyperoperations are introduced by De Salvo [15]. sophic subgroup of G(I) if A(I) is a neutrosophic This class of hyperrings has been further studied by group, that is, A(I) contains a proper subset which Asokkumar and Velrajan [5,22] and Davvaz and Leoranu- is a group; Fotea [14]. Further contributions to the theory of hyper- (2) A(I) is said to be a pseudo neutrosophic group if it structures can be found in [7,8,9,10,14,22]. does not contain a proper subset which is a group. Agboola and Davaaz introduced and studied neutro- sophic hypergroups in [4]. The present paper is concerned Definition 2.4. Let A(I) be a neutrosophic subgroup with the study of canonical hypergroups and hyperrings in (pseudo neutrosophic subgroup) of a neutrosophic group a neutrosophic environment. Basic properties of neutro- G(I). sophic canonical hypergroups and neutrosophic hyperrings (1) A(I) is said to be normal in G(I) if there exist are presented. Quotient neutrosophic canonical hyper- IGyx )(, such that xA IAyI )()( . groups and neutrosophic hyperrings are also presented. (2) G(I) is said to be simple if it has no non-trivial A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings 35 Neutrosophic Sets and Systems, Vol. 2, 2014 neutrosophic normal subgroup. ic subring) of R(I). Then, S(I) is called a neutrosophic ide- al (pseudo neutrosophic ideal) of R(I) if for all r R() I Example 1. [3] Let G(I)={e, a, b, c, I, aI, bI, cI} be a and s S() I , r s, s r S ( I ). set, where a2=b2=c2=e, bc=cb=a, ac=ca=b, ab=ba=c. Then (G(I),.) is a commutative neutrosophic group, and Definition 2.11. Let (RI1 ( ), , ) and (RI2 ( ), , ) H(I)={e,a,I,aI}, K(I)={e,b,I,bI} and PI={e,c,I,cI} are neu- be two neutrosophic rings and let :()()RIRI12 be a trosophic subgroups of G(I). mapping of RI1()into RI2 (). Then, is said to be a Theorem 2.5. [3] Let H(I) be a non-empty proper sub- homomorphism if the following conditions hold: set of a neutrosophic group (G(I), ). Then, H(I) is a neu- (1) is a group of homomorphism; trosophic subgroup of G(I) if and only if the following (2) . conditions hold: Moreover, if is a bijection, then is called a neu- (1) IHba )(, implies that (IHba ); trosophic ring isomorphism and we write RIRI12()() . (2) There exists a proper subset A of H(I) such that (A, The kernel of denoted by Ker is the set ) is a group. {x R1 ( I ): ( x ) 0} and the image of denoted by Im is the set { (x ): x R1 ( I )} . Definition 2.6. Let (G1(I), ) and (G1(I), ’) be two It should be noted that is a neutrosophic subring neutrosophic groups and let :()()GIGI12 be a of RI2 ()and is always a subring of R1 and never a mapping of G1(I) into G2(I). Then, is said to be a ho- neutrosophic subring (ideal) of R1(I). momorphism if the following conditions hold: (1) is a group of homomorphism; Definition 2.12. A map :*()SSPS is called (2) ()II . hyperoperation on the set S, where S is non-empty set and In addition, if is a bijection, then is called a neu- PS*() denotes the set of all non-empty subsets of S. trosophic group isomorphism and we write A hyperstructure or hypergroupoid is the pair (,)S , GIGI12()() . where is a hyperoperation on the set S. Definition 2.7. Let (R,+,.) be any ring. A neutrosophic Definition 2.13. A hyperstructure is called a ring is a triple (R(I),+,.) generated by R and I, that is, semihypergroup if for all RIRI(). x,,,()() y z S x y z x y z , which means that Indeed, R(){:,} I x a bI a b R , where if x=a+bI and y=c+dI are elements of R, then u z x v. xy ()()()() abI cdI ac bdI , u x y v y z x y()() a bI c dI ()()ac ad bc bd I . Definition 2.14. A non-empty subset A of a semihy- pergroup is called a subsemihypergroup. In other Example 2. Let n be a ring of integers modulo n. words, a non-empty subset A of a semihypergroup is Then, nn(){:,}I x a bI a b is a neutrosoph- a subsemihypergroup if AAA. ic ring of integers modulo n. If xS and A,B are non-empty subsets of S, then Theorem 2.8. [27] Let (R(I),+,.) be a neutrosophic , and ring. Then, (R(I),+,.) is a ring. A B a b,{} A x A x aA bB Definition 2.9. Let (R(I),+,.) be a neutrosophic ring. A non-empty subset S(I) of R(I) is said to be a neutrosophic x B {} x B . subring if (S(I),+,.) is a neutrosophic ring. It is essential that S(I) must contain a proper subset which is a ring. Oth- Definition 2.15. A hypergroupoid (,)H is called a erwise, S(I) is called a pseudo neutrosophic subring of R(I). quasihypergroup if for all a of H we have a H H a H . This condition is also called the re- Example 3. Let (12 (I ), , ) be a neutrosophic ring production axiom. of integers modulo 12 and let S(I) and T(I) be subsets of 12 ()I given by S(I)={0, 6, I, 2I, 3I, ..., 11I, 6+I, 6+2I, Definition 2.16. A hypergroupoid which is both 6+3I, ..., 6+11I} and T(I)={0, 2I, 4I, 6I, 10I}. Then, a semihypergroup and a quasihypergroup is called a hy- (S(I),+,.) is a neutrosophic subring of while pergroup. (T(I),+,.) is a pseudo neutrosophic ring of . Definition 2.17. Let H be a non-empty set and let + be Definition 2.10. Let (R(I),+,.) be a neutrosophic ring a hyperoperation on H. The couple (,)H is called ca- and let S(I) be a neutrosophic subring (pseudo neutrosoph- nonical hypergroup if the following conditions hold: A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings Neutrosophic Sets and Systems, Vol. 2, 2014 36 (1) x+y=y+x, for all x, y H ; y a d b c b d}. (2) x+(y+z)=(x+y)+z, for all x,, y z H ; (3) there exists a neutral element 0H such that 3 Development of Neutrosophic Canonical Hy- x+0={x}=0+x, for all xH ; pergroups and Neutrosophic Hyperrings (4) for every , there exists a unique element In this section, we develop the concepts of neutrosoph- xHsuch that 0x ( x ) ( x ) x ; ic canonical hypergroups and neutrosophic hyperrings. (5) z x y implies y x z and x z y , Necessary definitions are given and examples are provided. for all . A non-empty subset A of H is called a subcanonical Definition 3.1. Let (,)H be any canonical hyper- hypergroup if A is canonical hypergroup under the same group and let I be an indeterminate. Let hyperaddition as that of H that is, for every a, b A , H( I ) H I {( a , bI ) : a , b H }be a set gener- a b A. In addition, if a A a A for all aH , ated by H and I. The hyperstructure (HI ( ), ) is called a A is said to be normal. neutrosophic canonical hypergroup, where for all (a , bI ),( c , dI ) H ( I ) with b 0 or d 0 , we define Definition 2.18. A hyperring is a triple (,,)R satis- (,)(,){(,):abI cdI xyI xac , fying the following axioms: y a d b c b d} (1) (,)R is a canonical hypergroup; and (2) (,)R is a semihypergroup such that (x ,0) ( y ,0) {( u ,0): u x y }. xx0 0 0 for all xR , that is, ) is a bilat- The element I is represented by (0,I) in H(I) and any eral absorbing element; element xH is represented by (x,0) in H(I). For any (3) x() y z x y x z and non-empty subset A[I] of H(I) , we define AI[] ()x y z x z y z , for all x,, y z R . {(,)(,):,}a bI a bI a b H . Definition 2.19. Let be a hyperring and A be Lemma 3.2. Let H {0}be a canonical hypergroup a non-empty subset of R. Then, A is said to be subhyper- and let H(I) be the corresponding neutrosophic canonical ring of R if (,,)A is itself a hyperring. hypergroup. Then, (0,0) the neutral element of H is not a neutral element of H(I). Definition 2.20. Let A be a subhyperring of a hyper- Proof. Suppose that (0,0) is the neutral element of H(I) ring R. and suppose that (,)()a bI H I such that b is non-zero (1) A is called a left hyperideal of R if r a Afor and ab . all r R, a A . Then (2) A is called a right hyperideal of R if a r A (abI , ) (0,0) {( uvIua , ): 0, va 0 b 0 b 0} for all . {(,u vI ): u {}, a v {,}} a b (3) A is called a hyperideal of R if A is both left and (a , bI ), right hyperideal of R. a contradiction. Hence, (0,0) is not a neutral element of (4) A hyperideal A is said to be normal if H(I). r a r Afor all rR . Definition 3.3. Let (H(I),+) be a neutrosophic canoni- Definition 2.21. Let (,)H and (,)H be two cal hypergroup. 1 2 (1) A non-empty subset A[I] of H(I) is called a neu- canonical hypergroups. A mapping : HH12 , is called trosophic subcanonical hypergroup of H(I) if (A[I],+) is itself a neutrosophic canonical hyper- (1) a homomorphism if (i) for all x, y H1 , ()()()x y x y and (ii) (0) 0 . group. It is essential that A[I] must contain a prop- (2) a good or strong homomorphism if (i) for all er subset which is a subcanonical hypergroup of H. , ()()()x y x y and (ii) If A[I] does not contain a proper subset which is a . subcanonical hypergroup of H, then it is called a (3) an isomorphism (strong isomorphism) if is a bi- pseudo neutrosophic subcanonical hypergroup of jective homomorphism (strong homomorphism). H(I). (2) If A[I] is a neutrosophic subcanonical hypergroup Definition 2.22. [4] Let (,)H be any hypergroup and (pseudo neutrosophic subcanonical hypergroup), let H I {( a , bI ) : a , b H } . The couple then A[I] is said to be normal in H(I) if for all HIHI()(,) is called a neutrosophic hyper- , group generated by H and I under the hyperoperation , (,)a bI A [](,) I a bI A []. I where for all (a , bI ),( c , dI ) H ( I ) , the composition element of H(I) is defined by Lemma 3.4. Let (H(I),+) be a neutrosophic canonical (,)(,){(,):abI cdI xyI x ac , hypergroup and let A[I] be a non-empty proper subset of A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings 37 Neutrosophic Sets and Systems, Vol. 2, 2014 H(I). Then, A[I] is a neutrosophic subcanonical hyper- Definition 3.10. Let (RI ( ), , ) be a neutrosophic group if and only if the following conditions hold: hyperring and let A[I] be a non-empty subset of R(I). Then, (1) for all A[I] is called a neutrosophic subhyperring of R(I) if (,abI ),(, cdI ) AI [],(, abI )(, cdI ) AI [], (AI [ ], , ) is itself a neutrosophic hyperring. It is essen- tial that A[I] must contain a proper subset which is a hy- (2) A[I] contains a proper subset which is a canonical perring. Otherwise, A[I] is called a pseudo neutrosophic hypergroup of H. subhyperring of R(I). Lemma 3.5. Let (H(I),+) be a neutrosophic canonical Definition 3.11. Let (RI ( ), , ) be a neutrosophic hypergroup and let A[I] be a non-empty proper subset of hyperring and let A[I] be a neutrosophic subhyperring of H(I). Then, A[I] is a pseudo neutrosophic subcanonical R(I). hypergroup if and only if the following conditions hold: (1) A[I] is called a left neutrosophic hyperideal if for (1) for all all (,r sI ) R (),(, I a bI ) A [], I (r , sI ) ( a , bI ) A [ I ]. (2) A[I] is called a right neutrosophic hyperideal if for (2) A[I] does not contain a proper subset which is a all canonical hypergroup of H. (a , bI ) ( r , sI ) A [ I ]. (3) A[I] is called a neutrosophic hyperideal if A[I] is Definition 3.6. Let A[I] and B[I] be any two neutro- both a left and right neutrosophic hyperideal. sophic subcanonical hypergroups of a neutrosophic canon- A neutrosophic hyperideal A[I] of R[I] is said to be ical hypergroup H(I). The sum of A[I] and B[I] denoted by normal in R(I) if for all (r , sI ) R ( I ), A[I]+B[I] is defined as the set: (,)[](,)[]r sI A I r sI A I . A[] I B [] I (,)(,). a bI c dI Lemma 3.12. Let be a neutrosophic hy- (,)[]a bI A I (,)[]c dI B I perring and let A[I] be a non-empty subset of R(I). Then, A[I] is a neutrosophic hyperideal if and only if the follow- Definition 3.7. Let H(I) be a neutrosophic canonical ing conditions hold: hypergroup and let A[I] be a neutrosophic subcanonical (1) For all hypergroup of H(I). If K is a subcanonical hypergroup of H, (,abI ),(, cdI ) AI [],(, abI )(, cdI ) AI []; we define the set (2) For all (,r sI ) R (),(, I a bI ) A [], I A[] I K ( a , bI ) ( k ,0). (,)(,)[]a bI r sI A I and (,)[]a bI A I (kK ,0) (r , sI ) ( a , bI ) A [ I ]; (3) A[I] contains a proper subset which is a hyperring. Definition 3.8. Let (,,)R be any hyperring and let I be an indeterminate. The hyperstructure (RI ( ), , ) Lemma 3.13. Let be a neutrosophic hy- generated by R and I, that is, R(I)= RI , is called a perring and let A[I] be a non-empty subset of R(I). Then, neutrosophic hyperring, where for all A[I] is a pseudo neutrosophic hyperideal if and only if the (a , bI ),( c , dI ) R ( I ), following conditions hold: (,)(,){(,):abI cdI xyI xacybd , }; (1) For all for all (a , bI ),( c , dI ) R ( I ) with b 0 or d 0, (,abI ) (, cdI ){(, xyI ): x ac ., y a...} d b c b d (2) For all and and (x ,0) ( y ,0) {( u ,0): u x y }. We usually use + and instead of and . (3) A[I] does not contain a proper subset which is a hyperring. Lemma 3.9. Let R(I) be a neutrosophic hyperring. Then, (0,0)RI ( ) is bilaterally absorbing element. Definition 3.14. Let A[I] and B[I] be any two neutro- Proof. Suppose that (a , bI ) R ( I ). Then, sophic hyperideals of a neutrosophic hyperring R(I). The (a , bI ).(0,0) sum of A[I] and B[I] denoted by A[I]+B[I] is defined as {(,):u vI u a 0, v a 0 b 0 b 0} the set {(,x yI ):(, x yI ) (, a bI ) (, c dI ), where {(u , vI ): u {0}, v {0}} {(0,0)}. (,a bI ) A [],(, I c dI ) B []}. I Hence, is a bilaterally absorbing ele- ment. Definition 3.15. Let R(I) be a neutrosophic hyperring and let A[I] be a neutrosophic hyperideal of R(I). If K is a A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings Neutrosophic Sets and Systems, Vol. 2, 2014 38 hyperideal (pseudo hyperideal) of R, the sum of A[I] and K for all denoted by A[I]+K is defined as the set ((abI , ),( cdI , )),(( efI , ),( ghI , )) HI ( ) GI ( ). {(,x yI ):(, x yI ) (, a bI ) (,0), k where (,a bI ) A [],(,0) I k K }. Proposition 4.4. Let (H(I),+) be a neutrosophic canon- ical hypergroup and let (K,+’) be a canonical hypergroup. Example 4. Let R(I)={(0,0),(x,0),(0,xI),(x,xI)} be a set Then, H(I)xK is a neutrosophic canonical hypergroup, and let + and be hyperoperations on R(I) defined in the where tables below. ((a , bI ),( m ,0) "(( c , dI ),( n ,0)) + (0,0) (x,0) (0,xI) (x,xI) {((,x yI ),(,0)):(, k x yI ) (, a bI ) (, c dI ), (0,0) {(0,0)} {(x,0)} {(0,0), {(x,xI)} (k ,0) ( m ,0) '( n ,0)}, (0,xI)} for all (x,0) {(x,0)} {(0,0), {(x,0), R(I) ((abIm , ),( ,0)),(( cdIn , ),( ,0)) HI ( ) K . (x,0)} (x,xI)} (0,xI) {(0,0), {(x,0), {(0,0), {(x,0), Proposition 4.5. Let A[I] and B[I] be any two neutro- (0,xI)} (x,xI)} (0,xI)} (x,xI)} sophic subcanonical hypergroups of a neutrosophic canon- (x,xI) {(x,xI)} R(I) {(x,0), R(I) ical hypergroup H(I). Then, (x,xI)} (1) A[I]+B[I] is a neutrosophic subcanonical hyper- Table 1. group of H(I). (2) A[I]+A[I]=A[I]. (0,0) (x,0) (0,xI) (x,xI) (3) AIBI[][] is a neutrosophic subcanonical hy- (0,0) {(0,0)} {(0,0)} {(0,0)} {(0,0)} pergroup of H(I). (x,0) {(0,0)} {(0,0), {(0,0), R(I) (x,0)} (0,xI)} Proposition 4.6. Let H(I) be a neutrosophic canonical (0,xI) {(0,0)} {(0,0), {(0,0), {(0,0), hypergroup and let A[I] and B[I] be any neutrosophic sub- canonical hypergroup and pseudo neutrosophic subcanon- (0,xI)} (0,xI)} (0,xI)} ical hypergroup of H(I), respectively. Then, (x,xI) {(0,0)} R(I) {(0,0), R(I) (1) A[I]+B[I] is a neutrosophic subcanonical hyper- (0,xI)} group of H(I). Table 2. (2) is a pseudo neutrosophic subcanon- ical hypergroup of H(I). It is clear from the tables that (R(I),+) is a neutrosophic canonical hypergroup and is a neutrosophic (RI ( ), , ) Proposition 4.7. Let H(I) be a neutrosophic canonical hyperring. hypergroup and let A[I] and B[I] be any neutrosophic sub- canonical hypergroup and pseudo neutrosophic subcanon- 4 Properties of Neutrosophic Canonical Hyper- ical hypergroup respectively. If K is any subcanonical hy- groups pergroup of H, then In this section, we present some basic properties of (1) A[I]+K is a neutrosophic subcanonical hyper- neutrosophic canonical hypergroups. group of H(I). (2) B[I]+K is a neutrosophic subcanonical hyper- Proposition 4.1. Let (H(I),+) be a neutrosophic canon- group of H(I). ical hypergroup. Then, (1) (H(I),+) in general is not a canonical hypergroup. Proposition 4.8. Let (H(I),+) be a neutrosophic canon- (2) (H(I),+) always contain a canonical hypergroup. ical hypergroup and let A be a subcanonical hypergroup of H. If A is normal in H, A[I] is not necessarily normal to Lemma 4.2. Let (H(I),+) be a neutrosophic canonical H(I). hypergroup. (1) –(0,0)=(0,0). Proposition 4.9. Let (H(I),+) be a neutrosophic canon- (2) ( (a , bI )) ( a , bI ) for all (a , bI ) H ( I ). ical hypergroup and let A be a normal neutrosophic subca- (3) ((,)(,a bI c dI ) (,)(, a bI c dI ) for all nonical hypergroup of H. Then, (a , bI ),( c , dI ) H ( I ). (,)(,)[]a aI A a aI A I for all (a , aI ) H ( I ). Proof. Suppose that A is normal in H. Let (h,0) be an Proposition 4.3. Let (H(I),+) and (G(I),+’) be any two arbitrary element of A. Then, for all (,)()a aI H I with neutrosophic canonical hypergroups. Then, HIGI()() a 0 , we have is a neutrosophic canonical hypergroup, where (a , aI ) ( h ,0) ( a , aI ) ((a , bI ),( c , dI ) "(( e , fI ),( g , hI )) (,){(,):aaI xyIxhayha , 0} a {((,p qI ),(, x yI )):(, p qI ) (, a bI ) (, e fI ), {(u , v 1): u a x , v a y a x, (,x yI ) (, c dI ) '(, g hI )}, x h a,} y h a a A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings 39 Neutrosophic Sets and Systems, Vol. 2, 2014 {(u , vI ): u a h a , Proposition 5.1. Let (RI ( ), , ) be a neutrosophic v a h a a a a h a}, hyperring. Then, from which we obtain uA and vA . Therefore, (1) in general is not a hyperring. (u , vI ) A [ I ].Since (hA ,0) is arbitrary, the required (2) always contain a hyperring. results follow. Proof. (1) It has been presented in part (1) of Proposi- tion 4.1. that is not a canonical hypergroup. Definition 4.10. Let (H(I),+) be a neutrosophic canoni- Also, distributive laws are not valid in . Hence, cal hypergroup and let A[I] be a neutrosophic subcanonical is not a hyperring. (2) Follows from the defi- hypergroup of H(I). We consider the quotient nition. (HI ():[]) AI {(, abI ) AI []:(, abI ) HI []} Proposition 5.2. Let and (SI ( ), ', ')be and we put (,a bI ) A [] I [(, a bI )]. For all any two neutrosophic hyperrings. Then, RISI()() is a [(a , bI )],[( c , dI )] ( H ( I ): A [ I ]), we define the neutrosophic hyperring, where hyperoperation on (HIAI ( ): [ ]) as ((a , bI ),( c , dI ) "(( e , fI ),( g , hI )) [(a , bI )] [( c , dI )] {[( e , fI )]:( e , fI ) {((,p qI ),(, x yI )):(, p qI ) (, a bI ) (, e fI ), (a , bI ) ( c , dI )}. (,x yI ) (, c dI ) '(, g hI )}, Then the couple ((HIAI ( ): [ ]), ) is called the quo- and tient neutrosophic canonical hypergroup. If A[I] is a pseu- ((a , bI ),( c , dI ) "(( e , fI ),( g , hI )) do neutrosophic subcanonical hypergroup, then we call {((p , qI ),( x , yI )):( p , qI ) ( a , bI ).( e , fI ), a pseudo quotient neutrosophic ca- (x , yI ) ( c , dI ).'( g , hI )}, nonical hypergroup. for all ((abI , ),( cdI , )),(( efI , ),( ghI , )) RI ( ) SI ( ). Proposition 4.11. Let H(I) be a neutrosophic canoni- cal hypergroup and A[I] be a neutrosophic subcanonical Proposition 5.3. Let be a neutrosophic hypergroup (pseudo neutrosophic subcanonical hype- hyperring and let (K , ', ') be a hyperrings. Then, group) of H(I). Then, is generally not RIK() is a neutrosophic hyperring, where a canonical hypergroup. ((a , bI ),( m ,0) "(( c , dI ),( n ,0)) {((,x yI ),(,0)):(, k x yI ) (, a bI ) (, c dI ), Definition 4.12. Let (HI1 ( ), ) and (HI2 ( ), ) be (k ,0) ( m ,0) '( n ,0)}, two neutrosophic canonical hypergroups and let and :()()HIHI12 be a mapping from HI1() into ((a , bI ),( m ,0) "(( c , dI ),( n ,0)) HI2 (). {((x , yI ),( k ,0)):( x , yI ) ( a , bI ).( c , dI ), (1) is called a homomorphism if (k ,0) ( m ,0).'( n ,0)}, a. is a canonical hypergroup homomor- for all phism; ((abIm , ),( ,0)),(( cdIn , ),( ,0)) RI ( ) K . b. ((0,II )) (0, ). (2) is called a good or strong homomorphism if Lemma 5.4. Let A[I] be any neutrosophic hyperideal a. is a good or strong canonical hyper- of a neutrosophic hyperring R(I). Then, group homomorphism; (1) A[I]+A[I]=A[I]. b. (2) (a,bI)+A[I]=A[I] for all (,)[]a bI A I . (3) is called a isomorphism (strong isomorphism) if is a bijective homomorphism (strong homo- Proposition 5.5. Let be a neutrosophic morphism). hyperring and let A[I] and B[I] be left (right) neutrosophic ideals of R(I). Then, Definition 4.13. Let be a ho- (1) AIBI[][] is a left (right) neutrosophic hyper- momorphism from a neutrosophic canonical hypergroup ideal of R(I). into a neutrosophic canonical hypergroup . (2) AIBI[][] is a left (right) neutrosophic hyper- (1) The kernel of denoted by Ker is the set ideal of R(I). {(a , bI ) H1 ( I ): (( a , bI )) (0,0)}. (2) The kernel of denoted by Im is the set Proposition 5.6. Let R(I) be a neutrosophic hyperring {((, a bI )):(, a bI ) H1 ()}. I and let A[I] and B[I] be any neutrosophic hyperideal and pseudo neutrosophic hyperideal of R(I) respectively. Then, 5 Properties of Neutrosophic Hyperrings (1) is a neutrosophic hyperideal of R(I). (2) is a pseudo neutrosophic hyperideal In this section, we present some basic properties of of R(I). neutrosophic hyperrings. Proposition 5.7. Let R(I) be a neutrosophic hyperring A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings Neutrosophic Sets and Systems, Vol. 2, 2014 40 and let A[I] and B[I] be any neutrosophic hyperideal and momorphism from a neutrosophic hyperring into a pseudo neutrosophic hyperideal, respectively. If K is any neutrosophic hyperring . Then, subhyperring of R, then (1) Ker is a subhyperring of R1 and never be a (1) AIK[] is a neutrosophic hyperideal of R(I). neutrosophic hyperring (neutrosophic hyperideal) (2) BIK[] is a neutrosophic hyperideal of R(I). of . (2) Im is a neutrosophic subhyperring of . Proposition 5.8. Let (RI ( ), , ) be a neutrosophic hyperring and let A be a hyperideal of R. If A is normal in Question 1: Does there exist: R, A[I] is not necessarily normal in R(I). (1) A neutrosophic canonical hypergroup with normal Proposition 5.9. Let be a neutrosophic neutrosophic subcanonical hypergroups? hyperring and let A be a normal hyperideal of R. Then, (2) A neutrosophic hyperring with normal neutrosoph- (,)(,)[]a aI A a aI A I ic hyperideals? for all (a , aI ) R ( I ). (3) A simple neutrosophic canonical hypergroup? (4) A simple neutrosophic hyperring? Definition 5.10. Let be a neutrosophic hyperring and let (AI ( ), , ) be a neutrosophic hyperideal 6 Conclusion of R(I). We consider the quotient (RIAI ( ): [ ]) {(,a bI ) A []:(, I a bI ) R ()} I and we put In this paper, we have introduced and studied neutro- (a,bI)+A[I]=[(a,bI)]. sophic canonical hypergroups and neutrosophic hyperrings. For all [(a , bI )],[( c , dI )] (( R ( I ): A [ I ]), we con- We presented elementary properties of neutrosophic ca- sider the hyperoperation as defined in the Definition nonical hypergroups and neutrosophic hyperrings. Also, 4.10 and we define the hyperoperation on we studied quotient neutrosophic canonical hypergroups ((RIAI ( ): [ ]) as [(a , bI )] [( c , dI )] {[( e , fI )]: and quotient neutrosophic hyperrings. (e , fI ) ( a , bI ).( c , dI )}. Then, the triple ((RIAI (): []), , ) is called the References quotient neutrosophic hyperring. If A[I] is a pseudo neu- [1] A. A. A. Agboola, A. D. Akinola, and O. Y. Oyebola. Neu- trosophic hyperideal, then we call trosophic Rings I, Int. J. Math. Comb. 4 (2011), 1-14. a pseudo neutrosophic hyperring. [2] A. A. A. Agboola, E. O. Adeleke, and S. A. Akinleye. Neu- trosophic Rings II, Int. J. Math. Comb. 2 (2012), 1-8. Proposition 5.11. Let R(I) be a neutrosophic hyperring [3] A. A. A. Agboola, A. O. Akwu, and Y. T. Oyebo. Neutro- and let A[I] be a neutrosophic hyperideal (pseudo neutro- sophic Groups and Neutrosophic Subgroups, Int. J. Math. sophic hyperideal) of R(I). Then, Comb. 3 (2012), 1-9. is generally not a hyperring. [4] A. A. A. Agboola and B. Davvaz. Introduction to Neutro- sophic Hypergroups (To appear in ROMAI Journal of Definition 5.12. Let (RI ( ), ) and (RI ( ), ) be Mathematics). 1 2 [5] A. Asokkumar and M. Veelrajan. Characterization of regu- two neutrosophic hyperrings and let :()()RIRI12 lar hyperrings, Italian J. Pure and App. Math. 22 (2007), be a mapping from RI1() into RI2 (). 115-124. (1) is called a homomorphism if [6] A. R. Bargi. A class of hyperrings, J. Disc. Math. Sc. And a. is a hyperring homomorphism; Cryp. 6 (2003), 227-233. b. ((0,II )) (0, ). [7] P. Corsini. Prolegomena of Hypergroup Theory. Second (2) is called a good or strong homomorphism if edition, Aviain Editore, 1993. a. is a good or strong hyperring homo- [8] P. Corsini and V. Leoreanu. Applications of Hyperstructure morphism; Theory. Advances in Mathematics. Kluwer Academic Pub- b. lisher, Dordrecht, 2003. (3) is called a isomorphism (strong isomorphism) if [9] B. Davvaz. Polygroup Theory and Related Systems. World is a bijective homomorphism (strong homo- Sci. Publ., 2013. morphism). [10] B. Davvaz. Isomorphism theorems of hyperrings, Indian J. Pure Appl. Math. 35(3) (2004), 321-333. [11] B. Davvaz. Approximation in hyperrings, J. Mult.-Valued Definition 5.13. Let :()()RIRI12 be a homo- morphism from a neutrosophic hyperring RI() into a Soft. Comput. 15(5-6) (2009), 471-488. 1 [12] B. Davvaz and A. Salasi. A realization of hyperrings, neutrosophic hyperring RI(). 2 Comm. Algebra 34 (2006), 4389-4400. (1) The kernel of denoted by Ker is the set [13] B. Davvaz and T. Vougiouklis. Commutative rings obtained {(a , bI ) R1 ( I ): (( a , bI )) (0,0)} . from hyperrings (Hv-rings) with α*-relations, Comm. Alge- (2) The kernel of denoted by Im is the set bra 35 (2007), 3307-3320. {((, a bI )):(, a bI ) R1 ()}. I [14]B. Davvaz and V. Leoreanu-Fotea. Hyperring Theory and Applications. International Academic Press, USA, 2007. Proposition 5.14. Let be a ho- [15]M. De Salvo. Hyperrings and hyperfields, Annales Scien- A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings 41 Neutrosophic Sets and Systems, Vol. 2, 2014 tifiques de l’Universite de Clermont-Ferrand II, 22 (1984), 89-107. [16]F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Prob- ability (3rd Ed.). American Research Press, Rehoboth, 2003. URL: http://fs.gallup.unm.edu/eBook-Neutrosophic4.pdf. [17]K. Krasner, A class of hyperrings and hyperfields, Int. J. Math. And Math. Sci., 6(2) (1983), 307-311. [18]F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves, Stockholm, Sweden, (1934), 45-49. [19]S. Mirvakili, S.M. Anvariyeh and B. Davvaz, On α-relation and transitivity conditions of α, Comm. Algebra, 36 (2008), 1695-1703. [20]S. Mirvakili and B. Davvaz, Applications of the α*-relation to Krasner hyperrings, J. Algebra, 362 (2012), 145-146. [21]J. Mittas, Hypergroups canoniques, Math. Balkanica 2 (1972), 165-179. [22]M. Velrajan and A. Asokkumar, Note on Isomorphism Theorems of Hyperrings, Int. J. Math. And Math. Sci., Hindawi Publishing Corporation (2010), 1-12. [23]A. Nakassis, Expository and survey article of recent results in hyperring and hyperfield theory, Int. J. Math. And Math. Sci. 11 (1988), 209-220. [24]R. Rotta, Strongly distributive multiplicative hyperrings, J. Geo. 39(1-2) (1990), 130-138. [25]M. Stefanascu, Constructions of hyperfields and hyperrings, Stud. Cercet. Stiint. Ser. Mat. Univ. Bacau No. 16, suppl. (2006), 563-571. [26]W.B. Vasantha Kandasamy and F. Smarandache. Some Neutrosophic Algebraic Structures and Neutrosophic N- Algebraic Structures. Hexis, Phoenix, Arizona, 2006. URL: http://fs.gallup.unm.edu/NeutrosophicN- AlgebraicStructures.pdf. [27] W.B. Vasantha Kandasamy and F. Smarandache. Neutro- sophic Rings, Hexis, Phoenix, Arizona, 2006. URL: http://fs.gallup.unm.edu/NeutrosophicRings.pdf. [28]T. Vougiouklis. Hyperstructures and their Representations, Hadronic Press, Inc., 113 Palm Harber, USA, 1994. [29] T. Vougiouklis. The fundamental relation in hyper- rings.The general hyperfields. Proc. Fourth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scientific (1991), 203-221. Received: February 3rd, 2014. Accepted: February 27th, 2014. A.A.A. Agboola & B. Davaaz, On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings Neutrosophic Sets and Systems, Vol. 2, 2013 Neutrosophic Lattices Vasantha Kandasamy1 and Florentin Smarandache2 1 Department of Mathematics, Indian Institute of Technology, Madras, India. E-mail: [email protected] 2 Mathematics and Science Department, University of Mexico, 70S, Gurley Ave. Gallup NM 87301, USA. E-mail: [email protected] Abstract. In this paper authors for the first time define a introduces the concept of partially ordered neutrosophic new notion called neutrosophic lattices. We define few set and neutrosophic lattices. Section two introduces properties related with them. Three types of neutrosophic different types of neutrosophic lattices and the final sec- lattices are defined and the special properties about these tion studies neutrosophic Boolean algebras. Conclusions new class of lattices are discussed and developed. This and results are provided in section three. paper is organised into three sections. First section Keywords: Neutrosophic set, neutrosophic lattices and neutrosophic partially ordered set. 1 Introduction to partially ordered neutrosophic I ∪ 1 set • Here we define the notion of a partial order on a neutrosophic set and the greatest element and the least 1 • • I element of it. Let N(P) denote a neutrosophic set which must contain I, 0, 1 and 1 + I; that is 0, 1, I and 1+ I ∈ N(P). We call 0 to be the least element so 0 < 1 and 0 < I a • • a I is assumed for the working. Further by this N(P) becomes a partially ordered set. We define 0 of N(P) to be the least • 0 element and I ∪ 1 = 1 + I to be the greatest element of Figure 1.1 N(P). Suppose N(P) = {0, 1, I, 1 + I, a1, a2, a3, a1I, a2I, a3I} Example 1.2: Let N(P) = {0, 1, I, 1 ∪ I, a1, a2, a1I, a2I} be then N(P) with 0 < ai, 0 < aiI, 1 ≤ i ≤ 3. Further 1 > ai; I > a neutrosophic lattice associated with the following Hasse aiI; 1 ≤ i ≤ 3 ai Example 1.1: Let N(P) = {0, 1, I, I ∪ 1 = 1 + I, a, aI} be a Example 1.3: Let N(P) = {0 1, I, 1 ∪ I} be a neutrosophic partially ordered set; N(P) is a neutrosophic lattice. lattice given by the following neutrosophic Hasse diagram. We know in case of usual lattices [1-4]. Hasse defined the notion of representing finite lattices by diagrams 1∪I known as Hasse diagrams [1-4]. We in case of • Neutrosophic lattices represent them by the diagram which 1 • I will be known as the neutrosophic Hasse diagram. • The neutrosophic lattice given in example 1.1 will have the • following Hasse neutrosophic diagram. 0 Figure 1.3 Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices Neutrosophic Sets and Systems, Vol. 2, 2013 43 has only neutrosophic coordinates or equivalently all the It is pertinent to observe that if N(P) is a neutrosophic co ordinates (vertices) are neutrosophic barring 0. lattice then 0, 1, I, 1 ∪ I ∈ N(P) and so that N(P) given in In the example 1.5 we see the pure neutrosophic part of example 1.3 is the smallest neutrosophic lattice. the neutrosophic lattice figure 2.1; I Example 1.4: Let N(P) = {0, 1, I, 1 ∪ I = 1 + I, a1, a2, a1I, • a2I, a1 < a2} be the neutrosophic lattice. The Hasse diagram of the neutrosophic lattice N(P) is as follows: I ∪ 1 • a1I • a2I • a3I • • a4I 1 • • I • a2 • • a I 2 0 a1 • • a1I Figure 2.1 whose Hasse diagram is given is the pure neutrosophic • sublattice lattice from figure 1.5. Likewise we can have 0 the Hasse diagram of the usual lattice from example 1.5. Figure 1.4 1 • We can have neutrosophic lattices which are different. Example 1.5: Let N(P) = {0, 1, I, a1, a2, a3, a4, a1I, a2I, a3I, a4I, 1 + I = I ∪ 1} be the neutrosophic lattice of finite order. • (ai is not comparable with aj if i ≠ j, 1 ≤ i, j ≤ 4). a1 • a2 a3 • • a4 I ∪ 1 • • 0 1 • • I Figure 2.2 a1 • a2 • a3 • a4 • a1I • • • • a3I a2I a4I We see the diagrams are identical as diagrams one is pure neutrosophic where as the other is a usual lattice. As we have no method to compare a neutrosophic number and a • non neutrosophic number, we get two sublattices identical 0 in diagram of a neutrosophic lattice. For the modular Figure 1.5 identity, distributive identity and the super modular identity and their related properties refer [1-4]. We see N(P) is a neutrosophic lattice with the above The neutrosophic lattice given in example 1.5 has a neutrosophic Hasse diagram. sublattice which is a modular pure neutrosophic lattice and In the following section we proceed onto discuss sublattice which is a usual modular lattice. various types of neutrosophic lattices. The neutrosophic lattice given in example 1.3 is a distributive lattice with four elements. However the neutrosophic lattice given in example 1.5 is not distributive 2. Types of Neutrosophic Lattices as it contains sublattices whose homomorphic image is isomorphic to the neutrosophic modular lattice N(M4); The concept of modular lattice, distributive lattice, where N(M ) is a lattice of the form super modular lattice and chain lattices can be had from [1- 4 4]. We just give examples of them and derive a few properties associated with them. In the first place we say a neutrosophic lattice to be a pure neutrosophic lattice if it Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices 44 Neutrosophic Sets and Systems, Vol. 2, 2014 I • interesting and in general does not yield modular neutrosophic lattices. We define the strong neutrosophic set of a set S as • follows a1I • a2I a3I • • a4I Let A = {a1, a2, …, an}, the strong neutropshic set of A; SN(A) = {ai, ajI, ai ∪ ajI = ai + ajI; 0, 1, I, 1 + I, 1 ≤ i, j ≤ n}. • S(L) the strong neutrosophic lattice is defined as 0 follows: S(L) = {0, 1, I, 1 + I, ai, ajI, Figure 2.3 I ∪ ai = I + ai a1I ∪ 1 = aiI + 1, ai + ajI = ai ∪ ajI 0 < ≠ Likewise by N(Mn) we have a pure neutrosophic lattice of ai < 1; 0 < ajI < I, 1 ≤ i, j ≤ n}. the form given below in figure 2.4. ≠ S(L) with max, min is defined as the strong neutrosophic lattice. I • We will illustrate this situation by some examples. Example 2.1: Let S(L) = {0, 1, I, 1 + I, a, aI, a + aI, 1 + aI, I + a} a I • a I • • 1+I 1 2 …. a4I • 1+aI • • I+a • 0 1 • • a+aI • I Figure 2.4 a • • aI I • • 0 • cI Figure 2.6 aI • be a strong neutrosophic lattice. bI • We have several sublattices both strong neutrosophic sublattice as well as usual lattice. For • 0 • I Figure 2.5 • a The neutrosophic pentagon lattice is given in figure 2.5 which is neither distributive nor modular. • 0 The lattice N(M4) is not neutrosophic super modular we see the neutrosophic lattice in example 1.5 is not modular Figure 2.7 for it has sublattices whose homomorphic image is is the usual lattice. isomorphic to the pentagon lattice. • I So we define a neutrosophic lattice N(L) to be a quasi modular lattice if it has atleast one sublattice (usual) which • aI is modular and one sublattice which is a pure neutrosophic modular lattice. • 0 Thus we need to modify the set S and the neutrosophic set N(S) of S. For if S = {a , …, a } we define N(S) = {a I, 1 n 1 Figure 2.8 a2I, …, anI} and take with S ∪ N(S) and the elements 0, 1, I, and 1 ∪ I = 1 + I. Thus to work in this way is not is the pure neutrosophic lattice. Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices Neutrosophic Sets and Systems, Vol. 2, 2013 45 1+I This is a edge neutrosophic lattice as the edge connecting 0 • to a2 is an indeterminate. 1 • • I Example 2.4: Let us consider the following Hasse diagram of a lattice L. • 1 0 • Figure 2.9 is the strong neutrosophic lattice. • • a • a a1 • a2 • a3 4 5 These lattices have the edges to be real. Only vertices are indeterminates or neutrosophic numbers. However we can • have lattices where all its vertices are real but some of the 0 lines (or edges) are indeterminates. Figure 2.12 Example 2.2: For consider L is a edge neutrosophic modular lattice. 1 The edges connecting 0 to a and 1 to a are neutrosophic • 3 4 edges and the rest of the edges are reals. However all the vertices are real and it is a partially ordered set. We take some of the edges to be an indeterminate. a a1 • • a2 • 3 Example 2.5: Let L be the edge neutrosophic lattice whose Hassee diagram is as follows: • a4 1 • • 0 a1 • a2 • • a3 • a4 Figure 2.10 • Such type of lattices will be known as edge neutrosophic a5 lattices. a a In case of edge neutrosophic lattices, we can have edge a6 • a7 • • a8 • 9 • 10 • a11 neutrosophic distributive lattices, edge neutrosophic modular lattices and edge neutrosophic super modular lattices and so on. • 0 We will only illustrate these by some examples. Figure 2.13 Example 2.3: Consider the following Hasse diagram. 1 Clearly L is not a distributive edge neutrosophic lattice. • However L has modular edge neutrosophic sublattices as well as modular lattices which are not neutrosophic. Inview of this we have the following theorem. a2 • • a1 THEOREM 2.1: Let L be a edge neutrosophic lattice. Then L in general have sublattices which are not edge neutrosophic. • 0 Proof follows from the simple fact that every vertex is a Figure 2.11 sublattice and all vertices of the edge neutrosophic lattice Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices 46 Neutrosophic Sets and Systems, Vol. 2, 2014 which are not neutrosophic; but real is an instance of a not the containment relation of subsets as the partial order an edge neutropshic lattice. relation on P(S). We can have pure neutrosophic lattice which have the 1 edges as well the vertices to be neutrosophic. • The following lattices with the Hasse diagram are pure {0,1,I} {0,I,1+I} {1,I+1+I} neutrosophic lattices. • • {0,1,1+I}• • 1 • {0,I) • • {0,1+I} • • •{I,1+I} • {0,1} {1+I} {1,1+I} {0} • {1} • {I+1} • {I} • a1 I • • a2I • • Φ 0 Figure 3.1 Figure 2.14 Example 3.2: Let S = {0, 1, I, 1+I, a, aI, a+I, aI+1, aI+a} be the neutrosophic set; 0 < a < 1. P(S) be the power set of • 9 1+I S. |P(S)| = 2 . P(S) is a neutrosophic Boolean algebra of order 29. • a1I Example 3.3: Let S = {0, 1, I, 1+I, a1, a2, a1I, a2I, a1+I, a2+I, 1+a1I, 1+a2I, 1+a1I+a2, a1+a2, 1 + a1I + a2I, …} be the • a2I neutrosophic set with a1 Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices Neutrosophic Sets and Systems, Vol. 2, 2013 47 [2]. Gratzer G, General Lattice Theory, Birkhauser Verlage 1 Basel and Stullgart, 1978, • [3]. Iqbal Unnisa and W.B. Vasantha, On a certain class of Modular lattices, J. of Madras Univ, Vol. 41, 166-169, 1978, [4]. Iqbal Unnisa, W.B. Vasantha Kandasamy and Florentin a1 • a2 • a3 • a4 • Smarandache, Super modular Lattices, Educational Publisher, Ohio, 2012. a • a7 • a6 • 5 • Received: January 14, 2014. Accepted: February 12, 2014. 0 Figure 3.3 Clearly a1 and a6 are not comparable, a2 and a5 are not comparable a4 and a7 are not comparable. We can have the following Hasse diagram which has neutrosophic edges. 1 • a1 • • a2 • a3 • a4 a • a7 • a6 • 5 • 0 Figure 3.4 Clearly L is a edge neutrosophic lattice where we have some neutrosophic edges which are not comparable in the original lattice. So we can on usual lattices L remake it into a edge neutrosophic lattice this is done if one doubts that a pair of elements {a1, a2} of L with a1 ≠ a2, min {a1, a2} ≠ a1 or a2 or max {a1, a2} ≠ a1 or a2. If some experts needs to connect a1 with a2 by edge then the resultant lattice becomes a edge neutrosophic lattice. Conclusion: Here for the first time we introduce the concept of neutrosophic lattices. Certainly these lattices will find applications in all places where lattices find their applications together with some indeterminancy. When one doubts a connection between two vertices one can have a neutrosophic edge. References: [1]. Birkhoff G, Lattice Theory, 2nd Edition, Amer, Math. Soc, 1948. Vasantha Kadasamy and Florentin Smarandache ~ Neutrosophic Lattices 48 Neutrosophic Sets and Systems, Vol. 2, 2014 Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making Jun Ye1 and Qiansheng Zhang2 1 Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang 312000, P.R. China. E-mail: yehjun@ aliyun.com 2 School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, P.R. China. E-mail: [email protected] Abstract. Similarity measures play an important role in data weights of the three independent elements (i.e., truth- mining, pattern recognition, decision making, machine learning, membership degree, indeterminacy-membership image process etc. Then, single valued neutrosophic sets degree, and falsity-membership degree) in a SVNV are (SVNSs) can describe and handle the indeterminate and considered in the decision-making method. In the inconsistent information, which fuzzy sets and intuitionistic decision making, we utilize the single-valued fuzzy sets cannot describe and deal with. Therefore, the paper neutrosophic weighted similarity measure between the proposes new similarity meas-ures between SVNSs based on the ideal alternative and an alternative to rank the minimum and maxi-mum operators. Then a multiple attribute alternatives corresponding to the measure values and to decision-making method based on the weighted similarity select the most desirable one(s). Finally, two practical measure of SVNSs is established in which attribute values for al- examples are provided to demonstrate the applications ternatives are represented by the form of single valued and effectiveness of the single valued neutrosophic neutrosophic values (SVNVs) and the attribute weights and the multiple attribute decision-making method. Keywords: Neutrosophic set, single valued neutrosophic set, similarity measure, decision making. 1 Introduction handle indeterminate information and inconsistent information, which fuzzy sets, IFSs, and IVIFSs Since fuzzy sets [1], intuitionistic fuzzy sets (IFSs) cannot describe and deal with. Recently, Ye [7-9] [2], interval-valued intuitionistic fuzzy sets (IVIFSs) [3] proposed the correlation coefficients of SVNSs were introduced, they have been widely applied in data and the cross-entropy measure of SVNSs and mining, pattern recognition, information retrieval, applied them to single valued neutrosophic decision making, machine learning, image process and so decision-making problems. Then, Ye [10] on. Although they are very successful in their respective introduced similarity measures based on the domains, fuzzy sets, IFSs, and IVIFSs cannot describe and distances between INSs and applied them to deal with the indeterminate and inconsistent information multicriteria decision-making problems with that exists in real world. To handle uncertainty, imprecise, interval neutrosophic information. Chi and Liu incomplete, and inconsistent information, Smarandache [11] proposed an extended TOPSIS method for [4] proposed the concept of a neutrosophic set. The the multiple attribute decision making problems neutrosophic set is a powerful general formal framework with interval neutrosophic information. which generalizes the concepts of the classic set, fuzzy Furthermore, Ye [12] introduced the concept of set, IFS, IVIFS etc. [4]. In the neutrosophic set, truth- simplified neutrosophic sets and presented membership, indeterminacy-membership, and falsity- simplified neutrosophic weighted aggregation membership are represented independently. However, the operators, and then he applied them to neutrosophic set generalizes the above mentioned sets multicriteria decision-making problems with from philosophical point of view and its functions TA(x), simplified neutrosophic information. Majumdar IA(x) and FA(x) are real standard or nonstandard subsets and Samanta [13] introduced several similarity of ]−0, 1+[, i.e., TA(x): X → ]−0, 1+[, IA(x): X → ]−0, measures between SVNSs based on distances, a 1+[, and FA(x): X → ]−0, 1+[. Thus, it is difficult to apply matching function, membership grades, and then in real scientific and engineering areas. Therefore, Wang proposed an entropy measure for a SVNS. et al. [5, 6] introduced a single valued neutrosophic set Broumi and Smarandache [14] defined the (SVNS) and an interval neutrosophic set (INS), which are distance between neutrosophic sets on the basis of the subclass of a neutrosophic set. They can describe and the Hausdorff distance and some similarity Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic48 Similarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 49 measures based on the distances, set theoretic approach, Definition 3 [6]. The complement of a SVNS A is c c and matching function to calculate the similarity degree denoted by A and is defined as TA (x) = FA(x), between neutrosophic sets. c c IA (x) = 1 − IA(x), FA (x) = TA(x) for any x in X. Because the concept of similarity is fundamentally Then, it can be denoted by important in almost every scientific field and SVNSs can describe and handle the indeterminate and inconsistent c A (, ), (1 ), AA |)( XxxTxIxFxA . information, this paper proposes new similarity measures between SVNSs based on the minimum and maximum Definition 4 [6]. A SVNS A is contained in the operators and establishes a multiple attribute decision- other SVNS B, A ⊆ B, if and only if TA(x) ≤ TB(x), making method based on the weighted similarity measure IA(x) ≥IB(x), FA(x) ≥ FB(x) for any x in X. of SVNSs under single valued neutrosophic environment. To do so, the rest of the article is organized as follows. Definition 5 [6]. Two SVNSs A and B are equal, Section 2 introduces some basic concepts of SVNSs. i.e., A = B, if and only if A ⊆ B and B ⊆ A. Section 3 proposes new similarity measures between SVNSs based on the minimum and maximum operators 3 Similarity measures of SVNSs and investigates their properties. In Section 4, a single valued neutrosophic decision-making approach is This section proposes several similarity proposed based on the weighted similarity measure of measures of SVNSs based on the minimum and SVNSs. In Section 5, two practical examples are given to maximum operators and investigates their demonstrate the applications and the effectiveness of the properties. proposed decision-making approach. Conclusions and further research are contained in Section 6. In general, a similarity measure between two SVNSs A and B is a function defined as S: N(X)2 2 Some basic concepts of SVNSs [0, 1] which satisfies the following properties: Smarandache [4] originally introduced the concept of a neutrosophic set from philosophical point of view, (SP1) 0 S(A, B) 1; which generalizes that of fuzzy set, IFS, and IVIFS etc.. (SP2) S(A, B) = 1 if A = B; Definition 1 [4]. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A (SP3) S(A, B) = S(B, A); in X is characterized by a truth-membership function (SP4) S(A, C) S(A, B) and S(A, C) S(B, C) TA(x), an indeterminacy-membership function IA(x) and a if A B C for a SVNS C. falsity-membership function FA(x). The functions TA(x), I (x) and F (x) are real standard or nonstandard subsets of A A Let two SVNSs A and B in a universe of − + − + − + ] 0, 1 [. That is TA(x): X → ] 0, 1 [, IA(x): X → ] 0, 1 [, discourse X = {xl, x2, …, xn} be − + and FA(x): X → ] 0, 1 [. Thus, there is no restriction on (, ), ( ), |)( iiAiAiAi XxxFxIxTxA and the sum of T (x), I (x) and F (x), so −0 ≤ sup T (x) + sup A A A A , where + (, ), ( ), |)( iiBiBiBi XxxFxIxTxB IA(x) + sup FA(x) ≤ 3 . TA(xi), IA(xi), FA(xi), TB(xi), IB(xi), FB(xi) [0, 1] Obviously, it is difficult to apply in real scientific and for every xi X. Based on the minimum and engineering areas. Hence, Wang et al. [6] introduced the maximum operators, we present the following definition of a SVNS. similarity measure between A and B: 1 n (min ), xTxT )( iBiA Definition 2 [6]. Let X be a universal set. A SVNS A in X 1 BAS ),( 3n i1 (max ), xTxT iBiA )( is characterized by a truth-membership function TA(x), an . (1) indeterminacy-membership function IA(x), and a falsity- (min ), xIxI )( (min ), xFxF )( iBiA iBiA membership function FA(x). Then, a SVNS A can be (max ), xIxI iBiA )( (max ), xFxF iBiA )( denoted by The similarity measure has the following proposition. (, ), ( ), AAA |)( XxxFxIxTxA , Proposition 1. Let A and B be two SVNSs in a where TA(x), IA(x), FA(x) [0, 1] for each point x in X. universe of discourse X = {x1, x2, …, xn}. The Therefore, the sum of TA(x), IA(x) and FA(x) satisfies the single valued neutrosophic similarity measure condition 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3. S1(A, B) should satisfy the following properties: Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic49 Similarity Measures for Multiple Attribute Decision-Making 50 Neutrosophic Sets and Systems, Vol. 2, 2014 (SP1) 0 S (A, B) 1; 1 n (min ), xTxT )( 1 iBiA 2 BAS ),( , (2) n i1 (max ), xTxT iBiA )( (SP2) S1(A, B) = 1 if A = B; (min ), xIxI iBiA )( (min ), xFxF iBiA )( (max ), xIxI iBiA )( (max ), xFxF iBiA )( (SP3) S1(A, B) = S1(B, A); where , , are the weights of the three (SP4) S1(A, C) S1(A, B) and S1(A, C) S1(B, C) if A independent elements (i.e., truth-membership, B C for a SVNS C. indeterminacy-membership, and falsity- membership) in a SVNS and + + = 1. Proof. It is easy to remark that S (A, B) satisfies the 1 Especially, when = = = 1/3, Eq. (2) reduces properties (SP1)-(SP3). Thus, we must prove the property to Eq. (1). (SP4). Then, the similarity measure of S2(A, B) also Let A B C, then, TA(xi) TB(xi) TC(xi), IA(xi) has the following proposition: IB(xi) IC(xi), and FA(xi) FB(xi) FC(xi) for every xi X. According to these inequalities, we have the following Proposition 2. Let A and B be two SVNSs in a similarity measures: universe of discourse X = {x1, x2, …, xn}. The single valued neutrosophic similarity measure 1 n xT )( xI )( xF )( S (A, B) should satisfy the following properties: iA iB iB , 2 1 BAS ),( 3n i1 xT iB )( xI iA )( xF iA )( (SP1) 0 S2(A, B) 1; 1 n xT )( xI )( xF )( CAS ),( iA iC iC 1 (SP2) S2(A, B) = 1 if A = B; 3n i1 xT iC )( xI iA )( xF iA )( (SP3) S2(A, B) = S2(B, A); 1 n xT )( xI )( xF )( iB iC iC . 1 CBS ),( 3n i1 xT iC )( xI iB )( xF iB )( (SP4) S2(A, C) S2(A, B) and S2(A, C) S2(B, C) if A B C for a SVNS C. xT )( xI )( xI )( Since there are xT iA )( iA , iB iC , xT iB )( xT ic )( xI iA )( xI iA )( By the similar proof method in Proposition 1, we can prove that the similarity measure of S2(A, xF iB )( xF iC )( and , we can obtain that S1(A, C) S1(A, B). B) also satisfies the properties (SP1)-(SP4) xF iA )( xF iA )( (omitted). xT )( xT )( xI )( Furthermore, if the important differences are Similarly, there are iB iA , iC considered in the elements in a universe of xT iC )( xT iC )( xI iB )( discourse X = {xl, x2, …, xn}, we need to take the xI iC )( xF iC )( xF iC )( , and . Then, we can obtain weight of each element xi (i = 1, 2,…, n) into xI iA )( xF iB )( xF iA )( account. Therefore, we develop a weighted that S1(A, C) S1(B, C). similarity measure between SVNSs. Thus S1(A, B) satisfies the property (SP4). Let wi be the weight for each element xi (i = 1, n 2,…, n), wi [0, 1], and w 1, and then we Therefore, we finish the proof. i1 i have the following weighted similarity measure: If the important differences are considered in the three n (min ), xTxT )( independent elements (i.e., truth-membership, ),( wBAS iBiA 3 i (max ), xTxT )( . (3) indeterminacy-membership, and falsity-membership) in a i1 iBiA (min ), xIxI )( (min ), xFxF )( SVNS, we need to take the weights of the three iBiA iBiA (max ), xIxI )( (max ), xFxF )( independent terms in Eq.(1) into account. Therefore, we iBiA iBiA develop another similarity measure between SVNSs: Similarly, the weighted similarity measure of S3(A, B) also has the following proposition: Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic50 Similarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 51 Proposition 3. Let A and B be two SVNSs in a universe of Hence, there are S3(A, C) S3(A, B) and S3(A, discourse X = {x1, x2, …, xn}. Then, the single valued C) S3(B, C). neutrosophic similarity measure S3(A, B) should satisfy 4 Decisions making method using the weighted the following properties: similarity measure of SVNSs (SP1) 0 S3(A, B) 1; In this section, we propose a multiple attribute decision-making method based on the weighted (SP2) S3(A, B) = 1 if A = B; similarity measures between SVNSs under single valued neutrosophic environment. (SP3) S3(A, B) = S3(B, A); Let A = {A1, A2,…, Am} be a set of alternatives (SP4) S3(A, C) S3(A, B) and S3(A, C) S3(B, C) if A and C = {C1, C2,…, Cn} be a set of attributes. B C for a SVNS C. Assume that the weight of the attribute Cj (j = 1, n 2,…, n) is wj with wj [0, 1], x 1 and the j1 j Similar to the proof method in Proposition 1, we can weights of the three independent elements (i.e., prove that the weighted similarity measure of S (A, B) also 3 truth-membership, indeterminacy-membership, satisfies the properties (SP1)–(SP4) (omitted). and falsity-membership) in a SVNS are , , and T If w = (1/n, 1/n,…, 1/n) , then Eq. (3) reduces to Eq. (2). and + + = 1, which are entered by the For Example, Assume that we have the following decision-maker. In this case, the characteristic of three SVNSs in a universe of discourse X = {xl, x2}: the alternative Ai (i = 1, 2,…, m) on an attribute Cj A = { B = { Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic 51Similarity Measures for Multiple Attribute Decision-Making 52 Neutrosophic Sets and Systems, Vol. 2, 2014 attribute among all alternatives. Therefore, we define an A1 with respect to an attribute C1, he/she may say ideal SVNV for a benefit attribute in the ideal alternative that the possibility in which the statement is good * A as is 0.4 and the statement is poor is 0.3 and the degree in which he/she is not sure is 0.2. For the **** for jH; jjjj (max,, ij ), (min ij ), fitfita ij )(min i i i neutrosophic notation, it can be expressed as a11 = 0.4, 0.2, 0.3. Thus, when the four possible while for a cost attribute, we define an ideal SVNV in the alternatives with respect to the above three ideal alternative A* as attributes are evaluated by the expert, we can **** for jL. obtain the following single valued neutrosophic jjjj (min,, ij ), (max ij ), fitfita ij )(max i i i decision matrix D: Thus, by applying Eq. (3), the weighted similarity measure between an alternative Ai and the ideal alternative 5.0,2.0,8.03.0,2.0,4.03.0,2.0,4.0 * A are written as 8.0,2.0,5.02.0,1.0,6.02.0,1.0,6.0 . D 8.0,3.0,5.03.0,2.0,5.03.0,2.0,3.0 n tt * ),min * ),( wAAS ij j 8.0,3.0,6.02.0,1.0,6.01.0,0.0,7.0 4 i j * , (4) j1 ij ,max tt j ii * ),min ff * ),min Without loss of generality, let the weight ij j ij j * * values of the three independent elements (i.e., ij ,max ii j ij ,max ff j truth-membership degree, indeterminacy- which provides the global evaluation for each alternative membership degree, and falsity-membership regarding all attributes. According to the weighted similarity measure between each alternative and the ideal degree) in a SVNV be = = = 1/3. Then we * alternative, the bigger the measure value S4(Ai, A ) (i = 1, utilize the developed approach to obtain the most 2, 3, 4), the better the alternative Ai. Hence, the ranking desirable alternative(s). order of all alternatives can be determined and the best one can be easily selected as well. Firstly, from the single valued neutrosophic decision matrix we can yield the following ideal 5 Practical examples alternative: This section provides two practical examples for * CA C C }8.0,3.0,5.0,,2.0,1.0,6.0,,1.0,0.0,7.0,{ . multiple attribute decision-making problems with single 1 2 3 valued neutrosophic information to demonstrate the Then, by using Eq. (4) we can obtain the applications and effectiveness of the proposed decision- values of the weighted similarity measure S4(Ai, making method. * Example 1. Let us consider the decision-making problem A ) (i =1, 2, 3, 4): * * adapted from [7, 8]. There is an investment company, S4(A1, A ) = 0.6595, S4(A2, A ) = 0.9805, which wants to invest a sum of money in the best option. * * S4(A3, A ) = 0.7944, and S4(A4, A ) = 0.9828. There is a panel with four possible alternatives to invest the money: (1) A1 is a car company; (2) A2 is a food Thus, the ranking order of the four company; (3) A3 is a computer company; (4) A4 is an arms alternatives is A4 A2 A3 A1. Therefore, the company. The investment company must take a decision alternative A4 is the best choice among the four according to the three attributes: (1) C1 is the risk; (2) C2 alternatives. is the growth; (3) C3 is the environmental impact, where C1 and C2 are benefit attributes and C3 is a cost attribute. From the above results we can see that the The weight vector of the three attributes is given by w = ranking order of the alternatives and best choice (0.35, 0.25, 0.4)T. The four possible alternatives are to be are in agreement with the results (i.e., the ranking evaluated under the above three attributes by the form of order is A4 A2 A3 A1 and the best choice is SVNVs. A4.) in Ye’s method [8], but not in agreement with the results (i.e., the ranking order is A2 A4 A3 For the evaluation of an alternative Ai (i =1, 2, 3, 4) A1 and the best choice is A2.) in Ye’s method with respect to an attribute Cj (j =1, 2, 3), it is obtained from the questionnaire of a domain expert. For example, [7]. The reason is that different measure methods when we ask the opinion of an expert about an alternative may yield different ranking orders of the alternatives in the decision-making process. Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic52 Similarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 53 Example 2. A multi-criteria decision making problem From the above two examples, we can see adopted from Ye [9] is concerned with a manufacturing that the proposed single valued neutrosophic company which wants to select the best global supplier multiple attribute decision-making method is according to the core competencies of suppliers. Now more suitable for real scientific and engineering suppose that there are a set of four suppliers A = {A1, A2, applications because it can handle not only A3, A4} whose core competencies are evaluated by means incomplete information but also the indeterminate of the four attributes (C1, C2, C3, C4): (1) the level of information and inconsistent information which technology innovation (C1), (2) the control ability of flow exist commonly in real situations. Especially, in (C2), (3) the ability of management (C3), (4) the level of the proposed decision-making method we service (C4), which are all benefit attributes. Then, the consider the important differences in the three weight vector for the four attributes is w = (0.3, 0.25, 0.25, independent elements (i.e., truth-membership 0.2)T. The four possible alternatives are to be evaluated degree, indeterminacy-membership degree, and under the above four attributes by the form of SVNVs. falsity-membership degree) in a SVNV and can For the evaluation of an alternative Ai (i =1, 2, 3, 4) adjust the weight values of the three independent with respect to an attribute Cj (j =1, 2, 3, 4), by the similar elements. Thus, the proposed single valued evaluation method in Example 1 it is obtained from the neutrosophic decision-making method is more flexible and practical than the existing decision- questionnaire of a domain expert. For example, when we making methods [7-9]. The technique proposed in ask the opinion of an expert about an alternative A1 with this paper extends the existing decision-making respect to an attribute C1, he/she may say that the methods and provides a new way for decision- possibility in which the statement is good is 0.5 and the makers. statement is poor is 0.3 and the degree in which he/she is 6 Conclusion not sure is 0.1. For the neutrosophic notation, it can be This paper has developed three similarity expressed as a = 0.5, 0.1, 0.3 . Thus, when the four 11 measures between SVNSs based on the minimum possible alternatives with respect to the above four and maximum operators and investigated their attributes are evaluated by the similar method from the properties. Then the proposed weighted similarity expert, we can establish the following single valued measure of SVNSs has been applied to multiple neutrosophic decision matrix D: attribute decision-making problems under single valued neutrosophic environment. The proposed 1.0,2.0,3.02.0,1.0,7.04.0,1.0,5.03.0,1.0,5.0 . method differs from previous approaches for 2.0,3.0,5.01.0,0.0,9.04.0,2.0,3.03.0,2.0,4.0 D single valued neutrosophic multiple attribute 2.0,2.0,6.04.0,0.0,5.03.0,1.0,5.01.0,3.0,4.0 decision making not only due to the fact that the 1.0,2.0,7.02.0,3.0,4.05.0,2.0,2.02.0,1.0,6.0 proposed method use the weighted similarity measure of SVNSs, but also due to considering Without loss of generality, let the weight values of the the weights of the truth-membership, indeter- three independent elements (i.e., truth-membership minacy-membership, and falsity-membership in degree, indeterminacy-membership degree, and falsity- SVNSs, which makes it have more flexible and membership degree) in a SVNV be = = = 1/3. Then practical than existing decision making methods the proposed decision-making method is applied to solve [7-9] in real decision making problems. Through the weighted similarity measure between each this problem for selecting suppliers. alternative and the ideal alternative, we can obtain From the single valued neutrosophic decision matrix, the ranking order of all alternatives and the best we can yield the following ideal alternative: alternative. Finally, two practical examples demonstrated the applications and effectiveness * CA C ,3.0,1.0,5.0,,1.0,1.0,6.0,{ 1 2 . of the decision-making approach under single C3 C4 }1.0,2.0,7.0,,1.0,0.0,9.0, valued neutrosophic environments. The proposed decision-making method can effectively deal with By applying Eq. (4), the weighted similarity measure decision-making problems with the incomplete, values between an alternative Ai (i = 1, 2, 3, 4) and the indeterminate, and inconsistent information which ideal alternative A* are as follows: exist commonly in real situations. Furthermore, * * S4(A1, A ) = 0.7491, S4(A2, A ) = 0.7433, by the similar method we can easily extend the S (A , A*) = 0.7605, and S (A , A*) = 0.6871. proposed weighted similarity measure of SVNSs 4 3 4 4 and its decision-making method to that of INSs. According to the measure values, the ranking order of In the future, we shall investigate similarity the four suppliers is A3 A1 A2 A4. Hence, the best measures between SVNSs and between INSs in supplier is A3. From the results we can see that the ranking the applications of other domains, such as pattern order of the alternatives and best choice are in agreement recognition, clustering analysis, image process, with the results in Ye‟s method [9]. and medical diagnosis. Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity53 Measures for Multiple Attribute Decision-Making 54 Neutrosophic Sets and Systems, Vol. 2, 2014 ACKNOWLEDGEMENTS [8] J. Ye. Another form of correlation coefficient between single valued neutrosophic sets and its This work was supported by the National Social multiple attribute decision-making method. Science Fund of China (13CGL130), the Guangdong Neutrosophic Sets and Systems, 1(1) (2013), 8-12. Province Natural Science Foundation (S2013010013050), the Humanities and Social Sciences Research Youth [9] J. Ye. Single valued neutrosophic cross-entropy Foundation of Ministry of Education of China for multicriteria decision making problems. (12YJCZH281), and the National Statistical Science Applied Mathematical Modelling, 38 (2014), Research Planning Project (2012LY159). 1170–1175. [10] J. Ye. Similarity measures between interval References neutrosophic sets and their applications in multicriteria decision-making. Journal of [1] L.A. Zadeh. Fuzzy Sets. Information and Control, 8 (1965), Intelligent and Fuzzy Systems, 26 (2014), 165- 338-353. 172. [2] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and [11] P. P. Chi and P.D. Liu. An extended TOPSIS Systems, 20 (1986), 87-96. method for the multiple attribute decision making problems based on interval neutrosophic sets. [3] K. Atanassov and G. Gargov. Interval valued intuitionistic Neutrosophic Sets and Systems, 1(1) (2013), 63- fuzzy sets. Fuzzy Sets and Systems, 31 (1989), 343-349. 70. [4] F. Smarandache. A unifying field in logics. neutrosophy: [12] J. Ye. A multicriteria decision-making method Neutrosophic probability, set and logic. Rehoboth: using aggregation operators for simplified American Research Press, 1999 neutrosophic sets. Journal of Intelligent and Fuzzy Systems, (2013), doi: 10.3233/IFS-130916. [5] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunderraman. Interval neutrosophic sets and logic: Theory [13] P. Majumdar and S.K. Samanta. On similarity and and applications in computing, Hexis, Phoenix, AZ, 2005 entropy of neutrosophic sets. Journal of Intelligent and Fuzzy Systems, (2013), doi: 10.3233/IFS- [6] H. Wang, F. Smarandache, Y. Q. Zhang, and R. 130810. Sunderraman. Single valued neutrosophic sets. Multispace and Multistructure, 4 (2010), 410-413. [14] B. Said and F. Smarandache. Several Similarity Measures of Neutrosophic Sets. Neutrosophic [7] J. Ye. Multicriteria decision-making method using the Sets and Systems, 1 (2013), 54-62. correlation coefficient under single-valued neutrosophic environment. International Journal of General Systems 42(4) (2013), 386-394. Received: February 20th, 2014. Accepted: March 3rd, 2014. Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic54 Si milarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 55 Soft Neutrosophic Bigroup and Soft Neutrosophic N-Group Mumtaz Ali1*, Florentin Smarandache2, Muhammad Shabir3, Munazza Naz4 1,3Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000,Pakistan. E-mail: [email protected], [email protected] 2University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA E-mail: [email protected] 4Department of Mathematical Sciences, Fatima Jinnah Women University, The Mall, Rawalpindi, 46000, Pakistan. E-mail: [email protected] Abstract. Soft neutrosophic group and soft neutrosophic neutrosophic N-group are given. The structural properties subgroup are generalized to soft neutrosophic bigroup and theorems have been discussed with a lot of examples and soft neutrosophic N-group respectively in this paper. to disclose many aspects of this beautiful man made Different kinds of soft neutrosophic bigroup and soft structure. Keywords: Neutrosophic bigroup, Neutrosophic N-group, soft set, soft group, soft subgroup, soft neutrosophic bigroup, soft neu- trosophic subbigroup, soft neutrosophic N-group, soft neutrosophic sub N-group. 1 Introduction duced in 2,9,10 . Some more exciting properties and Neutrosophy is a new branch of philosophy which is in fact the birth stage of neutrosophic logic first found by algebra may be found in . Feng et al. introduced the soft Florentin Smarandache in 1995 . Each proposition in neu- semirings5 . By means of level soft sets an adjustable trosophic logic is approximated to have the percentage of approach to fuzzy soft sets based decision making can be truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F so that seen in6 . Some other new concept combined with fuzzy this neutrosophic logic is called an extension of fuzzy log- sets and rough sets was presented in7,8 . AygÄunoglu ic. In fact neutrosophic set is the generalization of classical sets, conventional fuzzy set 1 , intuitionistic fuzzy set et al. introduced the Fuzzy soft groups 4 . This paper is a mixture of neutrosophic bigroup,neutrosophic N -group 2 and interval valued fuzzy set 3 . This mathematical and soft set theory which is infact a generalization of soft tool is handling problems like imprecise, indeterminacy neutrosophic group. This combination gave birth to a new and inconsistent data etc. By utilizing neutrosophic theory, and fantastic approach called "Soft Neutrosophic Bigroup Vasantha Kandasamy and Florentin Smarandache dig out and Soft Neutrosophic -group". neutrosophic algebraic structures in 11 . Some of them are neutrosophic fields, neutrosophic vector spaces, neu- 2.1 Neutrosophic Bigroup and N-Group trosophic groups, neutrosophic bigroups, neutrosophic N- Definition 1 Let groups, neutrosophic semigroups, neutrosophic bisemi- BGBGBGN 1 2 ,, 1 2 groups, neutrosophic N-semigroup, neutrosophic loops, be a non empty subset with two binary operations neutrosophic biloops, neutrosophic N-loop, neutrosophic on BGsatisfying the following conditions: groupoids, and neutrosophic bigroupoids and so on. N Molodtsov in laid down the stone foundation of a 1) BGBGBGN 12 where BG1 richer structure called soft set theory which is free from the and BG are proper subsets of . parameterization inadequacy, syndrome of fuzzy se theory, 2 rough set theory, probability theory and so on. In many ar- 2) BG , is a neutrosophic group. eas it has been successfully applied such as smoothness of 11 functions, game theory, operations research, Riemann inte- . 3) BG22 , is a group gration, Perron integration, and probability. Recently soft set theory has attained much attention since its appearance Then we define BGN ,,12 to be a neutrosophic and the work based on several operations of soft sets intro- Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 56 Neutrosophic Sets and Systems, Vol. 2, 2014 bigroup. If both BG and BG are neutrosophic 1 2 be a neutrosophic bigroup. If both and groups. We say BGN is a strong neutrosophic are commutative groups, then we call bigroup. If both the groups are not neutrosophic group, we say is just a bigroup. to be a commutative bigroup. Example 1 Let BGBGBGN 12 Definition 4 Let 9 be a neutrosophic where B G1 g/1 g be a cyclic group of order bigroup. If both and are cyclic, we 9 and BGII2 1,2, ,2 neutrosophic group un- der multiplication modulo 3 . We call a neu- call a cyclic bigroup. trosophic bigroup. Definition 5 Let be a neutrosophic Example 2 Let bigroup. PGPGPG 1 2 ,, 1 2 be a neu- Where BG1 1,2,3,4,IIII ,2 ,3 ,4 a neutrosoph trosophic bigroup. ic group under multiplication modulo . 5 is said to be a neutrosophic normal subbigroup of if PG is a neutrosophic subbigroup and BGIIIIII2 0,1,2, ,2,1 ,2 ,1 2,2 2 both PG and PG are normal subgroups of is a neutrosophic group under multiplication modulo 1 2 and respectively. . Clearly is a strong neutrosophic bi Definition 6 Let group. be a neutrosophic Definition 2 Let BGBGBGN 1 2 ,, 1 2 bigroup of finite order. Let be a neutrosophic be a neutrosophic bigroup. A proper subset subbigroup of . If o P G / o BN G PPP1 2,, 1 2 is a neutrosophic subbi then we call a Lagrange neutrosophic sub- group of if the following conditions are bigroup, if every neutrosophic subbigroup P is such that satisfied is a neutroso o P / o BN G then we call to be a La- grange neutrosophic bigroup. phic bigroup under the operations 12, i.e. Definition 7 If has atleast one Lagrange neu- P , is a neutrosophic subgroup of B , 11 11 trosophic subbigroup then we call to be a weak and P22, is a subgroup of B22, . Lagrange neutrosophic bigroup. Definition 8 If has no Lagrange neutrosophic PPB11 and PPB22 are subgroups of B1 and B2 respectively. If both of P1 and P2 subbigroup then is called Lagrange free neutro- sophic bigroup. are not neutrosophic then we call PPP12 to Definition 9 Let be just a bigroup. be a neutrosophic bigroup. Suppose Definition 3 Let PPGPG 1 2 ,, 1 2 and Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 57 free strong neutrosophic bigroup. KKGKG 1 2 ,, 1 2 be any two neutro- Definition 15 Let GI,, be a strong neutro- sophic subbigroups. we say P and K are conjugate if each PGi is conjugate with K Gi , i 1,2 , then sophic bigroup with GIGIGI 12 . we say P and are neutrosophic conjugate sub- Let H,, be a neutrosophic subbigroup where bigroups of BGN . HHH12. We say is a neutrosophic normal Definition 10 A set GI,, with two binary subbigroup of G if both H1 and H2 are neutrosoph- operations ` ' and ` ' is called a strong neutrosophic ic normal subgroups of GI and GI re- bigroup if 1 2 spectively. 1) GIGIGI 12 , Definition 16 Let GGG12 ,, , be a neutro- 2) is a neutrosophic group and GI1 , sophic bigroup. We say two neutrosophic strong sub- bigroups HHH and KKK are conju- 3) GI2 , is a neutrosophic group. 12 12 gate neutrosophic subbigroups of Example 3 Let GI,, 12 be a strong neutro- GIGIGI 12 if H1 is conjugate to sophic bigroup where K and is conjugate to K as neutrosophic sub- GIZI 0,1,2,3,4,IIII ,2 ,3 ,4 . 1 2 groups of GI1 and GI1 respectively. ZI under ` ' is a neutrosophic group and Definition 17 Let GI, 1 ,..., N be a nonempty 0,1,2,3,4,IIII ,2 ,3 ,4 under multiplication modulo set with N -binary operations defined on it. We say 5 is a neutrosophic group. Definition 11 A subset H of a strong neutrosoph- GI is a strong neutrosophic N -group if the follow- ing conditions are true. ic bigroup GI,, is called a strong neutrosoph- 1) GIGIGIGI 12 ... N where ic subbigroup if H itself is a strong neutrosophic bigroup under ` ' and ` ' operations defined on GIi are proper subsets of GI . GI . 2) GIii, is a neutrosophic group, Definition 12 Let GI,, be a strong neutro- iN 1,2,..., . sophic bigroup of finite order. Let H be a strong 3) If in the above definition we have a. GIGGIGIGIG ... ... neutrosophic subbigroup of GI,, . If 1 2 k k1 N o H / o G I then we call H, a Lagrange strong b. Gii, is a group for some i or neutrosophic subbigroup of GI . If every strong 4) GIjj, is a neutrosophic group for some neutrosophic subbigroup of GI is a Lagrange j . Then we call GI to be a neutrosophic N - strong neutrosophic subbigroup then we call GI a group. Lagrange strong neutrosophic bigroup. Example 4 Let GIGIGIGIGI ,,,, Definition 13 If the strong neutrosophic bigroup has at 1 2 3 4 1 2 3 4 least one Lagrange strong neutrosophic subbigroup then be a neutrosophic 4 -group where we call GI a weakly Lagrange strong neutrosophic GI1 1,2,3,4,IIII ,2 ,3 ,4 bigroup. neutrosophic group under multiplication modulo 5 . Definition 14 If GI has no Lagrange strong neu- GIIIIIII2 0,1,2, ,2,1 ,2 ,1 2,2 2 trosophic subbigroup then we call GI a Lagrange a neutrosophic group under multiplication modulo 3 , Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 58 Neutrosophic Sets and Systems, Vol. 2, 2014 GIZI3 , a neutrosophic group under addi- Definition 20 If GI has atleast one Lagrange sub -group then we call a weakly Lagrange neu- tion and G4 I a, b : a , b 1, I ,4,4 I , GI trosophic N-group. component-wise multiplication modulo 5. Definition 21 If GI has no Lagrange sub - Hence GI is a strong neutrosophic 4 -group. group then we call GI to be a Lagrange free - Example 5 Let group. GIGIGIGG ,,,, 1 2 3 4 1 2 3 4 Definition 22 Let be a neutrosophic -group, where GIGIGIGI 1 2 ... NN , 1 ,..., GI1 1,2,3,4,IIII ,2 ,3 ,4 a neutrosophic be a neutrosophic -group. Suppose group under multiplication modulo 5 . HHHH 1 2 ... NN , 1 ,..., and GIII 0,1, ,1 ,a neutrosophic group under 2 KKKK 1 2 ... NN , 1 ,..., are two sub - multiplication modulo 2 . GS33 and GA45 , the groups of GI , we say K is a conjugate alternating group. GI is a neutrosophic 4 -group. to H or H is conjugate to K if each Hi is conjugate to Definition 18 Let Ki iN 1,2,..., as subgroups of Gi . GIGIGIGI 1 2 ... NN , 1 ,..., 2.2 Soft Sets be a neutrosophic N -group. A proper subset Throughout this subsection U refers to an initial uni- P,1 ,..., N is said to be a neutrosophic sub -group verse, E is a set of parameters, PU() is the power set of of GI if PPP ... and each P, U , and AE . Molodtsov defined the soft set in the 1 N ii following manner: is a neutrosophic subgroup (subgroup) of Gii, ,1 i N . Definition 23 A pair (,)FA is called a soft set over U where F is a mapping given by FAPU:(). It is important to note P,i for no i is a neutrosophic group. In other words, a soft set over U is a parameterized fami- Thus we see a strong neutrosophic -group can have 3 ly of subsets of the universe U . For xA , Fx() types of subgroups viz. may be considered as the set of x -elements of the soft set 1) Strong neutrosophic sub -groups. (,)FA , or as the set of e-approximate elements of the 2) Neutrosophic sub N -groups. soft set. 3) Sub -groups. Example 6 Suppose that U is the set of shops. E is the Also a neutrosophic N -group can have two types of sub -groups. set of parameters and each parameter is a word or sentence. 1) Neutrosophic sub N -groups. Let high rent,normal rent, 2) Sub -groups. E . Definition 19 If GI is a neutrosophic -group in good condition,in bad condition Let us consider a soft set (,)FAwhich describes the at- and if GI has a proper subset T such that is a tractiveness of shops that Mr. Z is taking on rent. Suppose neutrosophic sub -group and not a strong neutrosophic that there are five houses in the universe sub -group and o T/ o G I then we call U{,,,,} s1 s 2 s 3 s 4 s 5 under consideration, and that a Lagrange sub N -group. If every sub -group of A{,,} x1 x 2 x 3 be the set of parameters where GI is a Lagrange sub -group then we call x1 stands for the parameter 'high rent, GI a Lagrange -group. x2 stands for the parameter 'normal rent, Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 59 x3 stands for the parameter 'in good condition. We write (,)(,)(,)FAKBHC . Suppose that Definition 28 The restricted union of two soft sets (,)FA and (,)KB over a common universe U is the F(){,} x s s , 1 1 4 soft set (,)HC , where CAB , and for all F(){,} x2 s 2 s 5 , cC , Hc() is defined as H()()() c F c G c for all cC . We write it as F( x ) { s , s , s }. 3 3 4 5 (FAKBHC , ) ( , ) ( , ). The soft set (,)FA is an approximated family R Definition 29 The extended union of two soft sets {F ( ei ), i 1,2,3} of subsets of the set U which gives (,)FA and (,)KB over a common universe U is the us a collection of approximate description of an object. soft set (,)HC , where CAB , and for all Then (,)FA is a soft set as a collection of approxima- cC , Hc() is defined as tions over U , where F(x ) high rent { s , s }, F( c ) if cAB , 1 1 2 H( c ) G ( c ) if cBA , F( x2 ) normal rent { s2 , s 5 }, F( c ) G ( c ) if cAB . F( x3 ) in good condition { s3 , s 4 , s 5 }. We write (,)(,)(,)FAKBHC . 2.3 Soft Groups Definition 24 For two soft sets (,)FA and (,)HB over U , (,)FA is called a soft subset of (,)HB if Definition 30 Let (,)FA be a soft set over G . Then 1. AB and (,)FA is said to be a soft group over G if and only if 2. F()() x H x , for all xA . F() x G for all xA . This relationship is denoted by (,)(,)FAHB . Simi- Example 7 Suppose that larly (,)FA is called a soft superset of (,)HB if G A S{ e ,(12),(13),(23),(123),(132)}. (,)HB is a soft subset of (,)FA which is denoted by 3 Then is a soft group over where (,)(,)FAHB . (,)FA S3 Definition 25 Two soft sets (,)FA and (,)HB over F( e ) { e }, U are called soft equal if (,)FA is a soft subset of Fe(12) { ,(12)}, (,)HB and (,)HB is a soft subset of (,)FA. Fe(13) { ,(13)}, Definition 26 Let (,)FA and (,)KB be two soft sets Fe(23) { ,(23)}, over a common universe U such that AB . F(123) F (132) { e ,(123),(132)}. Then their restricted intersection is denoted by (,)(,)(,)FAKBHC where (,)HC is de- R Definition 31 Let (,)FA be a soft group overG . Then fined as H()()) c F c c for all 1. (,)FA is said to be an identity soft group c C A B . over G if F(){} x e for all xA, Definition 27 The extended intersection of two soft sets (,)FA and (,)KB over a common universe U is the where e is the identity element of G and 2. (,)FA is said to be an absolute soft group if soft set (,)HC , where CAB , and for all F() x G for all xA . cC , Hc() is defined as F( c ) if cAB , 3 Soft Neutrosophic Bigroup Definition 32 Let H( c ) G ( c ) if cBA , BGBGBG ,, F( c ) G ( c ) if cAB . N 1 2 1 2 Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 60 Neutrosophic Sets and Systems, Vol. 2, 2014 be a neutrosophic bigroup and let FA, be a soft set sophic bigroups FA, and KD, over BGN over BGN . Then FA, is said to be soft neutro- is not a soft neutrosophic bigroup over BGN . To prove it, see the following example. sophic bigroup over BGN if and only if Fx is a Example 9 Let BGBGBGN 1 2 ,, 1 2 , subbigroup of BGN for all xA . Example 8 Let where BGIIII1 1,2,3,4 ,2 ,3 ,4 and BGBGBGN 1 2 ,, 1 2 BGS23 . be a neutrosophic bigroup, where Let PGPGPG 1 2 ,, 1 2 be a neutro- BG1 0,1,2,3,4,IIII ,2 ,3 ,4 sophic subbigroup where PGII 1,4, ,4 and is a neutrosophic group under multiplication modulo 5 . 1 12 P G e, 12 . B G2 g:1 g is a cyclic group of order 12. 2 Also QGQGQG ,, be another Let PGPGPG 1 2 ,, 1 2 be a neutro- 1 2 1 2 neutrosophic subbigroup where QGI 1, and sophic subbigroup where PGII1 1,4, ,4 and 1 2 4 6 8 10 Q G e, 123 , 132 . P G2 1, g , g , g , g , g . 2 Then FA, is a soft neutrosophic bigroup over Also QGQGQG 1 2 ,, 1 2 be another BGN , where neutrosophic subbigroup where QGI1 1, and 3 6 9 F x1 1,4, I ,4 I , e , 12 Q G2 1, g , g , g . Then FA, is a soft neutrosophic bigroup over F x2 1, I , e , 123 , 132 . BGN , where Again let RGRGRG 1 2 ,, 1 2 be anoth- 2 4 6 8 10 er neutrosophic subbigroup where RGII 1,4, ,4 F e1 1,4,,4,1, I I g , g , g , g , g 1 3 6 9 and R G2 e, 13 . F e2 1, I ,1, g , g , g . Also TGTGTG 1 2 ,, 1 2 be a neutro- Theorem 1 Let FA, and HA, be two soft sophic subbigroup where TGI1 1, and neutrosophic bigroup over BGN . Then their intersec- T G2 e, 23 . tion FAHA,, is again a soft neutrosophic Then KD, is a soft neutrosophic bigroup over bigroup over BGN . BGN , where Proof Straight forward. K x2 1,4, I ,4 I , e ,13 , Theorem 2 Let FA, and HB, be two soft K x3 1, I , e , 23 . neutrosophic bigroups over BGN such that The extended union FAKDHC,,, such AB , then their union is soft neutrosophic bigroup that CAD and for xC2 , we have over BGN . Proof Straight forward. H x2 F x2 K x2 1,4, I ,4 I , e , 13 123 , 132 Proposition 1 The extended union of two soft neutro- is not a subbigroup of BGN . Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 61 Proposition 2 The extended intersection of two soft neu- BGBGBG 1 2 ,, 1 2 where trosophic bigroups FA, and KD, over 0,1,2,3,4,IIIIIIII ,2,3,4,1 ,2 ,3 ,4 , BGN is again a soft neutrosophic bigroup over BGIIIIII1 1 2,2 2,3 2,4 2,1 3,2 3, BG . N 3 3,4IIIIII 3,1 4,2 4,3 4,4 4 Proposition 3 The restricted union of two soft neutro- be a neutrosophic group under multiplication modulo 5 sophic bigroups FA, and KD, over BGN is 16 and B G2 g:1 g a cyclic group of order not a soft neutrosophic bigroup over BGN . 16 . Let PGPGPG 1 2 ,, 1 2 be a neu- Proposition 4 The restricted intersection of two soft trosophic subbigroup where neutrosophic bigroups FA, and KD, over PG1 0,1,2,3,4,IIII ,2 ,3 ,4 and BG is a soft neutrosophic bigroup over BG. N N be another neutrosophic subbigroup where P G g2, g 4 , g 6 , g 8 , g 10 , g 12 , g 14 ,1 . Proposition 5 The AND operation of two soft neu- 2 Also trosophic bigroups over BGN is again soft neutro- QGQGQG 1 2 ,, 1 2 sophic bigroup over BGN . QGII1 0,1,4, ,4 and Proposition 6 The OR operation of two soft neutro- 4 8 12 Q G2 g, g , g ,1 . sophic bigroups over BG may not be a soft nuetro- N Again let RGRGRG ,, be a neu- sophic bigroup. 1 2 1 2 trosophic subbigroup where Definition 33 Let FA, be a soft neutrosophic 8 RGI1 0,1, and R G2 1, g . bigroup over BGN . Then Let FA, be a soft neutrsophic bigroup over BG 1) FA, is called identity soft neutrosophic bigroup N where if F x e12, e for all xA , where e1 and 2 4 6 8 10 12 14 F x1 0,1,2,3,4,,2,3,4,IIIIggggggg , , , , , , ,1, e are the identities of BG and BG re- 2 1 2 4 8 12 spectively. F x2 0,1,4, I ,4, I g , g , g ,1, 2) FA, is called Full-soft neutrosophic bigroup if 8 F x3 0,1, I , g ,1 . F x BN G for all xA . Theorem 3 Let BGN be a neutrosophic bigroup of Let HK, be another soft neutrosophic bigroup over prime order P , then FA, over BGN is either BGN , where identity soft neutrosophic bigroup or Full-soft neutrosoph- H x 0,1,2,3,4,g4 , g 8 , g 12 ,1 , ic bigroup. 1 Definition 34 Let FA, and HK, be two soft 8 H x2 0,1, I , g ,1 . neutrosophic bigroups over . Then is BGN HK, Clearly HKFA,,. soft neutrosophi subbigroup of FA, written as Definition 35 Let BGN be a neutrosophic bigroup. HKFA,, , if Then FA, over BGN is called commutative soft 1) KA , neutrosophic bigroup if and only if Fx is a commuta- 2) K x F x for all xA . tive subbigroup of BG for all xA . Example 10 Let N Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 62 Neutrosophic Sets and Systems, Vol. 2, 2014 over . Example 11 Let BGBGBG 1 2 ,, 1 2 BGN 10 3) Their restricted union FAKD,, over be a neutrosophic bigroup where B G1 g:1 g R be a cyclic group of order 10 and BGN is not commutative soft neutrosophic BG 1,2,3,4,III ,2I,3 ,4 be a neutrosophic 2 bigroup over BGN . group under mltiplication modulo 5 . 4) Their restricted intersection FAKD,, R Let PGPGPG 1 2 ,, 1 2 be a commuta- over BGN is commutative soft neutrosophic 5 tive neutrosophic subbigroup where P G1 1, g bigroup over BGN . and PGII 1,4, ,4 . Also 2 Proposition 8 Let FA, and KD, be two com- QGQGQG 1 2 ,, 1 2 be another commu- mutative soft neutrosophic bigroups over BGN . Then tative neutrosophic subbigroup where 1) Their operation is com- 2 4 6 8 AND FAKD,, Q G1 1, g , g , g , g and QGI2 1, . mutative soft neutrosophic bigroup over BGN . Then FA, is commutative soft neutrosophic bigroup 2) Their OR operation FAKD,, is not com- over BGN , where mutative soft neutrosophic bigroup over BGN . 5 F x1 1, g ,1,4, I ,4 I , Definition 36 Let BGN be a neutrosophic bigroup. F x 1, g2 , g 4 , g 6 , g 8 ,1, I . 2 Then FA, over BGN is called cyclic soft neutro- Theorem 4 Every commutative soft neutrosophic sophic bigroup if and only if Fx is a cyclic sub- bigroup FA, over BGN is a soft neutrosophic bigroup of BG for all xA . bigroup but the converse is not true. N Example 12 Let BGBGBG 1 2 ,, 1 2 Theorem 5 If is commutative neutrosophic BGN 10 be a neutrosophic bigroup where B G1 g:1 g bigroup. Then FA, over BG is commutative N be a cyclic group of order 10 and soft neutrosophic bigroup but the converse is not true. BGIIIIII2 0,1,2, ,2,1 ,2 ,1 2,2 2 Theorem 6 If BGN is cyclic neutrosophic bigroup. be a neutrosophic group under multiplication modulo 3 . Le PGPGPG 1 2 ,, 1 2 be a cyclic neu- Then FA, over BGN is commutative soft neu- trosophic bigroup. 5 trosophic subbigroup where P G1 1, g and Proposition 7 Let FA, and KD, be two commu- 1,1 I . Also QGQGQG ,, be another tative soft neutrosophic bigroups over BGN . Then 1 2 1 2 cyclic neutrosophic subbigroup where 1) Their extended union FAKD,, over 2 4 6 8 Q G1 1, g , g , g , g and BGN is not commutative soft neutrosophic QGI2 1,2 2 . bigroup over BGN . Then FA, is cyclic soft neutrosophic bigroup over 2) Their extended intersection FAKD,, over BGN , where BGN is commutative soft neutrosophic bigroup Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 63 5 Example 13 Let BGBGBG ,, F x1 1, g ,1,1 I , 1 2 1 2 2 4 6 8 be a neutrosophic bigroup, where F x 1, g , g , g , g ,1,2 2 I . 22 2 e,,,,,,, y x x xy x y I BG Theorem 7 If BG is a cyclic neutrosophic soft 1 22 N Iy,,,, Ix Ix Ixy Ix y bigroup, then FA, over BGN is also cyclic soft is a neutrosophic group under multiplaction and neutrosophic bigroup. 6 B G2 g:1 g is a cyclic group of order 6 . Theorem 8 Every cyclic soft neutrosophic bigroup Let PGPGPG 1 2 ,, 1 2 be a normal FA, over BG is a soft neutrosophic bigroup but N neutrosophic subbigroup where P G e, y and the converse is not true. 1 24 P G2 1, g , g . Proposition 9 Let FA, and KD, be two cyclic Also QGQGQG 1 2 ,, 1 2 be another soft neutrosophic bigroups over BGN . Then normal neutrosophic subbigroup where 1) Their extended union over 2 3 FAKD,, Q G1 e,, x x and Q G2 1, g . BGN is not cyclic soft neutrosophic bigroup over Then FA, is a normal soft neutrosophic bigroup over BG. N BGN where 24 2) Their extended intersection FAKD,, over F x1 e, y ,1, g , g , is cyclic soft neutrosophic bigroup over BGN 23 F x2 e, x , x ,1, g . BG . N Theorem 9 Every normal soft neutrosophic bigroup 3) Their restricted union FAKD,, R over FA, over BGN is a soft neutrosophic bigroup but BGN is not cyclic soft neutrosophic bigroup over the converse is not true. BGN . Theorem 10 If BGN is a normal neutrosophic 4) Their restricted intersection FAKD,, R bigroup. Then FA, over BGN is also normal soft over BGN is cyclic soft neutrosophic bigroup neutrosophic bigroup. over BGN . Theorem 11 If BGN is a commutative neutrosophic Proposition 10 Let FA, and KD, be two cyclic bigroup. Then FA, over BGN is normal soft neu- soft neutrosophic bigroups over BG. Then N trosophic bigroup. 1) Their AND operation FAKD,, is cyclic Theorem 12 If BGN is a cyclic neutrosophic soft neutrosophic bigroup over BGN . bigroup. Then FA, over BG is normal soft neu- 2) Their OR operation FAKD,, is not cyclic N trosophic bigroup. soft neutrosophic bigroup over BG. N Proposition 11 Let FA, and KD, be two nor- Definition 37 Let BGN be a neutrosophic bigroup. mal soft neutrosophic bigroups over BGN . Then Then FA, over BGN is called normal soft neutro- 1) Their extended union FAKD,, over sophic bigroup if and only if Fx is normal subbigroup BGN is not normal soft neutrosophic bigroup over of BGN for all xA . Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 64 Neutrosophic Sets and Systems, Vol. 2, 2014 BGN . QGI2 0,1 . 2) Their extended intersection FAKD,, over Then FA, is Lagrange soft neutrosophic bigroup over BGN is normal soft neutrosophic bigroup over BGN , where BGN . F x2 e, y ,0,1 , 3) Their restricted union FAKD,, R over F x2 e, yI ,0,1 I . BGN is not normal soft neutrosophic bigroup over Theorem 13 If BGN is a Lagrange neutrosophic BGN . bigroup, then FA, over BGN is Lagrange soft neu- trosophic bigroup. 4) Their restricted intersection FAKD,, R over BGN is normal soft neutrosophic bigroup Theorem 14 Every Lagrange soft neutrosophic bigroup FA, over BGN is a soft neutrosophic bigroup but over BGN . the converse is not true. Proposition 12 Let FA, and KD, be two nor- Proposition 13 Let FA, and KD, be two La- mal soft neutrosophic bigroups over BGN . Then 1) Their AND operation FAKD,, is normal grange soft neutrosophic bigroups over BGN . Then 1) Their extended union FAKD,, over soft neutrosophic bigroup over BGN . 2) Their OR operation FAKD,, is not nor- BGN is not Lagrange soft neutrosophic bigroup over BG. mal soft neutrosophic bigroup over BGN . N Definition 38 Let FA, be a soft neutrosophic bigroup 2) Their extended intersection FAKD,, over BG is not Lagrange soft neutrosophic bigroup over BGN . If for all xA each Fx is a La- N over BG. grange subbigroup of BGN , then FA, is called N 3) Their restricted union FAKD,, over Lagrange soft neutosophic bigroup over BGN . R BGN is not Lagrange soft neutrosophic bigroup Example 14 Let BGBGBG 1 2 ,, 1 2 be a neutrosophic bigroup, where over BGN . 22 e,,,,,,, y x x xy x y I 4) Their restricted intersection FAKD,, R BG1 22 over BG is not Lagrange soft neutrosophic Iy,,,, Ix Ix Ixy Ix y N is a neutrosophic symmetric group of and bigroup over BGN . BGII 0,1, ,1 be a neutrosophic group under ad- 2 Proposition 14 Let FA, and KD, be two La- dition modulo 2 . Let grange soft neutrosophic bigroups over BGN . Then PGPGPG 1 2 ,, 1 2 be a neutrosophic 1) Their operation FAKD,, is not La- subbigroup where P G1 e, y and grange soft neutrosophic bigroup over BGN . PG2 0,1 . 2) Their operation FAKD,, is not La- Also QGQGQG 1 2 ,, 1 2 be another grange soft neutrosophic bigroup over BGN . neutrosophic subbigroup where Q G1 e, Iy and Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 65 Definition 39 Let FA, be a soft neutrosophic 3) Their restricted union FAKD,, R over bigroup over BGN . Then FA, is called weakly BGN is not weakly Lagrange soft neutrosophic Lagrange soft neutosophic bigroup over BGN if at- bigroup over BGN . least one Fx is a Lagrange subbigroup of BGN , 4) Their restricted intersection FAKD,, R for some xA . over BGN is not weakly Lagrange soft neutrosoph- Example 15 Let BGBGBG 1 2 ,, 1 2 ic bigroup over BGN . be a neutrosophic bigroup, where Proposition 16 Let FA, and KD, be two weak- 0,1,2,3,4,IIIIIIII ,2,3,4,1 ,2 ,3 ,4 , ly Lagrange soft neutrosophic bigroups over BGN . BGIIIIII1 1 2,2 2,3 2,4 2,1 3,2 3, Then 3 3,4IIIIII 3,1 4,2 4,3 4,4 4 1) Their AND operation FAKD,, is not is a neutrosophic group under multiplication modulo 5 weakly Lagrange soft neutrosophic bigroup over 10 and B G2 g:1 g is a cyclic group of order BGN . . Let PGPGPG ,, be a neu- 10 1 2 1 2 2) Their OR operation FAKD,, is not weakly trosophic subbigroup where PGII 0,1,4, ,4 1 Lagrange soft neutrosophic bigroup over BGN . 2 4 6 8 and P G2 g, g , g , g ,1 . Also Definition 40 Let FA, be a soft neutrosophic QGQGQG 1 2 ,, 1 2 be another neutro- bigroup over BGN . Then FA, is called Lagrange free soft neutrosophic bigroup if each is not La- sophic subbigroup where QGII1 0,1,4, ,4 and Fx 5 grange subbigroup of BG, for all xA . Q G2 g ,1 .Then FA, is a weakly Lagrange N Example 16 Let BGBGBG ,, soft neutrosophic bigroup over BGN , where 1 2 1 2 2 4 6 8 be a neutrosophic bigroup, where F x1 0,1,4,,4, I I g , g , g , g ,1, BGII1 0,1, ,1 is a neutrosophic group under 5 F x2 0,1,4, I ,4 I , g ,1. addition modulo 2 of order 4 and Theorem 15 Every weakly Lagrange soft neutrosophic 12 B G2 g:1 g is a cyclic group of order 12. bigroup FA, over BG is a soft neutrosophic N Let PGPGPG ,, be a neutro- bigroup but the converse is not true. 1 2 1 2 sophic subbigroup where PGI1 0, and Proposition 15 Let FA, and KD, be two weak- 48 P G2 g, g ,1 . Also ly Lagrange soft neutrosophic bigroups over BGN . QGQGQG ,, be another neutro- Then 1 2 1 2 1) Their extended union FAKD,, over sophic subbigroup where QGI1 0,1 and 3 6 9 BGN is not weakly Lagrange soft neutrosophic Q G2 1, g , g , g . Then FA, is Lagrange bigroup over BGN . free soft neutrosophic bigroup over BGN , where 2) Their extended intersection FAKD,, over 48 F x1 0, I ,1, g , g , BGN is not weakly Lagrange soft neutrosophic F x0,1 I ,1, g3 , g 6 , g 9 . bigroup over BGN . 2 Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 66 Neutrosophic Sets and Systems, Vol. 2, 2014 Theorem 16 If is Lagrange free neutrosophic BGN Example 17 Let BGBGBG 1 2 ,, 1 2 be a soft neutrosophic bigroup, where bigroup, and then FA, over BGN is Lagrange free 22 soft neutrosophic bigroup. B G1 e,,,,, y x x xy x y is Klien 4 -group and Theorem 17 Every Lagrange free soft neutrosophic 0,1,2,3,4,5,IIIII ,2 ,3 ,4 ,5 , bigroup FA, over BG is a soft neutrosophic N BG2 bigroup but the converse is not true. 1IIII ,2 ,3 ,...,5 5 Proposition 17 Let FA, and KD, be two La- be a neutrosophic group under addition modulo 6 . Let PGPGPG 1 2 ,, 1 2 be a neutrosoph- grange free soft neutrosophic bigroups over BGN . Then ic subbigroup of BGN , where P G1 e, y and 1) Their extended union FAKD,, over PGII2 0,3,3 ,3 3 . Again BGN is not Lagrange free soft neutrosophic letQGQGQG 1 2 ,, 1 2be another neu- bigroup over BGN . trosophic subbigroup of BGN , where 2) Their extended intersection FAKD,, 2 Q G1 e,, x x and over BGN is not Lagrange free soft neutro- QG2 0,2,4,2 2,4IIII 4,2,4 . Then sophic bigroup over BG . N FA, is conjugate soft neutrosophic bigroup over 3) Their restricted union FAKD,, R over BGN , where BGN is not Lagrange free soft neutrosophic F x1 e, y ,0,3,3,3 I 3 I , bigroup over BGN . 2 F x2 e, x , x ,0,2,4,2 2,4IIII 4,2,4 . 4) Their restricted intersection FAKD,, R Theorem 18 If BGN is conjugate neutrosophic over BGN is not Lagrange free soft neutro- bigroup, then FA, over BGN is conjugate soft neu- sophic bigroup over BGN . trosophic bigroup. Proposition 18 Let FA, and KD, be two La- Theorem 19 Every conjugate soft neutrosophic bigroup FA, over BG is a soft neutrosophic bigroup but grange free soft neutrosophic bigroups over BG . N N the converse is not true. Then Proposition 19 Let FA, and KD, be two conju- 1) Their AND operation FAKD,, is not La- gate soft neutrosophic bigroups over BGN . Then grange free soft neutrosophic bigroup over BGN . 1) Their extended union FAKD,, over 2) Their OR operation FAKD,, is not La- BGN is not conjugate soft neutrosophic bigroup grange free soft neutrosophic bigroup over BGN . over BGN . Definition 41 Let BGN be a neutrosophic bigroup. 2) Their extended intersection FAKD,, over Then FA, is called conjugate soft neutrosophic BGN is conjugate soft neutrosophic bigroup over bigroup over BGN if and only if Fx is neutrosophic BGN . conjugate subbigroup of BGN for all xA . Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 67 3) Their restricted union FAKD,, R over F x1 0, 2, 4,...,1,4,II ,4 , BGN is not conjugate soft neutrosophic bigroup F x2 0, 3, 6,...,1,IIII ,2 ,3 ,4 . over BG . N Theorem 20 Every soft strong neutrosophic bigroup 4) Their restricted intersection FA, is a soft neutrosophic bigroup but the converse is FAKD,, over BG is conjgate soft R N not true. neutrosophic bigroup over BGN . Theorem 21 If GI,, 12 is a strong neutro- Proposition 20 Let FA, and KD, be two conju- sophic bigroup, then FA, over GI,, 12 is gate soft neutrosophic bigroups over BGN . Then soft strong neutrosophic bigroup. 1) Their AND operation FAKD,, is conju- Proposition 21 Let FA, and KD, be two soft gate soft neutrosophic bigroup over BGN . strong neutrosophic bigroups over GI,, 12 . 2) Their OR operation FAKD,, is not conju- Then 1) Their extended union FAKD,, over gate soft neutrosophic bigroup over BGN . GI,, 12 is not soft strong neutrosophic 3.3 Soft Strong Neutrosophic Bigroup bigroup over GI,,. 12 Definition 42 Let GI,, 12 be a strong neutro- 2) Their extended intersection FAKD,, over sophic bigroup. Then FA, over GI,, 12 is GI,, 12 is soft strong neutrosophic bigroup called soft strong neutrosophic bigroup if and only if over GI,,. Fx is a strong neutrosophic subbigroup of 12 3) Their restricted union FAKD,, R over GI,, 12 for all xA . GI,, 12 is not soft strong neutrosophic Example 18 Let GI,, 12 be a strong neutro- bigroup over GI,,. 12 sophic bigroup, where GIGIGI 12 4) Their restricted intersection FAKD,, R with GIZI1 , the neutrosophic group un- over GI,, 12 is soft strong neutrosophic der addition and GI2 0,1,2,3,4,IIII ,2 ,3 ,4 a neutrosophic group under multiplication modulo Let 5. bigroup over GI,,. 12 HHHbe a strong neutrosophic subbigroup of 12 Proposition 22 Let FA, and KD, be two soft GI,,, 12 where HZI1 2, is a strong neutrosophic bigroups over GI,, 12 . neutrosophic subgroup and HII2 0,1,4, ,4 is a neu- Then trosophic subgroup. Again let KKK12be another 1) Their AND operation FAKD,, is soft strong neutrosophic subbigroup of GI,,, 12 strong neutrosophic bigroup over GI,,. 12 where KZI1 3, is a neutrosophic subgroup 2) Their OR operation FAKD,, is not soft and KIIII2 0,1, ,2 ,3 ,4 is a neutrosophic subgroup. strong neutrosophic bigroup over GI,,. 12 Then clearly FA, is a soft strong neutrosophic bigroup Definition 43 Let GI,, 12 be a strong neutro- over GI,,, 12 where sophic bigroup. Then FA, over GI,, 12 is Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 68 Neutrosophic Sets and Systems, Vol. 2, 2014 called Lagrange soft strong neutrosophic bigroup if and GI,,. 12 Then only if Fx is Lagrange subbigroup of 1) Their extended union FAKD,, over GI,, 12 for all xA . GI,, 12 is not Lagrange soft strong neu- Example 19 Let GI,, 12 be a strong neutro- trosophic bigroup over GI,,. 12 sophic bigroup of order 15, where 2) Their extended intersection FAKD,, over GIGIGI 12 with GI,, is not Lagrange soft strong neutro- GI1 0,1,2,1 IIIIII ,,2,2 ,2 2,1 2 , 12 the neutrosophic group under mltiplication modulo 3 and sophic bigroup over GI,,. 12 22. G23 I A I e,,,,, x x I xI x I Let 3) Their restricted union FAKD,, R over HHH12 be a strong neutrosophic subbigroup of GI,, 12 is not Lagrange soft strong neutro- GI,,, 12 where HI1 1,2 2 is a neutro- sophic bigroup over GI,,. 12 2 sophic subgroup and H2 e,, x x is a neutrosophic 4) Their restricted intersection FAKD,, R subgroup. Again let KKK12be another strong neu- over GI,, 12 is not Lagrange soft strong trosophic subbigroup of GI,,, 12 where neutrosophic bigroup over GI,,. 12 KI1 1,1 is a neutrosophic subgroup and Proposition 24 Let FA, and KD, be two La- K I,, xI x2 I is a neutrosophic subgroup. Then 2 grange soft strong neutrosophic bigroups over clearly is Lagrange soft strong neutrosophic FA, GI,,. 12 Then bigroup over GI,,, 12 where 1) Their AND operation FAKD,, is not La- 2 grange soft strong neutrosophic bigroup over F x1 1,2 2 I , e , x , x , GI,,. 12 2 F x2 1,1 I , I , xI , x I . 2) Their OR operation FAKD,, is not La- Theorem 22 Every Lagrange soft strong neutrosophic grange soft strong neutrosophic bigroup over bigroup FA, is a soft neutrosophic bigroup but the GI,,. converse is not true. 12 Definition 44 Let GI,, 12 be a strong neutro- Theorem 23 Every Lagrange soft strong neutrosophic bigroup FA, is a soft strong neutrosophic bigroup but sophic bigroup. Then FA, over GI,, 12 is the converse is not true. called weakly Lagrange soft strong neutrosophic bigroup if atleast one Fx is a Lagrange subbigroup of Theorem 24 If GI,, 12 is a Lagrange strong GI,, 12 for some xA . neutrosophic bigroup, then FA, over Example 20 Let GI,, 12 be a strong neutro- GI,, is a Lagrange soft strong neutrosophic 12 sophic bigroup of order , where soft bigroup. GIGIGI 12 with Proposition 23 Let FA, and KD, be two La- GI 0,1,2,1 IIIIII ,,2,2 ,2 2,1 2 , grange soft strong neutrosophic bigroups over 1 the neutrosophic under mltiplication modulo 3 and Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 69 22. 4) Their restricted intersection FAKD,, G2 I e,,,,, x x I xI x I Let R HHH12 be a strong neutrosophic subbigroup of over GI,, 12 is not weakly Lagrange soft GI,,, where HII 1,2, ,2 is a neu- 12 1 strong neutrosophic bigroup over GI,,. 12 2 trosophic subgroup and H2 e,, x x is a neutrosoph- Proposition 26 Let FA, and KD, be two weak- ly Lagrange soft strong neutrosophic bigroups over ic subgroup. Again let KKK12 be another strong GI,, 12 . Then neutrosophic subbigroup of GI,, 12 , where 1) Their AND operation FAKD,, is not KI1 1,1 is a neutrosophic subgroup and weakly Lagrange soft strong neutrosophic bigroup 2 K2 e,,, I xI x I is a neutrosophic subgroup. over GI,,. 12 . Then clearly FA, is weakly Lagrange soft strong neu- 2) Their OR operation FAKD,, is not weakly trosophic bigroup over GI,,, 12 where Lagrange soft strong neutrosophic bigroup over 2 GI,,. 12 F x1 1,2, I ,2 I , e , x , x , Definition 45 Let GI,, be a strong neutro- 2 12 F x2 1,1 I , e , I , xI , x I . sophic bigroup. Then FA, over GI,, 12 is Theorem 25 Every weakly Lagrange soft strong neutro- called Lagrange free soft strong neutrosophic bigroup if sophic bigroup FA, is a soft neutrosophic bigroup but and only if Fx is not Lagrange subbigroup of the converse is not true. GI,, 12 for all xA . Theorem 26 Every weakly Lagrange soft strong neutro- Example 21 Let GI,, 12 be a strong neutro- sophic bigroup FA, is a soft strong neutrosophic sophic bigroup of order 15, where bigroup but the converse is not true. GIGIGI 12 with GI0,1,2,3,4,IIII ,2 ,3 ,4 , the neutrosoph- Proposition 25 Let FA, and KD, be two weak- 1 ly Lagrange soft strong neutrosophic bigroups over ic under mltiplication modulo 5 and G I e,,,,,, x x22 I xI x I a neutrosophic sym- GI,, 12 . Then 2 metric group. Let HHH be a strong neutro- 1) Their extended union FAKD,, over 12 sophic subbigroup of GI,,, where GI,, 12 is not weakly Lagrange soft 12 HII1 1,4, ,4 is a neutrosophic subgroup and strong neutrosophic bigroup over GI,,. 12 2 . H2 e,, x x is a neutrosophic subgroup. Again let 2) Their extended intersection FAKD,, over KKK12 be another strong neutrosophic sub- GI,, is not weakly Lagrange soft strong 12 bigroup of GI,,, 12 where neutrosophic bigroup over GI,,. 12 KIIII1 1, ,2 ,3 ,4 is a neutrosophic subgroup and 3) Their restricted union FAKD,, over 2 R K2 e,, x x is a neutrosophic subgroup. GI,, 12 is not weakly Lagrange soft strong Then clearly FA, is Lagrange free soft strong neutro- neutrosophic bigroup over GI,,. . 12 sophic bigroup over GI,,, 12 where Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 70 Neutrosophic Sets and Systems, Vol. 2, 2014 2 1) Their AND operation FAKD,, is not La- F x1 1,4, I ,4 I , e , x , x , grange free soft strong neutrosophic bigroup over F x 1,,2,3,4,,, I I I I e x x2 . 2 GI,,. 12 2) Their OR operation FAKD,, is not La- Theorem 27 Every Lagrange free soft strong neutro- grange free soft strong neutrosophic bigroup over sophic bigroup FA, is a soft neutrosophic bigroup but the converse is not true. GI,,. 12 Definition 46 Let GI,, be a strong neutro- Theorem 28 Every Lagrange free soft strong neutrosoph- 12 ic bigroup FA, is a soft strong neutrosophic bigroup sophic bigroup. Then FA, over GI,, 12 is but the converse is not true. called soft normal strong neutrosophic bigroup if and only if Fx is normal strong neutrosophic subbigroup of Theorem 29 If GI,, 12 is a Lagrange free GI,, 12 for all xA . strong neutrosophic bigroup, then FA, over Theorem 30 Every soft normal strong neutrosophic GI,, is also Lagrange free soft strong neu- 12 bigroup FA, over GI,, 12 is a soft neutro- trosophic bigroup. sophic bigroup but the converse is not true. Proposition 27 Let FA, and KD, be weakly Theorem 31 Every soft normal strong neutrosophic Lagrange free soft strong neutrosophic bigroups over bigroup FA, over GI,, 12 is a soft strong neutrosophic bigroup but the converse is not true. GI,,. 12 Then Proposition 29 Let FA, and KD, be two soft 1) Their extended union FAKD,, over normal strong neutrosophic bigroups over GI,, 12 is not Lagrange free soft strong GI,, 12 . Then neutrosophic bigroup over GI,,. 12 1) Their extended union FAKD,, over 2) Their extended intersection FAKD,, GI,, 12 is not soft normal strong neutro- over GI,, 12 is not Lagrange free soft sophic bigroup over GI,,. 12 strong neutrosophic bigroup over GI,,. 12 2) Their extended intersection FAKD,, 3) Their restricted union FAKD,, R over over GI,, 12 is soft normal strong neutro- GI,, 12 is not Lagrange free soft strong sophic bigroup over GI,,. 12 neutrosophic bigroup over GI,,. 12 3) Their restricted union FAKD,, R over 4) Their restricted intersection FAKD,, R GI,, 12 is not soft normal strong neutro- over GI,, 12 is not Lagrange free soft sophic bigroup over GI,,. 12 strong neutrosophic bigroup over 4) Their restricted intersection FAKD,, R GI,,. 12 over GI,, 12 is soft normal strong neutro- Proposition 28 Let FA, and KD, be two La- grange free soft strong neutrosophic bigroups over sophic bigroup over GI,,. 12 GI,,. 12 Then Proposition 30 Let FA, and KD, be two soft Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 71 normal strong neutrosophic bigroups over Proposition 32 Let FA, and KD, be two soft GI,,. 12 Then conjugate strong neutrosophic bigroups over 1) Their AND operation FAKD,, is soft GI,,. 12 Then normal strong neutrosophic bigroup over 1) Their operation FAKD,, is soft GI,,. 12 conjugate strong neutrosophic bigroup over 2) Their OR operation FAKD,, is not soft GI,,. 12 normal strong neutrosophic bigroup over 2) Their operation FAKD,, is not soft GI,,. 12 conjgate strong neutrosophic bigroup over GI,,. Definition 47 Let GI,, 12 be a strong neutro- 12 sophic bigroup. Then FA, over GI,, is 12 4.1 Soft Neutrosophic N-Group called soft conjugate strong neutrosophic bigroup if and Definition 48 Let GI, 1 ,..., N be a neutro- only if Fx is conjugate neutrosophic subbigroup of sophic N -group. Then FA, over GI,, 12 for all xA . GI, ,..., is called soft neutrosophic N - Theorem 32 Every soft conjugate strong neutrosophic 12 group if and only if is a sub -group of bigroup FA, over GI,, 12 is a soft neutro- Fx N sophic bigroup but the converse is not true. GI, 12 ,..., for all xA . Theorem 33 Every soft conjugate strong neutrosophic Example 22 Let bigroup FA, over GI,, 12 is a soft strong GIGIGIGI 1 2 3 ,,, 1 2 3 neutrosophic bigroup but the converse is not true. Proposition 31 Let FA, and KD, be two soft be a neutrosophic 3 -group, where GIQI1 conjugate strong neutrosophic bigroups over a neutrosophic group under multiplication. GI2 0,1,2,3,4,IIII ,2 ,3 ,4 neutrosophic GI,,. 12 Then group under multiplication modulo 5 and 1) Their extended union FAKD,, over GI3 0,1,2,1 IIIIII ,2 , ,2,1 2,2 2 GI,, is not soft conjugate strong neu- 12 a neutrosophic group under multiplication modulo 3. Let trosophic bigroup over GI,,. 12 11n n P ,2, ,2IIIIII ,,1 ,1,4,,4 ,1,2,,2 , 2n 2I n 2) Their extended intersection FAKD,, over GI,, 12 is soft conjugate strong neutro- TQ \ 0 , 1,2,3,4 , 1,2 and sophic bigroup over GI,,. 12 XQIIII \ 0 ,1,2, ,2 ,1,4, ,4 are sub - groups. 3) Their restricted union FAKD,, R over Then FA, is clearly soft neutrosophic -group over GI,, 12 is not soft conjugate strong neu- GIGIGIGI 1 2 3 ,,,, 1 2 3 trosophic bigroup over GI,,. 12 where 4) Their restricted intersection FAKD,, R over GI,, 12 is soft conjugate strong neu- trosophic bigroup over GI,,. 12 Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 72 Neutrosophic Sets and Systems, Vol. 2, 2014 neutrosophic N -group over GI, ,..., . 11n n 1 N F x1 ,2,n ,2IIIIII ,,1 ,1,4,,4 ,1,2,,2 , 2n 2) Their OR operation FAKD,, is not soft 2I neutrosophic -group over GI, ,..., . F x2 Q \ 0 ,1,2,3,4 ,1,2 , 1 N Definition 49 Let FA, be a soft neutrosophic N - F x3 Q\ 0,1,2, I ,2 I ,1,4, I ,4 I . group over GI, 1 ,..., N . Then Theorem 34 Let FA, and HA, be two soft neu- 1) FA, is called identity soft neutrosophic N -group trosophic N -groups over GI, 1 ,..., N . Then if F x e1,..., eN for all xA , where their intersection FAHA,, is again a soft neu- ee1,..., N are the identities of trosophic -group over GI, ,..., . 1 N GIGI1 ,..., N respectively. Proof The proof is straight forward. 2) FA, is called Full soft neutrosophic N -group if Theorem 35 Let FA, and HB, be two soft neu- F x G I , 1 ,..., N for all xA . trosophic -groups over GI, 1 ,..., N such Definition 50 Let FA, and KD, be two soft neu- that AB , then their union is soft neutrosophic trosophic N -groups over GI, 1 ,..., N . Then N -group over GI, 1 ,..., N . Proof The proof can be established easily. KD, is soft neutrosophic sub -group of FA, Proposition 33 Let FA, and KD, be two soft written as KDFA,, , if 1) DA , neutrosophic -groups over GI, 1 ,..., N . Then 2) K x F x for all xA . 1) Their extended union is not soft FAKD,, Example 23 Let FA, be as in example 22. Let neutrosophic N -group over GI, 1 ,..., N . KD, be another soft neutrosophic soft -group over 2) Their extended intersection FAKD,, is GIGIGIGI 1 2 3 ,,,, 1 2 3 soft neutrosophic -group over where GI, ,..., . 1 N 1 n K x1 ,2 ,1,4,IIII ,4 ,1,2, ,2 , 2n 3) Their restricted union FAKD,, R is not soft K x Q \ 0,1,4,1,2 . neutrosophic -group over GI, 1 ,..., N . 2 Clearly KDFA,,. 4) Their restricted intersection FAKD,, R is soft neutrosophic -group over Thus a soft neutrosophic N -group can have two types of soft neutrosophic sub -groups, which are following GI, 1 ,..., N . Proposition 34 Let FA, and KD, be two soft Definition 51 A soft neutrosophic sub -group neutrosophic -groups over KD, of a soft neutrosophic -group FA, is called soft strong neutrosophic sub -group if GI, 1 ,..., N . Then 1) DA , 1) Their AND operation FAKD,, is soft 2) Kx is neutrosophic sub -group of Fx for Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 73 all xA . 6 1234 1234 1234 1234 Definition 52 A soft neutrosophic sub N -group F x1 0,3,3,33,1,, I I g , , , , 1234 2143 4321 3412 KD, of a soft neutrosophic N -group FA, is called soft sub N -group if 3 6 9 1234 1234 1234 1234 1) DA , F x2 0,3,3,33,1,,,, I I g g g , , , . 1234 2143 4321 3412 2) Kx is only sub -group of Fx for all Theorem 36 Every soft Lagrange neutrosophic -group xA . FA, over GI, 1 ,..., N is a soft neutrosoph- Definition 53 Let GI, ,..., be a neutro- 1 N ic -group but the converse is not true. sophic N -group. Then FA, over Theorem 37 If GI, 1 ,..., N is a Lagrange GI, 1 ,..., N is called soft Lagrange neutrosoph- neutrosophic -group, then FA, over ic -group if and only if Fx is Lagrange sub N - GI, 1 ,..., N is also soft Lagrange neutrosoph- ic -group. group of GI, 1 ,..., N for all xA . Example 24 Let Proposition 35 Let FA, and KD, be two soft Lagrange neutrosophic N -groups over GIGIGG 1 2 3,,, 1 2 3 be neutro- GI, 1 ,..., N . Then sophic N -group, where GIZI16 is a 1) Their extended union FAKD,, is not soft group under addition modulo 6 , GA24 and 12 Lagrange neutrosophic -group over G3 g: g 1 , a cyclic group of order 12, GI, 1 ,..., N . o G I 60. 2) Their extended intersection FAKD,, is Take PPIPP 1 2 3,,,, 1 2 3 a neutro- not soft Lagrange neutrosophic -group over sophic sub GI, 1 ,..., N . 3 -group where 3) Their restricted union FAKD,, R is not soft TIII1 0,3,3 ,3 3 , Lagrange neutrosophic -group over 1234 1234 1234 1234 GI, ,..., . P2 ,,,, 1 N 1234 2143 4321 3412 4) Their restricted intersection FAKD,, R is Pg 1,6 . Since P is a Lagrange neutrosophic sub 3 - 3 not soft Lagrange neutrosophic -group over group where order of P 10. GI, 1 ,..., N . Let us Take TTITT 1 2 3,,,, 1 2 3 Proposition 36 Let FA, and KD, be two soft where TIIITP {0,3,3 ,3 3 }, and 1 2 2 Lagrange neutrosophic -groups over T g3, g 6 , g 9 ,1 is another Lagrange sub 3 -group 3 GI, 1 ,..., N . Then where oT 12. 1) Their AND operation FAKD,, is not soft Let FA, is soft Lagrange neutrosophic -group over Lagrange neutrosophic -group over GI, 1 ,..., N . GIGIGG 1 2 3,,, 1 2 3 , where 2) Their OR operation FAKD,, is not soft Lagrange neutrosophic -group over GI, 1 ,..., N . Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 74 Neutrosophic Sets and Systems, Vol. 2, 2014 Definition 54 Let GI, 1 ,..., N be a neutro- Theorem 39 If GI, 1 ,..., N is a weakly La- sophic N -group. Then FA, over grange neutrosophi -group, then FA, over GI, 1 ,..., N is called soft weakly Lagrange neu- GI, 1 ,..., N is also soft weakly Lagrange neu- trosophic -group if atleast one Fx is Lagrange sub trosophic -group. Proposition 37 Let FA, and KD, be two soft -group of GI, 1 ,..., N for some xA . weakly Lagrange neutrosophic -groups over Examp 25 Let GI, 1 ,..., N . Then GIGIGG 1 2 3,,, 1 2 3 be neutro- 1. Their extended union FAKD,, is not sophic N -group, where GIZI16 is a soft weakly Lagrange neutrosophic -group over group under addition modulo 6 , GA24 and GI, 1 ,..., N . 12 G3 g: g 1 , a cyclic group of order 12, 2. Their extended intersection FAKD,, is not soft weakly Lagrange o G I 60. neutrosophic -group over Take PPIPP 1 2 3,,,, 1 2 3 a neutro- GI, 1 ,..., N . sophic sub 3 -group where 3. Their restricted union FAKD,, R is not TIII1 0,3,3 ,3 3 , soft weakly Lagrange neutrosophic -group over 1234 1234 1234 1234 GI, 1 ,..., N . P2 ,,,, 4. Their restricted intersection 1234 2143 4321 3412 FAKD,, R is not soft weakly Lagrange Pg 1,6 . Since P is a Lagrange neutrosophic sub 3 - 3 neutrosophic -group over group where order of P 10. GI, 1 ,..., N . Let us Take TTITT 1 2 3,,,, 1 2 3 where TIIITP1 {0,3,3 ,3 3 }, 2 2 and Proposition 38 Let FA, and KD, be two soft 48 weakly Lagrange neutrosophic -groups over T3 g, g ,1 is another Lagrange sub 3 -group. GI, ,..., . Then Then FA, is soft weakly Lagrange neutrosophic - 1 N group over 1) Their AND operation FAKD,, is not soft weakly Lagrange neutrosophic -group over GIGIGG 1 2 3,,, 1 2 3 , where GI, ,..., . 1234 1234 1234 1234 1 N F x0,3,3,33,1,, I I g 6 , , , , 1 2) Their OR operation FAKD,, is not soft 1234 2143 4321 3412 weakly Lagrange neutrosophic -group over 481234 1234 1234 1234 GI, 1 ,..., N . F x2 0,3,3,33,1,,, I I g g , , , . 1234 2143 4321 3412 Definition 55 Let GI, 1 ,..., N be a neutro- Theorem 38 Every soft weakly Lagrange neutrosoph- sophic -group. Then FA, over ic -group FA, over GI, 1 ,..., N is a soft GI, 1 ,..., N is called soft Lagrange free neutro- neutrosophic N -group but the converse is not tue. Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 75 sophic N -group if Fx is not Lagrange sub -group Proposition 39 Let FA, and KD, be two soft Lagrange free neutrosophic N -groups over of GI, 1 ,..., N for all xA . Example 26 Let GI, 1 ,..., N . Then GIGIGG 1 2 3,,, 1 2 3 be neutro- 1) Their extended union FAKD,, is not soft Lagrange free neutrosophic -group over sophic 3 -group, where GIZI16 is a GI, 1 ,..., N . group under addition modulo 6 , GA24 and 2) Their extended intersection 12 G3 g: g 1 , a cyclic group of order 12, FAKD,, is not soft Lagrange free o G I 60. neutrosophic -group over GI, 1 ,..., N . Take PPIPP 1 2 3,,,, 1 2 3 a neutro- sophic sub 3 -group where 3) Their restricted union FAKD,, R is not soft Lagrange free neutrosophic -group over P1 0,2,4 , GI, ,..., . 1234 1234 1234 1234 1 N P2 ,,,, 4) Their restricted intersection 1234 2143 4321 3412 FAKD,, R is not soft Lagrange free 6 Pg3 1, . Since P is a Lagrange neutrosophic sub 3 - neutrosophic -group over group where order of P 10. GI, 1 ,..., N . Let us Take TTITT 1 2 3,,,, 1 2 3 Proposition 40 Let FA, and KD, be two soft where TIIITP1 {0,3,3 ,3 3 }, 2 2 and 48 Lagrange free neutrosophic -groups over T3 g, g ,1 is another Lagrange sub 3 -group. GI, 1 ,..., N . Then Then FA, is soft Lagrange free neutrosophic 3 -group 1) Their AND operation FAKD,, is not over GIGIGG ,,, , 1 2 3 1 2 3 soft Lagrange free neutrosophic -group over where GI, 1 ,..., N . 1234 1234 1234 1234 6 2) Their OR operation FAKD,, is not F x1 0,2,4,1,g , , , , , 1234 2143 4321 3412 soft Lagrange free neutrosophic -group over GI, ,..., . 1234 1234 1234 1234 1 N F x0,3,3,33,1,,, I I g48 g , , , 2 Definition 56 Let GI, 1 ,..., N be a neutro- 1234 2143 4321 3412 sophic -group. Then FA, over Theorem 40 Every soft Lagrange free neutrosophic N - GI, ,..., is called soft normal neutrosoph- group FA, over GI, 1 ,..., N is a soft neu- 1 N trosophic -group but the converse is not true. ic N -group if Fx is normal sub -group of Theorem 41 If GI, 1 ,..., N is a Lagrange GI, 1 ,..., N for all xA . free neutrosophic -group, then FA, over Example 27 Let GIGIGGI1 1 2 3 ,,, 1 2 3 be GI, 1 ,..., N is also soft Lagrange free neutro- sophic -group. a soft neutrosophic -group, where 2 2 2 2 G1 I e,,,,,,,,,,, y x x xy x y I yI xI x I xyI x yI Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 76 Neutrosophic Sets and Systems, Vol. 2, 2014 is a neutrosophic group under multiplaction, normal neutrosophic -groups over G g: g 6 1 , a cyclic group of order 6 and 2 GI, 1 ,..., N . Then G38 I Q I 1, i , j , k , I , iI , jI , kI 1) Their AND operation FAKD,, is soft nor- is a group under multiplication. Let mal neutrosophic -group over PPIPPI ,,,, a normal 1 2 3 1 2 3 GI, 1 ,..., N . 24 sub 3 -group where P1 e,,,, y I yI P2 1, g , g 2) Their OR operation FAKD,, is not soft normal neutrosophic -group over and P3 1, 1 . Also GI, 1 ,..., N . TTITTI 1 2 3 ,,, 1 2 3 be another normal sub 3 -group where Definition 56 Let GI, 1 ,..., N be a neutro- 23 T12 I e, I , xI , x I , T 1, g and sophic -group. Then FA, over Then is a soft normal neu- T3 I 1, i . FA, GI, 1 ,..., N is called soft conjugate neutrosoph- trosophic N -group over ic -group if Fx is conjugate sub -group of GIGIGGI1 1 2 3 ,,,, 1 2 3 GI, 1 ,..., N for all xA . where 24 F x1 e, y , I , yI ,1, g , g , 1 , Theorem 43 Every soft conjugate neutrosophic N - 23 F x2 e, I , xI , x I ,1, g , 1, i . group FA, over GI, 1 ,..., N is a soft neu- Theorem 42 Every soft normal neutrosophic N -group trosophic -group but the converse is not true. FA, over GI, ,..., is a soft neutrosoph- 1 N Proposition 43 Let FA, and KD, be two soft ic N -group but the converse is not true. conjugate neutrosophic N -groups over Proposition 41 Let FA, and KD, be two soft GI, 1 ,..., N . Then normal neutrosophic N -groups over 1) Their extended union FAKD,, is not soft GI, 1 ,..., N . Then conjugate neutrosophic N -group over 1) Their extended union FAKD,, is not soft GI, 1 ,..., N . normal neutrosophic soft -group over 2) Their extended intersection FAKD,, is GI, 1 ,..., N . soft conjugate neutrosophic N -group over 2) Their extended intersection FAKD,, is GI, 1 ,..., N . soft normal neutrosophic -group over 3) Their restricted union FAKD,, R is not soft GI, 1 ,..., N . conjugate neutrosophic -group over 3) Their restricted union FAKD,, R is not soft GI, 1 ,..., N . normal neutrosophic -group over 4) Their restricted intersection FAKD,, R is GI, 1 ,..., N . soft conjugate neutrosophic N -group over 4) Their restricted intersection FAKD,, R is GI, 1 ,..., N . soft normal neutrosophic -group over Proposition 44 Let FA, and (,)KD be two soft GI, 1 ,..., N . conjugate neutrosophic -groups over Proposition 42 Let FA, and KD, be two soft Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 77 GI, ,..., . Then 1 N Theorem 44 Every soft strong neutrosophic soft - 1) Their AND operation FAKD,, is soft group FA, is a soft neutrosophic N -group but the conjugate neutrosophic N -group over converse is not true. GI, 1 ,..., N . Theorem 89 FA, over GI, 1 ,..., N is soft 2) Their OR operation FAKD,, is not soft strong neutrosophic N -group if GI, 1 ,..., N is conjugate neutrosophic N -group over a strong neutrosophic -group. GI, 1 ,..., N . Proposition 45 Let FA, and KD, be two soft 4.2 Soft Strong Neutrosophic N-Group strong neutrosophic N -groups over Definition 57 Let GI, 1 ,..., N be a neutro- GI, 1 ,..., N . Then sophic -group. Then FA, over 1) Their extended union FAKD,, is not soft strong neutrosophic -group over GI, 1 ,..., N is called soft strong neutrosophic GI, 1 ,..., N . -group if and only if Fx is a strong neutrosophic 2) Their extended intersection FAKD,, is sub N -group for all xA . not soft strong neutrosophic -group over Example 28 Let GI, 1 ,..., N . GIGIGIGI 1 2 3 ,,, 1 2 3 3) Their restricted union FAKD,, is not soft be a neutrosophic 3 -group, where R strong neutrosophic -group over GIZIII12 0,1, ,1 , a neutrosophic GI, 1 ,..., N . group under multiplication modulo 2 . 4) Their restricted intersection FAKD,, is GIOIIII2 ,1,2,3,4, ,2 ,3 ,4 , neutrosophic R group under multiplication modulo 5 and not soft strong neutrosophic N -group over GI, ,..., . GIII3 0,1,2, ,2 ,a neutrosophic group under 1 N multiplication modulo 3 . Let Proposition 46 Let FA, and KD, be two soft strong neutrosophic -groups over 11n n P n ,2,n ,2IIIIII ,,1 ,1,4,,4 ,1,2,,2 , 2 GI, 1 ,..., N . Then 2I 1) Their AND operation FAKD,, is not soft and XQIII \ 0,1,2, ,2 ,1, are neutrosophic strong neutrosophic N -group over sub 3 -groups. GI, 1 ,..., N . Then FA, is clearly soft strong neutrosophic 3 -group over 2) Their OR operation FAKD,, is not soft strong neutrosophic -group over GIGIGIGI 1 2 3 ,,, 1 2 3 , GI, 1 ,..., N . where Definition 58 Let and be two soft strong neutrosophic 11n n FA, HK, F x1 n ,2,n ,2IIIII ,,1 ,1,4,,4 ,1, , 2 N -groups over GI, 1 ,..., N . Then HK, is 2I called soft strong neutrosophic sub xA -group F x2 Q\ 0,1,2, I ,2 I ,1, I . of FA, written as HKFA,, , if Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 78 Neutrosophic Sets and Systems, Vol. 2, 2014 1) KA , 4) Their restricted intersection FAKD,, is not soft Lagrange strong 2) Kx is soft neutrosophic soft sub N -group of R neutrosophic -group over Fx for all xA . GI, 1 ,..., N . Theorem 45 If GI, 1 ,..., N is a strong neutro- Proposition 48 Let FA, and KD, be two soft sophic N -group. Then every soft neutrosophic sub N - Lagrange strong neutrosophic -groups over group of FA, is soft strong neutosophic sub N -group. GI, 1 ,..., N . Then Definition 59 Let GI, 1 ,..., N be a strong 1) Their AND operation FAKD,, is not neutrosophic -group. Then FA, over soft Lagrange strong neutrosophic -group GI, ,..., is called soft Lagrange strong neu- 1 N over GI, 1 ,..., N . trosophic -group if Fx is a Lagrange neutrosophic 2) Their OR operation FAKD,, is not soft Lagrange strong neutrosophic -group over sub -group of GI, 1 ,..., N for all xA . Theorem 46 Every soft Lagrange strong neutrosophic GI, 1 ,..., N . -group FA, over GI, 1 ,..., N is a soft Definition 60 Let GI, 1 ,..., N be a strong neutrosophic soft N -group but the converse is not true. neutrosophic -group. Then over Theorem 47 Every soft Lagrange strong neutrosoph- FA, ic -group FA, over GI, 1 ,..., N is a soft GI, 1 ,..., N is called soft weakly Lagrange srtong neutrosophic -group but the converse is not tue. strong neutrosophic soft -group if atleast one Fx is a Lagrange neutrosophic sub -group Theorem 48 If GI, 1 ,..., N is a Lagrange strong neutrosophic N -group, then FA, of GI, 1 ,..., N for some xA . Theorem 49 Every soft weakly Lagrange strong neutro- over GI, 1 ,..., N is also soft Lagrange strong sophic -group FA, over GI, ,..., is a neutrosophic -group. 1 N soft neutrosophic soft -group but the converse is not Proposition 47 Let FA, and KD, be two soft true. Lagrange strong neutrosophic N -groups over Theorem 50 Every soft weakly Lagrange strong neutro- Then GI, 1 ,..., N . sophic -group FA, over GI, 1 ,..., N is a soft strong neutrosophic -group but the converse is not 1) Their extended union FAKD,, is not true. soft Lagrange strong neutrosophic N -group over Proposition 49 Let and be two soft GI, 1 ,..., N . weakly Lagrange strong neutrosophic -groups over 2) Their extended intersection Then FAKD,, is not soft Lagrange strong neutrosophic -group over 1) Their extended union is not soft weakly Lagrange strong neutrosophic - GI, 1 ,..., N . group over 3) Their restricted union FAKD,, R is not 2) Their extended intersection soft Lagrange strong neutrosophic -group over is not soft weakly Lagrange GI, 1 ,..., N . strong neutrosophic -group over Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 79 GI, 1 ,..., N . GI, 1 ,..., N . Then 3) Their restricted union FAKD,, R is not 1) Their extended union FAKD,, is not soft weakly Lagrange strong neutrosophic N - soft Lagrange free strong neutrosophic -group group over GI, 1 ,..., N . over GI, 1 ,..., N . 4) Their restricted intersection 2) Their extended intersection FAKD,, R is not soft weakly Lagrange FAKD,, is not soft Lagrange free strong neutrosophic -group over strong neutrosophic -group over GI, 1 ,..., N . 3) Their restricted union is not Proposition 50 Let FA, and KD, be two soft soft Lagrange free strong neutrosophic -group weakly Lagrange strong neutrosophic -groups over over GI, ,..., . Then 1 N 4) Their restricted intersection 1) Their AND operation FAKD,, is not is not soft Lagrange free soft weakly Lagrange strong neutrosophic - strong neutrosophic -group over group over GI, 1 ,..., N . 2) Their OR operation FAKD,, is not Proposition 52 Let and be two soft soft weakly Lagrange strong neutrosophic - Lagrange free strong neutrosophic -groups over group over GI, 1 ,..., N . Then Definition 61 Let GI, 1 ,..., N be a strong neu- 1) Their operation is not trosophic N -group. Then FA, over soft Lagrange free strong neutrosophic -group over GI, 1 ,..., N is called soft Lagrange free strong 2) Their operation is not neutrosophic N -group if Fx is not Lagrange neutro- soft Lagrange free strong neutrosophic -group sophic sub N -group of GI, 1 ,..., N for all N . over Theorem 51 Every soft Lagrange free strong neutro- Definition 62 Let N be a strong neutrosophic N -group. sophic -group FA, over GI, 1 ,..., N is a Then FA, over GI, 1 ,..., N is called sofyt soft neutrosophic -group but the converse is not true. Theorem 52 Every soft Lagrange free strong neutro- normal strong neutrosophic -group if Fx is normal sophic -group over is a FA, GI, 1 ,..., N neutrosophic sub N -group of GI, 1 ,..., N for soft strong neutrosophic -group but the converse is not all xA . true. Theorem 54 Every soft normal strong neutrosophic - Theorem 53 If GI, 1 ,..., N is a Lagrange group FA, over GI, 1 ,..., N is a soft neutro- free strong neutrosophic -group, then FA, over sophic N -group but the converse is not true. Theorem 55 Every soft normal strong neutrosophic - GI, 1 ,..., N is also soft Lagrange free strong group FA, over GI, 1 ,..., N is a soft strong neutrosophic -group. neutrosophic -group but the converse is not true. Proposition 51 Let FA, and KD, be two soft Proposition 53 Let and be two soft Lagrange free strong neutrosophic N -groups over Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 80 Neutrosophic Sets and Systems, Vol. 2, 2014 normal strong neutrosophic N -groups over Proposition 55 Let FA, and KD, be two soft GI, 1 ,..., N . Then conjugate strong neutrosophic -groups over Then 1) Their extended union FAKD,, is not soft normal strong neutrosophic N -group over 1) Their extended union is not GI, 1 ,..., N . soft conjugate strong neutrosophic -group over 2) Their extended intersection is soft normal strong neutro- FAKD,, 2) Their extended intersection is soft conjugate strong neu- sophic -group over GI, 1 ,..., N . 3) Their restricted union FAKD,, R is not trosophic -group over soft normal strong neutrosophic -group over 3) Their restricted union is not GI, 1 ,..., N . soft conjugate strong neutrosophic -group over 4) Their restricted intersection FAKD,, R is soft normal strong neu- 4) Their restricted intersection is soft conjugate strong neu- trosophic -group over GI, 1 ,..., N . Proposition 54 Let FA, and KD, be two soft trosophic -group over normal strong neutrosophic -groups over Proposition 56 Let and be two soft GI, 1 ,..., N . Then conjugate strong neutrosophic -groups over 1) Their AND operation FAKD,, is soft Then normal strong neutrosophic -group over 1) Their operation is soft GI, 1 ,..., N . conjugate strong neutrosophic -group over 2) Their OR operation FAKD,, is not soft normal strong neutrosophic -group over 2) Their operation is not GI, 1 ,..., N . soft conjugate strong neutrosophic -group over Definition 63 Let GI, 1 ,..., N be a strong neu- trosophic N -group. Then FA, over Conclusion This paper is about the generalization of soft neutrosophic groups. We have extended the concept of soft neutrosophic GI, 1 ,..., N is called soft conjugate strong neu- group and soft neutrosophic subgroup to soft neutrosophic trosophic N -group if Fx is conjugate neutrosophic bigroup and soft neutrosophic N-group. The notions of soft normal neutrosophic bigroup, soft normal neutrosophic N- sub N -group of GI, 1 ,..., N for all xA . group, soft conjugate neutrosophic bigroup and soft conju- Theorem 56 Every soft conjugate strong neutrosophic gate neutrosophic N-group are defined. We have given var- ious examples and important theorems to illustrate the as- -group FA, over GI, 1 ,..., N is a soft pect of soft neutrosophic bigroup and soft neutrosophic N- neutrosophic -group but the converse is not true. group. Theorem 57 Every soft conjugate strong neutrosophic References -group FA, over GI, ,..., is a soft 1 N [1] Muhammad Shabir, Mumtaz Ali, Munazza Naz, strong neutrosophic N -group but the converse is not true. Florentin Smarandache, Soft neutrosophic group, Neutrosophic Sets and Systems. 1(2013) 5-12. Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group Neutrosophic Sets and Systems, Vol. 2, 2014 81 [2] A.A.A. Agboola, A.D. Akwu, Y.T.Oyebo, Neu- Received: January 2nd, 2014. Accepted: January 31st, 2014. trosophic groups and subgroups, Int. J. Math. Comb. 3(2012) 1-9. [3] Aktas and N. Cagman, Soft sets and soft group, Inf. Sci 177(2007) 2726-2735. [4] M.I. Ali, F. Feng, X.Y. Liu, W.K. Min and M. Shabir, On some new operationsin soft set theory, Comput. Math. Appl. (2008) 2621-2628. [5] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 64(2) (1986)87-96. [6] S.Broumi, F. Smarandache, Intuitionistic Neutro- sophic Soft Set, J. Inf. & Comput. Sc. 8(2013) 130-140. [7] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parameterization reduction of soft sets and its applications,Comput. Math. Appl. 49(2005) 757- 763. [8] M.B. Gorzalzany, A method of inference in ap- proximate reasoning based on interval- valuedfuzzy sets, Fuzzy Sets and Systems 21 (1987) 1-17. [9] P. K. Maji, Neutrosophic Soft Set, Annals of Fuzzy Mathematics and Informatics, 5(1)(2013), 2093-9310. [10] P.K. Maji, R. Biswas and R. Roy, Soft set theory, Comput. Math. Appl. 45(2003) 555-562. [11] P.K. Maji, A.R. Roy and R. Biswas, An applica- tion of soft sets in a decision making problem, Comput. Math. Appl. 44(2002) 1007-1083. [12] D. Molodtsov, Soft set theory first results, Com- put. Math. Appl. 37(1999) 19-31. [13] Z. Pawlak, Rough sets, Int. J. Inf. Compu. Sci. 11(1982) 341-356. [14] Florentin Smarandache,A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic. Rehoboth: American Research Press, (1999). [15] W. B. Vasantha Kandasamy & Florentin Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Struc- tures, 219 p., Hexis, 2006. [16] W. B. Vasantha Kandasamy & Florentin Smarandache, N-Algebraic Structures and S-N- Algebraic Structures, 209 pp., Hexis, Phoenix, 2006. [17] W. B. Vasantha Kandasamy & Florentin Smarandache, Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, 149 pp., 2004. [18] A. Sezgin & A.O. Atagun, Soft Groups and Nor- malistic Soft Groups, Comput. Math. Appl. 62(2011) 685-698. [19] L.A. Zadeh, Fuzzy sets, Inform. Control 8(1965) 338 -353. Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Soft Neutrosophic Bigroup and Soft Neutro- sophic N-group 82 Neutrosophic Sets and Systems, Vol. 2, 2013 Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir Surapati Pramanik1, Tapan Kumar Roy2 1 Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, Dist-North 24 Parganas, PIN Code-743126, West Bengal, India. E-mail: [email protected] 2 Bengal Engineering and Science University, P.O.-B. Garden, District – Howrah-711103, West Bengal, India. E-mail: roy [email protected] Abstract. The study deals with the enduring conflict between Jammu-Kashmir conflict between India and Pakistan. India and Pakistan over Jammu and Kashmir since 1947. The International Journal of Mathematical Archive (IJMA), ongoing conflict is analyzed as an enduring rivalry; characterized 4(8) (2013), 162-170.) studied game theoretic model toJammu by three major wars (1947-48), 1965, 1971, low intensity and Kashmir conflict in crisp environment. In the present study military conflict (Siachen), mini war at Kargil (1999), internal we have extended the concept of the game theoric model of the insurgency, cross border terrorism. We examine the progress and Jammu and Kashmir conflict in neutrosophic envirorment. We the status of the dispute, as well as the dynamics of the India have explored the possibilities and developed arguments for an Pakistan relationship by considering the influence of USA and application of principle of neutrosophic game theory to China in crisis dynamics. We discuss the possible solutions understand properly of the Jammu and Kashmir conflict in terms offered by the various study groups and persons. Most of the of goals and strategy of either side. Standard 2×2 zero-sum game studies were done in crisp environment. Pramanik and Roy (S. theoretic model used to identify an optimal solution. Pramanik and T.K. Roy, Game theoretic model to the Keywords: Conflict resolution, game theory, Jammu and Kashmir conflict, neutrosophic membership function, optimal solution sad- dle point, zero-sum game. 1 Introduction Our contribution to the literature is to discuss briefly the genesis of the conflict and apply neutrosophic game theory The purpose of this study is to develop neutrosophic game for conflict resolution. theoretic model to India-Pakistan (Indo-Pak) crisis dynamics and contribute to the neutrosophic analysis of Rest of the paper is organized in the following way. conflicts and their neutrosophic resolution. M. Intriligator Section 2 presents some basics of neutrosophy and [1] reviewed mathematical approaches to the study of neutrosophic sets and their operations. Section 3 describes conflict resolutions in crisp environment. He prepared a a brief history and the genesis of Jammu and Kashmir list of primary methodological thrusts as differential conflict. Section 4 is devoted to formulation neutrosophic equations, decision and control theory, game and game theoretic model to Jammu and Kashmir conflict bargaining theory, uncertainty analysis, stability theory, between India and Pakistan. Section 5 presents concluding action-reaction models and organization theory. remarks. Anandalingam and Apprey [2] proposed multilevel mathematical programming model in order to develop a 2. Basics of neutrosophy and neutrosophic sets conflict resolution theory based on the integration of non- In this section, we present some basic definitions of cooperative game within a mathematical paradigm. They neutrosophy, and neutrosophic sets and their operations postulated conflict problem as a Stackelberg [3] due to Smrandache [7] and Wang et al.[8]. optimization with leaders and followers. However, the Definition 1. Neutrosophy: A new branch of philosophy, model is suitable only for the normal version of introduced by Florentin Smarandache that presents the information distribution [4] when the strategy of all lower- origin, nature, and scope of neutralities, as well as their level players is completely known to the leader. Yakir interactions with different ideational spectra. Plessner [5] employed the game theoretic model to resolve Neutrosophy is the basis of neutrosophic set, neutrosophic the conflict between Israel and the Palestinians. Pramanik probability, and neutrosophic statistics. and Roy [6] studied game theoretic model to the J&K conflict between India and Pakistan in crisp environment. Definition 2. Infinitesimal number: ε is said to be But the situation and relation between India and Pakistan infinitesimal number if and only if for all positive integers are not static but dynamic and neutrosophic in nature. So n, ε < 1/n new approach is required to deal with the conflict. Definition 3. Hyper-real number: Let ε > 0 be an infinitesimal number. The hyper-real number set is an Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir Neutrosophic Sets and Systems, Vol. 2, 2013 83 ~ extension of the real number set, which includes classes of and when X is discrete a SVNSs N can be written as infinite numbers and classes of infinitesimal numbers. ~ m + = ∑ ~~ ~ x/)x(F),x(I),x(T , ∈∀ .Xx N N N N Definition 4. Non-standard finite number: 1 = 1+ ε , =1i - where “1” is its standard part and “ ε ” its non-standard SVNS is an instance of neutrosophic set that can be used in part real life situations like decision making, scientific and en- Definition 5. Non-standard finite number: -0 = 0- ε , and gineering applications. In case of SVNS, the degree of the “0” is standard part and “ ε ” is its non-standard part. truth membership ~ ),x(T the indeterminacy membership N Definition 6. A non-standard unit interval: A non-standard ~ )x(I and the falsity membership ~ )x(F values belong to - + - N N unit interval can be defined as ||- 0, 1 -||. Here 0 is non- [0, 1]. standard number infinitely small but less than 0 and 1+ is ~ non-standard number infinitely small but greater than 1. It should be noted that for a SVNS ,N 0 ≤ ~ + ~ + ~ )x(Fsup)x(Isup)x(Tsup ≤ 3 , ∈∀ .Xx (4) Main Principle: Between an idea < ψ > and its opposite N N N A ≥ B )x(Finf)x(Finf , A ≥ B )x(Fsup)x(Fsup (3) falsity membership functions are related as follows: ~ ~ ~ ))x(T),x(T(min=)x(T ; for all x in X. NC NA NB Definition 10. Single-valued neutrosophic set (SVNS): Let ~ ~ ~ ))x(I),x(I(min=)x(I ; X be a universal space of points (objects) with a generic NC NA NB element of X denoted by x. A single-valued neutrosophic ~ ~ ~ ))x(F),x(F(max=)x(F for all x in X. ~ NC NA NB set N⊂X is characterized by a true membership function 3. Brief history and the genesis of Jammu and ~ )x(T , a falsity membership function ~ )x(F and an N N Kashmir conflict indeterminacy function ~ )x(I with ~ ),x(T ~ ),x(I ~ )x(F ∈ N N N N It is said that Kashmir is more beautiful than the heaven, [0, 1] for all x in X. ~ and the benefactor of the supreme blessing and happiness. When X is continuous a SVNSs, N can be written as The account of Kashmir is found in the oldest extant book- ~ = ~~ ~ ,x)x(F),x(I),x(T ∈∀ .Xx “Nilamat Purana”. Kalhan, Kashmir’s greatest historian N ∫ N N N x scholarly depicted the history of Kashmir starting just Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 84 Neutrosophic Sets and Systems, Vol. 2, 2013 before the great Mahabharata War. According to Kalhan, peasants who had been cruelly fleeced by the rapacious the Mauryan emperor Ashoka annexed Kashmir in 250 sardars of Kabul. I do not mean to suggest that the Sikh B.C. He embraced Buddhism after the Kalinga war. He rule was benign or good, but it was at any rate better that made it a state religion. He built many Bihars, temples that of the Pathans.” specially Shiva temple. According to Chinese traveler, Huen Tsang over five thousand Buddhist Monks settled 3.1 Dogra rule (1846-1947) down in Kashmir during the reign of Ashoka. After the fall of Maurya dynasty, it is believed that Kashmir for over two Dogra dynesty played an in important role in developing hundred years was ruled by Indo-Greek Kings before the Jammu and Kahmir State. start of "Turushka" (Kushan ) rule in the state. Thus, the 3.1.1 Gulab Singh (1846-1857) people of Kashmir came in contact with the Greeks. The The State of Jammu was conferred on Gulab Singh with reflection of which is found on the beautiful architectural the title of Raja by Maharaja Ranjit Singh of Punjab in and sculptural style of old Kashmiri temples, and the 1820. He annexed Ladakh in September 1842. Some parts coinage of the later Kashmiri kings. of Gilgit and Baltistan were invaded before 1846. The The zenith of Buddhist power in Kashmir was reached in State of Jammu and Kashmir (J&K) is founded through the reign of king Kanishka. Influenced by Indian culture, Amritsar treaty in 1846 between the East India Company Kanishka adopted Buddhism and made it the state religion. and Raja Gulab Singh who buys Kashmir Valley from the During his reign, it is believed that the forth Buddhist East India Company for Rs. 7.5 million and annexes it to Council was held at Kundalavana in Kashmir. It was Jammu and Ladakh already under his rule. Thus the Dogra enthusiastically attended by a large number of scholars, dynesty establishes in 1846. Gulab Singh conquered theoreticians, and commentators. . During his reign, Muzaffarabad in 1854. Buddhism propagated in Tibet, China and Central Asia. 3.1.2 Ranbir Singh (1857-1885) However, Buddhism was followed by a revival of Ranbir Singh (1857-1885) ascended the throne after his Hinduism and Hindu rulers ruled Kashmir up to 1320. father death in 1857 A.D., who ruled from 1857 to 1885 th A.D. Lord Northbook’s Government recommended for a Rinchan (1320-1323) ascended the throne on 6 October political officer to reside permanently at the Maharaja’s 1320. He was the first converted Islam ruler in the history Court in September 26, 1873 A.D. A British Resident of Kashmir. Shah Mir ascended the throne with the title of remained permanently at the court of Maharaja relating to Sultan Shamsuddin (1339-1342) in 1339 A.D. and Shah- the external relations of British India from 1873. Mir dynasty (1339-1561) ruled the state for 222 years. 3.1.3 Maharaja Pratap Singh (1885-1925) Shah Mir dynasty is one of the most important in the annals of Kashmir, in as much as Islam was firmly Maharaja Pratap Singh (1885-1925) ascended the throne established here. During Chak rule (1561-1586) Sunnni after his father death in 1885. During his rule, British Muslims and Hindus alike were persecuted. power was deeply interested in Kashmir and through British Resident Maharaja Pratap Singh was kept under Akbar, the Mughal Emperor annexed Kashmir in 1586. It pressure. In September 1885 during the initial stage of is important to note that under the Mughal reign (1586- Pratap Singh’s rule, the British Government changed the 1752), people got slight relief and lived honorably. status of the British officer Special Duty in Kashmir to that However, the Mughal used forced labor in their visits to of a political Resident. Pratap Singh's Address in Durbar Kashmir in terms of a huge retinue of unpaid laborers to October 19, 1885 revealed that the position of political carry their goods and other supplies for the journey. officer in Kashmir has been placed on the same footing Afghan rule (1752-1819) succeeded in maintaining their with that of Residents in other Indian States in subordinate suzerainty over Kashmir for a span of sixty-seven years. alliance with the Government. British Government of India The Afghans were highly unscrupulous in the employment disposed Maharaja in 1889. Maharaja was offered an of forced labor. The common Kashmirian people were allowance, which was ungenerously described as sufficient tired of their ferocity, barbarity and persecution. It is true for dignity but not for extravagance, would be made to him. history of Kashmir that all sections of people suffered No specific period was set for this arrangement to come to during Afghan rule but the principal victims of these cruel an end. Colonel Nisbet, Resident of Kashmir became the were the peasants. During this era all cruel and inhuman virtual ruler because although the Council of minister measures of Afghan rulers could not put an end the basic would have full powers of administration, they would be tradition of Kashmiri. expected to exercise those powers under the guidance of The reign of Sikh Power (1819-1846) in Kashmir lasted for the British Resident. Without consulting with him, Council only 27 years. It is to be noted that the Sikhs continued would not take any important decision and the Council with the practice of forced labor in order to transport of would follow Resident’s advice whenever it was offered. goods and materials. According to Lawrence [9], "to all classes in Kashmir to see the downfall of the evil rule of In 1889, the British Government instituted Gilgit Agency Pathan, and to none was the relief greater than to the under the direct rule of British political agent. Colonel Algeron Durand [10], the first British Agent in Gilgit Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 85 Neutrosophic Sets and Systems, Vol. 2, 2013 records the Russian influence for creation of Gilgit Agency Kilam, Pandit Lal Saraf, Pandit Kasyap Bandhu. Under the in his Book, “The Making of a Frontier”. He remarked in a leadership of Sheik Abdullah AJKMC felt the necessity of statement “Why it has been asked should it be worth our common platform to struggle against the rule of Maharaja. while to interfere there whatever happened? The answer is After series of discussions and debates, the working of course Russia…Expensive as the Gilgit game might committee of AJKMC took the historic decision of re- have been, it was worth the Candle.” Viceroy Lord christening to Jammu and Kashmir National Conference Curzon reinstated Maharaja Pratap Singh in power in 1905 (or simply National Conference) on 24 June 1938. On 27 A.D. The State Council is abolished in May 1906 A.D. April 1939, National Conference came into being. Its secular credentials set a new pace for the politics of Jammu 3.1.4 Hari Singh (1925-1947) Kashmir. National Conference [12] consisted of many Hari Singh (1925-1947) ascended the throne after his leaders of minority communities like Hindu, Sikh etc grandfather, Pratap Singh’s death in 1925. During his rule during 1940s. the agitation against the Dogra rule started mainly against In the history of India subcontinent, the Pakistan resolution the misrule, corrupt administration, autocratic rule, demanding the creation of an independent state comprised repression on the subjects at the slightest excuse and lack of all regions in which Muslims are the majority is passed of administrative efficiency. Maharaja Hari ruthlessly at Iqbal Park, Lahore on March 23, 1940 by Muslim crushed a mass uprising in 1931. Hari Singh constituted League. Grievances Enquiry Commission headed by B .J. Glancy on 12 November 1931 for a probe into the complaints of The secularization of Kashmir politics and redefinition of the people of Kashmir. In April 1932, the commission the goal helped immensely National Conference to come in recommended its suggestions. Among these close contact with the Indian National Congress. In 1942 recommendations, the important one was the step to be ‘New Kashmir’ manifesto was formulated under the taken for propagating education for Kashmiri Muslims. leadership of Dr. N. N. Raina by a brilliant group of young The Commission recommended to give payment [11] communist operating within the National Conference who Kashmiri people for Government work. In the order dated were mostly responsible for introducing the nationalist 31 May 1932, Maharaja Hari Singh accepted the movement to the concept of socialist pattern of society recommendation of the President of the Kashmir based on equality, democracy and free from exploitation. It Constitutional Reforms Committee, B. J. Glancy and consists of two parts: a) the constitution of the state; b) the appointed a Franchise Committee to put forward tentative National Economic Plan. Under the sound leadership of suggestion regarding the important question of the Abdullah, National Conference led a powerful mass franchise and the composition of the assembly. In this movement in order to find a new political and economic background All Jammu and Kashmir Muslim Conference order in Kashmir and other parts of Jammu region. The (AJKMC) was formed under the leadership of Sheikh National Conference started agitation against the Dogra Abdullah in 1932 in October in Srinagar. The conference rule in 1945. In the grave political situation, offering him held from 15 to 17 October 1932. all charges, Ram Chandra Kak was appointed as Prime In 1934, the Muslim Conference demonstrated its secular Minister in order to bring the agitation in control. In May view when it forwarded memorandum drafted by Ghulam 1946 National Conference launched “Quit Kashmir” Abbus to the Maharaja demanding early implementation of movement following the “Quit India Movement” in 1942 the report of Glancy Commission and specifically urged led by the Indian National Congress. Mohamod Ali Jinnah for a system of joint electorates in the State. In 1934, was not interested in the ‘Quit Kashmir Movement’[13] Maharaja introduced a Legislative Assembly. However 35 rather blamed the movement as act of Gundas. In March of its 75 members were to be nominated. 8 per cent of the 1946 Crisps Mission came to visit India. Sheikh Abdullah population was allowed to cast vote. To become a voter sent a telegram by demanding freedom of people of literacy and property qualifications were specified. The Kashmir on withdrawal of British power from India. Assembly enjoyed only consultative powers. Maharaja Prime Minister of J&K Ram Chandra Kak declared further reformed making the provision for Council of emergency to crackdown the movement. Abdullah was Ministers and a judicial and legislative branch of public arrested on 20 May 1946. The State Government employed administration in 1939. However, Maharaja enjoyed most a wave of arrests and a policy of repression throughout the of the decision powers under the new reforms. State. The people protested strongly and several agitated Kashmiri people were killed and injured due to clash with On 26 March 1938 Sheik Abdullah iterated two important armed forces of Maharaja. The Indian National Congress points: i) to put an end communalism by ceasing to think in and the All India States peoples’ Conference supported terms of Muslims-non-Muslims when discussing political National Conference strongly. Sheik Abdullah was problems. ii) Universal suffrage on the basis of joint imprisoned for three years for antinational activities. electorate. It is to be noted that the national demand issued National Conference was banned. In January 1947, in August 1938 was signed among others by Pandit Jia Lal National Conference boycotted elections because of Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 86 Neutrosophic Sets and Systems, Vol. 2, 2013 repression. Muslim Conference grabbed the opportunity Sikh, one Christian-and one Parsee. It started functioning and won 16 out of 21 Muslim seats. on 2nd September 1946. The League joined the Interim 3.2 Muslim Conference Government in the last week of October 1946 but was not prepared to join the Constituent Assembly, which led every Muslim Conference did not support the ‘Quit Kashmir’ day a more and more difficult and delicate on account of agitation. Muslim Conference discouraged the people of the differences between the cabinet ministers of Congress Kashmir from joining the agitation in the same tune of and the Muslim League. 0n 26 November 1946, Mr. Atlee Muslim League. On 30 May 1946, Chaudhury Gulam invited Lord Wavell and representatives of the Congress Abbas the President of Muslim Conference stated that the and the Muslim League to meet in London to attempt to agitation had started at Congress leaders’ behest in order to resolve the deadlock The discussions were held from 3 to 6 “restore the lost prestige of the Nationalist.” The Muslim December 1946 but did not yield any agreed settlement. Conference adopts the Azad Kashmir Resolution on 26 The first meeting of the Constituent Assembly of India was July 1946 calling for the end of autocratic Dogra rule in the held in on 11 December 1946. The Muslim League region and claiming the right to elect their own constituent boycotted it and it developed a stake in sabotaging the assembly. He said that the primary task [14]was to restore Assembly’s work. the unity of the Muslim nation and there be “no other place On 20 February 1947, Prime Minister Atlee declared that for an honest and self-respecting Muslim but in his own Britain would transfer power by June 1948, by which time organization.” On 25 October 1946, State Government the Congress and the Muslim League were supposed to re- arrested and detained four top leaders of Muslim solve their differences. On 24 March Mountbatten was Conferences. sworn in as Viceroy and Governor General of India in 3.3 British Cabinet Mission place of Wavell. After negotiations with the leaders of dif- ferent political parties, Viceroy, Lord Mountbatten an- In March 1946, The British Cabinet Mission held nounced that long before June 1948, the Dominions of In- conference about a week at Simla with four representatives, dia and Pakistan would be created and that the question of two each of the Congress and the Muslim Leagues and the Indian states would be dealt with in the light of the Cabinet conference broke down on the issue of Pakistan and parity Mission's memorandum [16] of 12 May 1946.To approve in the proposed interim government. On 16 May 1946, the the Mountbatten plan accepted by British Cabinet, a confe- Cabinet Mission announced their own proposals, the rence between Mountbatten and representatives of the essence of which was the creation of a Constituent Congress and the Muslim League was held on 2 June 1947. Assembly to frame the Constitution of India, which was to On 3 June 1947 a White paper was issued which stated the be based on the principle that the Center would control detail procedure of the partition of India. Regarding the only three subjects, viz., Defense, Foreign Affairs and Princely States it declared that British policy towards In- Communications and the creation of three group of dian States contained in the Cabinet Mission’s memoran- provinces-two of the areas claimed by Muslim League for dum [17] of 12th may, 1946 remained unchanged. Pakistan in the east and the west and the third of the rest of the subcontinent [15]. 3.5 Partition Plan accepted by Congress 3.4 Interim Government announced On 14 June 1947, in a historic session of All India Congress Committee (AICC) in New Delhi, Pandit Ballabh On 25 June 1946, the Congress Working Committee Pant moved the resolution dealing with the Mountbatten announced their rejection of the plan of Interim plan for partition British India. Mahatma Gandhi Government. On June 26, 1946, Lord Wavell announced intervened in the debate in the second day and expressed that he would set up a temporary ‘caretaker’ Government that he was always against the partition but situation had of officials to carry on in the interim period. changed and appealed to support the resolution. On 15 In July 1946, the Muslim League withdrew its acceptance June 1947, the resolution was passed with 29 votes in favor of the Cabinet Mission’s plan and resolved that “now the and 15 against. time has come for the Muslim nation to resort to direct Mr. Jinnah clearly expressed Muslim League view [18] on action to achieve Pakistan, to assert their just rights, to the question of Princely States on 17 June 1947 by saying vindicate their honor and to get rid of the present British "Constitutionally and legally the Indian states will be slavery and the contemplated future ‘caste- Hindu independent sovereign states on the termination of paramountcy and they will be free to decide for themselves domination” at a meeting in Bombay. and adopt any course they like; it is open to them to join Accepting the invitation from the Viceroy to constitute an the Hindustan Constitutional Assembly or decide to remain interim Government, on 6 August 1946, Jawaharlal Nehru independent. In case they opt for independence they would formed it, which consisted of six Hindus, including one enter into such agreements or relationships with Hindustan Depressed Class member, three Muslims of whom two or Pakistan as they may choose". belonged neither to the Congress nor to the League, one 3.6 Partition and riots Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 87 Neutrosophic Sets and Systems, Vol. 2, 2013 Calcutta, capital of Bengal witnessed a beginning of region, hitherto administered by a British agency, to J&K holocaust on an unprecedented scale on 16 August 1946, without taking the verdict of the local people. which was declared a public holiday by the Muslim League Government of Bengal. It was estimated that 3.8 Standstill Agreement Jinnah’s direct action [18] caused death of more than 5000 Pakistan became independent on 14 august 1947. India and lives, and over 15000 people were injured, besides 100000 few princely states, which did not join either of India or being rendered homeless. After a fortnight 560 people Pakistan, became independent on 15 August 1947. In this were killed in Bombay. After Calcutta, on October 1946, way J&K attained the status of independent on 15 August serious anti- Hindu riots erupted in Noakhali in East 1947. On 15 August post offices in J&K hoisted the Bengal followed by massacred of Muslims in Bihar The Pakistani flags. Maharaja Hai Singh signed a standstill chain reaction of riots started in the Punjab causing large agreement with Pakistan on 16 August 1947 with regard to scale killings of Hindus, Sikhs, and Muslims shortly State’s postal services, railways, and communications and afterwards. hoped to sign similar agreement with India with regard to external affairs, control of state forces, defense etc. India 3.7 Development in Jammu and Kashmir [23] did not show any interest in the acceptance of the Based on two-nation theory, India was partitioned into offer of standstill agreement. . On 18 August 1947 a Pakistan and India in August 14, 1947. The princely states controversy came into light when Sir Cyril Radcliffe were offered the right under the ‘Indian Independent Act awarded a portion of Muslim majority Gurudaspur District 1947’ and ‘Government of Indian Act 1935’ [19] to to India causing fundamental differences in J&K’s accede either to India or Pakistan or remain independent. It geopolitical situation. The subcontinent experienced seemed that Hari Singh, the then Maharaja of Jammu and communal riots during these days. By this time, Muslim Kashmir hoped to create independent Kingdom or majority Poonch estate within the Jammu region autonomy from India and Pakistan. He did not accede to experienced serious troubles with regard to some local either of two successor dominions at the time of accession. demands like the rehabilitation of 60,000 demobilized All Jammu and Kashmir Rajya Hindu Sabha passed a soldiers of the British army belonging to the area. The resolution [20] expressing its faith in Maharaja Hari Singh agitation finally transformed into communal form having and extended its “support to whatever he was doing or mixed with other issues. The state army refused to fire on might do on the issue of accession” in 1947. On 15 June the demonstrators with whom they had religious and ethnic 1947, an important resolution [21] regarding the princely ties. The agitation turned to the form of armed revolt states saying the lapse of paramountcy does not lead to the because of mass desertion from army. The supply of arms independence of the princely states was adopted by AICC and ammunition and other assistance from outside the unanimously. Contrary to this, Mr. Jinnah clearly border magnified the revolt. The Kashmir Socialist party expressed the view [18] of Muslim League on the question passed a resolution on 18 September 1947 to join Pakistan of Princely States on 17 June 1947 by saying and not India. The party impressed on Maharaja that "Constitutionally and legally the Indian states will be without any further unnecessary delay he should make an independent sovereign states on the termination of announcement accordingly. It is to be noted here that a paramountcy and they will be free to decide for themselves convention of Muslim Conference workers formally asked and adopt any course they like; it is open to them to join for accession to Pakistan on 22 September 1947. the Hindustan Constitutional Assembly or the Pakistan Maharaja Hari Singh released Sheikh Abdullah from Constituent Assembly, or decide to remain independent. In prison along with some other National Conference workers the last case, they enter into such agreements or on 29 September 1947 but he did not release the workers of Muslim Conference due to grave situation of the state. relationship with Hindusthan or Pakistan as they may Pakistan termed Abdullah’s release as a conspiracy choose ". because workers of Muslim Conference were not On 19 July 1947, the working committee of Muslim simultaneously released. By October, communal riots Conference passed a modified resolution [22] in favor of spread all over J&K. The mass infiltration baked by independence, which respectfully and fervently appealed to Pakistani army jeopardized the environment of the state. the Maharaja to declare internal autonomy of the state and Pakistan violated the standstill agreement by stopping accede to Pakistan regarding to defense, communication regular supply of food, salt, petrol and essential and external affairs. Khurshid Ahmad, Jinnah’s personal commodities from Pakistan. The communication system Secretary during his stay in Kashmir on the crucial days controlled by Pakistani Government did not render proper for the question of accession gave Maharaja assurance [23] service. that “Pakistan would not touch a hair of his head or take On 21 October 1947, Pakistan decided to settle the future away a iota of his power”. Before partition British of Kashmir with the power of gun suspecting that Mahara- Government restored the Gilgit area, an important strategic ja was likely to accede to India. Jinnah, the Governor Gen- Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 88 Neutrosophic Sets and Systems, Vol. 2, 2013 eral of Pakistan personally authorized a plan [25] to launch have plenary power to control and supervise the plebiscite. “a clandestine invasion by a force comprised of Pathan Ultimately, the first direct bilateral talks broke down. (Afghan) tribesmen, ex-servicemen and soldiers on leave”. On 1 January 1948, based on the advice of Mountbatten, It was witnessed that charges and counter charges were India lodged a complaint with the Security Council being made by both the government of J&K and Pakistan invoking articles 35 of Chapter VI of the UN Charter to during the month of October and finally On 22 October “recommend appropriate procedures or methods of 1947, 2000 tribesmen from Northwest Frontier Province adjustment” for the pacific settlement of disputes and not (NWFP) of Pakistan and other Pakistani nationals fully for “action” with respect to acts of aggression as provided armed with modern arms, under the command of trained for in Chapter VII of the Charter [32]. India reiterated her generals, started invasion to capturing the state’s territory. pledge of her conditional commitment to a plebiscite under The Muslims in the Western part of Kashmir established international auspices once the aggressor was evicted. their own independent (Azad) Kashmir Government on 24 Pakistan contradicted the validity of the Maharaja’s October 1947. The State forces were wiped out in fighting. accession to India [33], and urged the Security Council to The tribesmen resorted to “indiscriminate slaughter of both appoint a commission for securing a cease-fire and Hindus and Muslims”[26]. They reached within 15 miles ensuring withdrawal of outside forces, and conducting a from capital Srinagar. Under this great emergency of the plebiscite in order to determine the future of J&K. situation, Maharaja sought Indian military assistance in his letter dated 26 October 1947 along with the ‘Instrument of 3.10 Role of the United Nation Security Council Accession’ [27] to Mountbatten, the Governor General of (UNSC) India. Thereafter the Maharaja signed the instrument of accession, which the Governor General Mountbatten Both India and Pakistan denied implementing the UN reso- accepted on 27 October 1947 by adding that the question lutions [34-36] for a free and impartial plebiscite in order of accession [28] should be settled by a referendum. Indian to put an end to the situation for the accession of J&K. forces [29] airlifted at Srinagar almost at the crucial Having taken note of the developments in J&K, the United moment, for, “a few minutes later the airfield might well Nations Commission for India and Pakistan UNCIP sub- have been in enemy hands”. Members of the National mitted a draft resolution [36] consisting of three parts to Conference provided logistical support for the Indian the council on 13 August 1948. forces. Infuriated by Indian intervention, on 27 October Part I of the resolution comprised of instruction for a cea- 1947, Pakistani Governor General, Mohammed Ali Jinnah sefire. ordered Lt. General Sir Douglas Gracey, Chief of the Part II of the resolution dealt with the principle of a truce Pakistani Army, to send Pakistani regular troops to agreement which called for Pakistan to withdraw tribes- Kashmir, but Field Marshall Auchinleck, the Supreme men, Pakistani nationals not normally resident therein who Commander of the transition period succeeded in had entered the State of J&K for the purpose of fighting, to persuading him to withdraw his orders. A message [30] evacuate the territory occupied by Pakistan and after the was sent to the Governor General and the Prime Minister notice of the implementation of the above stipulation by of India to go to Lahore for discussion regarding Kashmir. the UNCIP India was to withdraw the bulk of her forces in 3.9 Indo-Pak talks stages from J&K leaving minimum strength with the ap- proval of the commission in order to ensure law, order and On 1 November 1947, at a meeting of Governors General peace in the State. of India and Pakistan at Lahore, Mountbatten offered to re- Part III of the resolution appeared to be important as it solve the J&K issue by holding referendum. Rejecting the clearly expressed that both the Government of India and Mountbatten formula, M.A. Jinnah remarked that a plebis- the Government of Pakistan reaffirm their wish that the cite was “redundant and undesirable”. H.V. Hodson [31] future status of the State of J&K shall be determined in has recorded in his book, The Great Divide, that M.A. Jin- accordance with the will of the people. nah “objected that with Indian troops present and Sheikh The second resolution [37] specified the basic principle of Abdullah in power the people would be frightened to vote plebiscite was formally adopted on 5 January 1949 after for Pakistan”. Jinnah proposed a simultaneous withdrawal acceptance of India and Pakistan on 23 and 25 December of all forces- the Indian troops and the invading forces. 1948 respectively. Here it is interesting to note that when he was asked how An important development occurred when both India and anyone could guarantee that the latter would also be with- Pakistan agreed to the cease-fire line in 1949. This enabled drawn, Jinnah [30] replied “If you do this I will call the the UN to finally send a Military observer Group to super- whole thing off”. In connection with the steps to ascertain vise the line [38]. The ceasefire came into effect on 1 Jan- the wishes of the people of J&K, Mountbatten was in favor uary 1949. The most important long- term outcome of the of a plebiscite under the auspices of United Nations while first Indo-Pak war was the creation of ceasefire line. Thus M. L. Jinnah proposed that he and Mountbatten should UNCIP succeeded in implementing the important provi- sion of Part I of the resolution. In order to monitor to the Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir Neutrosophic Sets and Systems, Vol. 2, 2013 89 ceasefire line (CFL), the UNCIP sent a Monitoring Group two nations, without prejudice to their rights or claims to for India and Pakistan (UNMGIP) to J&K on 24 January prepare and execute within the stipulated period of five 1949 relying on its resolution of 13 august 1948. In Kara- months for the demilitarization of J&K based on proposals chi on 27 July 1949, the military representatives of India of McNaughton and for self determination [45] through an and Pakistan, duly authorized, approved CFL and thus ap- impartial plebiscite. The resolution terminated the UNCIP proved the presence of UNMGIP [39]. and transferred their powers and responsibilities to a UN representative. In March 1949, the conflicting attitudes came into light as India and Pakistan expressed their viewpoints before the 3.10.1 Dixon mediation truce subcommittee of the UNCIP. On 15 April 1949, Sir Owen Dixon, UN Representative submitted his rec- UNCIP transmitted to the governments of India and Pakis- ommendations to the UN on 15 September 1950. He sug- tan its own proposals [40], which were: gesting a unique proposal [46] limiting the plebiscite only i) to create a cease-fire line, eliminating all no to the Kashmir Valley claimed by both by Pakistan due to man’s lands and based on the factual position of its Muslim majority and the waters of Jhelum. India and the troops in January 1949. Pakistan rejected the plan. ii) to draw a phased program of withdrawal of Pa- UN representatives worked to negotiate for free and impar- kistani troops to be completed in seven weeks, tial plebiscite in J&K until 1953 but their efforts brought and the withdrawal of all Pakistani nationals. no fruit. The UN continued its efforts for a plebiscite [47], iii) to ask Indian forces also to withdraw in accor- but all attempts of UN failed due to the conflicting and di- dance with a phased program after the withdrawn vergent attitude of the Governments of India and Pakistan of tribesmen and Pakistani national and after the towards the dispute and the cold war [18]. The fifth report declaration of UNCIP’s satisfaction regarding the of Dr. Frank P. Graham [48] suggested direct negotiations troops withdrawal of Pakistan. between India and Pakistan. Thus the UN attempts at solv- iv) to release all prisoners of war within one month. ing the problem of J&K came to end which reflected the v) to repeal all emergency laws. limitations of the UN. vi) to release all political prisoners. The armies of India and Pakistan waged an inconclusive Both India and Pakistan [41] could not accept the propos- war (1947-48) for over a year in J&K. The Indian army als because of their own interest. occupied almost two third of J&K remaining 1/3 portion The UNCIP proposed arbitration on the issues regarding was under the control of Pakistan which is called Azad the part II of the resolution in a letter to the two Govern- Kashmir or Pakistan occupied Kashmir (POK). ments on 26 August 1949 and named Fleet Admiral Ches- ter Nimitz as the Arbitrator. Pakistan accepted the propos- 3.11 Indo-Pak negotiation (1962-1963) al on 7 September 1949 but India rejected this proposal of arbitration. The Czechoslovak representative of the UN- India experienced a huge defeat in 1962 war against China. CIP, Dr. Oldrich Chyle (Chyle took the post after resigna- The J&K dispute became the subject of Indo-Pak negotia- tion of Korbel) criticized the UNCIP’s work [42]. Accord- tion in late 1962 but no agreement could be signed for res- ing to him, the arbitration move was a pre-planned attempt olution of J& K question despite six round talks between on the part of the USA and UK to intervene in the dispute. an Indian delegation headed by Swaran Singh and a Pakis- tani delegation headed by Z.A. Bhutto from 27 December On 17 December 1949, the UNSC named its president 1962 and 16 May 1963. General A. G. L. McNaughton of Canada as the Informal 3.12 Sino-Pak border agreement 1963 Mediator [43], instead of commission to negotiate a demi- litarization plan in consultation with India and Pakistan. He On 2 March 1963, the Sino-Pakistan Border Agreement submitted his proposal on 22 December 1949. Pakistan ac- was signed in Peking and they had agreed that after the set- cepted the proposals, suggesting minor amendments while tlement of the Kashmir dispute between Pakistan and In- India suggested major amendments: one calling for the dia, the sovereign authority concerned would reopen nego- disbanding and disarming of Azad forces, and the other tiations with China on the boundary as described in Ar- dealing with the return of the Northern Areas to India for ticle. By this agreement Pakistan [49] succeeded in stabi- purposes of defense and administration of J&K. Pakistan lizing Pakistan’s position regarding Kashmir in the eyes of was unable to accept Indian amendments [44] as a clear re- Chinese Government and compelling her “to reject une- jection of the proposals. Pakistan agreed to simultaneous quivocally the contention that Kashmir belonged to India”. demilitarization but Indian rejected it on the grounds of the legal and moral aspects of the plan. 3.13 The Kutch conflict- a low intensity war The UNSC adopted another resolution introduced by C. In 19 April 1965, Pakistani permanent representative [50] Blanco of Cuba on behalf of four powers Cuba, Norway, in UN made claims about 8960 square kilometers area of UK and USA on 14 March 1950, which called upon the Rann. Pakistani claim to the Rann of Kutch was based on Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 90 Neutrosophic Sets and Systems, Vol. 2, 2013 the fact that the Rann was a lake and according to Holding a Plebiscite in Kashmir within three months of the international law [51], the boundary line between India and cease fire. Pakistan must be drawn through the middle of the Rann. Armies of both the countries engaged in large - On other hand India argued that the Rann of Kutch was a scale combat in a series of sharp and intense actions along “marsh” land rather than a lake. India asked Pakistan to the ceasefire line in J&K and the international border in restore the status quo ante. Tikka Khan, in command of the Punjab, Rajasthan, and Gujarat by employing import 18 Infantry Division, did painstakingly prepare for the weaponry system but outmoded war stretegies. They operations and succeeded in advancing inside Indian reached to the point of exhaustion, battle fatigue. The territories in strength, causing to the fall of the Indian representative of the Netherlands moved the draft forward post hastily positioned there. India and Pakistan resolution [58], which was accepted, by both India and fought a low intensity war. The important aspect of the Pakistan in the UNSC on 20 September 1965. It was conflict lies in the historic fact that both India and Pakistan adopted by ten votes to nil, with Jordan abstaining. On 20 accepted a ceasefire and arbitration on British intervention. September 1965, the super power USA concurred with On the other hand, India captured some Pakistani Posts in USSR in the Security Council on calling ceasefire within the Kargil area of Ladakh. The Kutch dispute [52] was 48 hours. Pakistan and India accepted the call [59] on 21 referred to a tribunal comprising of three members, one and 22 September 1965 respectively. The ceasefire, the nominated by India, another by Pakistan and a Chairman UN enforced became effective at 03:30 hours of 23 chosen by the UN Secretary General. After a long September 1965. Both India and Pakistan lost nearly 3000 deliberation the tribunal awarded Pakistan 317 square people each in the war. Economy of both the countries miles out of 3500 square miles claimed by her. India left suffered a setback. the occupied posts of Pakistan in Kargil. Although fighting ended inconclusive both India and Pakistan claimed victories,. China identified India as an 3.14 Indo-Pak war in 1965 aggressor and supported the Kashmiri’s right of self- The Pakistani Government was greatly emboldened by determination. presumably military success in the Rann of Kutch in 1965. In August 1965 infiltration had started in Jammu and 3.15 Tashkent agreement 1966 Kashmir to wage what Zulfikar Ali Bhutto called a “war of The Tashkent Declaration was signed between Indian liberation”. On 10 August 1965, Z. A. Bhutto [53] publicly Prime Minister L. B. Shastri and Pakistani President after declared his country’s full support to the people of Kash- six days of hard bargaining on 10 January 1966. They mir but denied his country’s involvement in the Kashmir agreed that all armed personnel of the two countries should trouble. On 1 September, 1965 Indian forces crossed the be drawn not later than 25 February 1966 to the position international border and sealed the borders of Kashmir. On they held prior to 5 August 1965, and both sides should 4 September, Malaysia moved a resolution co-sponsored observe the ceasefire terms on the ceasefire line. They by Bolivia, the Ivory Cost, Jordan, the Netherlands, and affirmed to employ peaceful means to solve their conflicts. Uruguay proposing an immediate ceasefire in Kashmir Neither side was allowed to enjoy the gains of war. without calling Pakistan as an aggressor in the UNSC [54]. Pakistan was not even mentioned as the aggressor nor did But it did not succeed in stopping the fighting. Ayub Khan it admit having engineered the infiltration in J&K. backed the infiltration with a full-fledged attack in the Chhamb sector by crossing the international border, lead- 3.16 Indo-Pak war in 1971 ing to effective progress to reach Jaurian. On 5 September In the general election held in Pakistan in 7 December 1965, Indian forces launched three-pronged thrust in of 1970, the Awami League led by Mujibur Rehman secured West Pakistan in Lahore Sector and in Sialkot sector a day majority in the national assembly by winning 158 seats out later. Following this development, Malaysian representa- of 300 seats. He demanded complete autonomy for East tive submitted another resolution [55] supported by Boli- Pakistan. The East Pakistanis formed Mukti Bahini via, the Ivory Cost, Jordan, the Netherlands, and Uruguay (Liberation Force) and civil war erupted in East Pakistan. calling upon both the countries to cease hostilities and India supported the Movement. Pakistan used armed forces withdraw their troops to the positions held by 5 August to curve the movement. The fighting forced 10 million 1965, which was passed unanimously on 5 September East Pakistanis to flee in Indian territories. India accused 1965. The goodwill mission to India and Pakistan by the the Government of Pakistan of committing brutal genocide U.N. Secretary General, U Thant did not succeed. Both in the East Pakistan. India asked Pakistan to negotiate with countries were requested by U Thant to stop fighting with- Rehaman for a political settlement. On 3 December 1971, out imposing any condition on each other [56]. India ac- Pakistan launched attack on Indian airfields along the cepted unconditional ceasefire but President Ayub Khan frontier of Punjab, Rajasthan, and J&K [60]. On the other [57] imposed certain pre-conditions: (i) Withdrawal of all hand, Pakistan alleged that Indian forces attacked on 21 forces of both India and Pakistan (ii) Induction of foreign November 1971 in the south- eastern sector of East forces, preferably Afro-Asian under UN auspicious, (iii) Pakistan. India is the first country who recognized formally Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 91 Neutrosophic Sets and Systems, Vol. 2, 2013 the birth of Bangladesh [61] on 6 December 1971 . The kilometers. Both the Governments approved the Indian Army along with the Mukti Bahini (Liberation delineation [70] almost next day. On completion of Army) fought the Pakistani armed forces. The news of adjustment in the line of control, India and Pakistan sending a naval task force from the US Seven Fleet [62] to withdrew troops from the occupied territories in order to the Bay of Bengal from the Indo-China theatre caught restore the status quo ante on the international border on 20 much attention. But the USSR [63] confirmed India that December 1972. Pakistan [71] has recognized Bangladesh the Soviet powerful naval fleet would follow the Seven in February 1974. The issue regarding prisoner of wars Fleet. On 15 December 1971 the Indian army reached the [72] closed with the repatriation of the last group along outskirts of Dacca. On 16 December 1971, 9000 Pakistani with Gen. Niazi at Wagah on 29 April 1974. East Pakistan forces along with their commander General Niazi crisis reflected that the two-nation theory failed miserably surrendered to the Joint Command of India and in the subcontinent. Bangladesh. India declared a unilateral ceasefire [64] effective from 20:00 hours on 17 December 1971 and 3.19 The conflict at Siachen (1984 onwards) Yahya Khan accepted it. Yahya Khan had to resign The conflict between India & Pakistan over Siachen origi- because of huge defeat in East Pakistan. He handed power nated due to the non-demarcations on the western side of to Z.A. Bhutto. Although India and Pakistan fought a the map beyond a grid point known as NJ 9842. The CFL, third war over East Pakistan, J&K dispute was only a which was established because of the first Indo-Pak war of peripheral issue but vital one in the case of J&K. At time 1947-48 and the intervention of the UN, runs along the in- of ceasefire, India occupied 204. 7 sq kms of territory of ternational Indo-Pak border and then north and northeast Pakistan administered Kashmir, 957.31 sq km of Punjab until map grid-point NJ 9842, located near the Shyok River and 12198.84 sq kms of Kutch while Pakistan occupied at the base of the Saltoro mountain range. Unfortunately, it 134.58 sq kms of territory of Indian administered J&K in was not delineated beyond the grid point known as NJ the Chhamb sector, 175.87 sq kms in Punjab and 1.48 sq 9842 as far as the Chinese border but both countries agreed kms in Rajasthan [65]. vaguely that the CFL extends to the terminal point NJ 9842, and "thence north to the Glaciers". After second In- 3.17 Role of UN do-Pak war in 1965, obeying the Tashkent Agreement both The UN intervened to arrange cease-fires during the war countries withdrew forces along the 1949 CFL. After third 1971. USSR exercised her veto power several times in Indo-Pak war 1971, the Simla Agreement of 1972 created favor of India. The Secretary General [66] was authorized a new LOC based on December 1971 cease-fire. However, to appointment, if necessary, a special representative to the Siachen Glacier region was left un-delineated where no help in the solution of humanitarian problem. The issue of hostilities occurred. The authorities of both countries Indo-Pak conflict came to an end on 25 December 1971 showed no interest to clarify the position of the LOC with the appointment by U Thant, the Secretary General, of beyond NJ 9842. Due to lack of strategic viewpoint and se- V. W. Guicciardi, as Secretary General’s special riousness the LOC was poorly described as running from representative for humanitarian problems in India and Nerlin (inclusive to India), Brilman (inclusive to Pakistan), Pakistan. up to Chorbat La in the Turtok sector. In April 1984, In- dian army occupied key mountain Passes and established 3.18 Simla agreement 1972 permanent posts at the Siachen heights. Indian troops The Prime Minister of India and President of Pakistan had brought control over two out of three passes on the Sia- talks in Simla from 28 June 1972 to 2 July 1972 and signed chen, Sia La and Bilfond La, while the third pass, Gyong the Simla Agreement [67] on 2 July, 1972. By signing the La remained under Pakistan's control. The Indian forces agreement, both India and Pakistan committed themselves are permanently deployed all along the 110-km long Ac- to settling their differences through bilateral negotiations tual Ground Position Line (AGPL). Armies of both India or by any other peaceful means mutually agreed upon and Pakistan fight in lethal 10-22000 foot altitude in Sia- between them. Hopefully, they also agreed that in “Jammu chen Glacier. Pakistan retaliates in the world’s highest war and Kashmir, the line of control (LOC) resulting from the zone. cease-fire of December 17, 1971, shall be respected by both sides without prejudice to the recognized position of 3.20 Kashmir insurgency in Indian administered either side”. Kashmir The Simla Agreement was ratified by both countries [68] The assembly elections in J&K on 23 March 1987 were and it came into force on 5 August 1972. To delineate the partly manipulated and rigged which the National line of control General Bhagat and General Hamid Khan Conference-Congress coalition won a landslide victory. had to hold ten meetings between 10 August to 7 The opposition party Muslim United Front (MUF) called December 1972. On 11 December 1972, they [69] met at the victory as blatantly fraudulent and rigged. A large Suchetgarh and jointly signed 19 maps delineating the line number young people of Kashmir were alienated by this of control from Chhamb to Turtuk, covering about 800 perception. State Government of J&K witnessed various Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 92 Neutrosophic Sets and Systems, Vol. 2, 2013 demonstration and agitation between mid-1987 and mid- J&K and J&K Disturbed Areas Act to put down the 1989 based manifestation of an accumulated anger militancy. Indian security forces allegedly committed a comprised of many components such as administrative (the series of human right abuses [75] in J&K. It is observed curtail number of Offices that move to the winter capital that the encounter between Indian security forces and the Jammu), the regional autonomy, economic policy (increase militants caused more than 50, 000 deaths [75] including of power tariffs), religious sentiments, civil liberties many hundreds of innocent civilians. Kashmiri militants (custodian death), and anti-India demonstration of 14 and have been also accused of killing moderate Muslim leaders, 15 August, 26 October (accession day) and 26 January. On Hindus, bombing passenger busses and railway bridges 8 December 1989, the militants kidnapped Rubaiye Sayeed, and public establishments. In September 1996, National daughter of Indian Home Minster Mufti Mohammed Conference had won a landslide victory in J&K Assembly Sayeed. The prestige of Farooq Abdullah led State election, although the 30-disparate party coalition, known Government suffered serious setbacks for repression of as All-Party Hurriyat Conference (APHC) did boycott the any form of protest Farooq Abdullah’s resignation with the election. Indian authorities formed several Muslim appointment of Jagmohan as Governor for the second time counterinsurgency groups to combat the insurgency along on 19 January 1990, brought Central Government into with Indian security forces. Due to the acute failure of direct confrontation with the various rebel groups. At 5 a. Indian authorities to address the socio-economic problems m. on 20 January 1990, Indian paramilitary forces cracked and ambition of autonomy to some extent of the people of down on a part of Srinagar city and began the most intense J&K and Pakistan’s active role in fostering cross border house-to-house search and rounded up over three hundred terrorism, the situation in J&K becomes more complex and people. Most of them, however, were later released and volatile and neutrosophic in nature. arrested persons complained to be beaten up or dragged out of their houses. People got frightened first, but discovering 3.21 Nuclear rivalry between India and Pakistan the courage of desperation, the people started pouring out India had conducted her first nuclear device in 1974. In into the street defying the curfew, to protest against the May 1998, India conducted several nuclear tests in the alleged excessive use of force in search operation in next desert of Rajasthan. Pakistan got the opportunity to con- day. The administration got completely unnerved and gave duct nuclear test, and hopefully grabbed the opportunity orders to fire at when most of the groups of demonstrators and conducted six tests in Baluchistan in order to balancing converged at Gau Kadal. The number of deaths [73, 74] is nuclear power with India. The arm race between Indo-Pak disputed; however, the press reported 35 dead. Then the caught international attention. The UNSC condemned both implicit support for the separatists for independence the countries for conducting nuclear tests and urged them transformed into explicit due to mainly the high-handed to stop all nuclear weapons program. On 23 September searches ordered by Jagmohan, the Governor of J&K. On 1998, new development occurred following at UN General 19 February 1990 Governor dissolved the State assembly Assembly session. Both India and Pakistan agreed to try to and Governor rule was imposed. The Jagmohan regime resolve the Kashmir question peacefully and to focus on [13] witnessed sadly the exodus of almost the entire small trade and “people to people contact”. Pakistan sent her Kashmiri Pandit community from the valley and 20000 cricket team in India as goodwill gesture on November thousand Muslim had been forced to migrate. The State 1998 after a decade absence. On the other hand, India assembly election of 1990 resulted in Abdullah downfall agreed to buy sugar and powder from Pakistan. In Febru- following the outbreak of a Muslim uprising. During the ary 1999, bus service between New Delhi to Lahore 1990s, several new militant groups were formed, having started. Accepting an invitation from Sharif, Vajpayee vi- radical Islamic views. The large numbers of Islamic Jihadis, sited Lahore by bus. His visit to Pakistan is known as bus who had fought in Afghanistan against the Soviet Union in diplomacy. It drew much attention and at end of the sum- the 1980s, joined the movement mit they issued Lahore Declaration that was backed by Many umbrella groups were responsible for the uprising in Memorandum of Understanding (MOU) [76]. J&K. Among them, the first umbrella group is tied to the Jammu and Kashmir Liberation Front (JKLF). They 3.22 Kargil conflict in 1999 demaned independent Kashmir. The second group comprised of Muslim fundamentalists and has links with In May 1999, Pakistan –backed militants together with Pa- the fundamentalist Pakistan party, Jammait-I- Islam. No kistani regular forces crossed the LOC and infiltrated into doubt the group has a pro-Pakistan Orientation. The third India-held Kargil region of North Kashmir. The militants group is Jammu and Kashmir People`s League that has a occupied covertly more than thirty well-fortified positions pro-Pakistan orientation. The groups demanded plebiscite the most inhospitable frigidly cold ridges at an altitude of so that people of J&K could exercise their right of self 16000 -18000 feet, in the Great Himalayan range facing determination. India adopted a multiple prolonged Dras, Kargil, Batalik and the Mushko Valley sectors. India approach to deal with the insurgency in J&K. In 1990, the retaliated by launching air attacks known as ‘Operation Vi- then Governor Jagmohan announced the implementation of jay (victory) on 26 May 1999. India identified Pakistan as Armed Forces Special Powers Act of 1958 (AFSPA) for an aggressor that violated the LOC. As the battle turned Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 93 Neutrosophic Sets and Systems, Vol. 2, 2013 more intense, the Clinton administration intervened to help tan. Musharraf regime closed down some of militants defuse the conflict. It was witnessed that on 15 June Clin- groups’ offices and recruiting centers. India welcomed ton made separate telephonic conversations with both the these measures cautiously and the tension was somewhat Prime Ministers of India and Pakistan. He asked Sharif to defused. On May 14, 2002, three militants disguised in In- withdraw infiltrators from across the LOC. He cordially dian Army uniforms shot passengers indiscriminately on a appreciated Vajpayee for his display of restraint in the con- public bus and then killed 40 people (mostly wives and flict. Pakistan was isolated from world community regard- children of army personnel) including eight bus passengers ing the Kargil-issue, only Saudi Arabia and United Emi- at Kaluchak of Jammu before militants were gunned down. rates supported Pakistan. On 4 July 1999, Sharif and Clin- India reacted by threatening to strike at the terrorist camps ton held a three-hour meeting and issued a joint communi- situated in POK and took tough stand declaring some qué in which Sharif agreed to withdraw the intruders. On measures: i) expelled the Ambassador of Pakistan to India, 11 July 1999, in accordance with the agreement the infil- ii) withdrew her diplomatic personnel from Pakistan, iii) trators started retreating from Kargil as India set 16 July, imposed ban on Pakistani commercial air flights from In- 1999 as the dead line for the total withdrawal. On 12 July dian air space, iv) mobilized 100000 Indian forces close to 1999, Sharif defended his 4 July agreement with Mr. Clin- LOC. ton and defended his Kargil policy that designed to draw On 22 May 2002, on his visit to the frontlines in J&K, the international attention of the international community Vajpayee called for a “decisive battle”. Pakistani authority to Kashmir issue. In the war [77], India lost more than 400 declared that it would defend Pakistani administered forces. 670 intruders and 30 Pakistani regular forces were Kashmir by any cost. Musharraf mobilized 50,000 troops also killed excluding the injured. to the borders. On 27 May 2002, Musharraf [78] warned India by declaring, “if war is thrust upon us, we will re- 3.23 Agra summit (14-16 July 2001) spond with full might”. Agra Summit was held between the Indian Prime Minister Even Pakistan threatened to use nuclear weapons against Atal Behari Vajpayee and Pakistan's President Pervez Mu- India. This threat drew pointed attention to the USA and sharraf in Agra, from July 14 to 16, 2001. The summit be- UK. The British Foreign Secretary, Jack Straw and US gan amid high hopes of resolving various disputes between Deputy of Secretary, Richard Armitage and Defense Secre- the two countries including complex J&K issue. Both sides tary, Donald Ramsfeld rushed to both India and Pakistan in remained inflexible on the core issue of J&K and ultimate- May-June 2002 in order to defuse tension. They were suc- ly the bilateral summit failed to produce any formal cessful in their mission to defuse tension and succeeded in agreement. getting promise from Musharraf to stop cross-border infil- tration into J&K. After witnessing a slowdown in infiltra- 3.24 The threat of war between India and Pakistan and tion, India tried to improve her relation with Pakistan by the role of Bush administration reestablishing diplomatic ties, recalling her naval ships to On 13 December 2001, five militants attacked Indian na- their Bombay base, and opening her airspace to Pakistani tional parliament house causing the deaths of 13 people in- commercial flights. cluding five terrorists. India held Pakistan responsible for 3.25 Musharraf’s proposals for J&K resolution the incidence. India immediately reacted by deploying a si- On 25 October 2004, while addressing an ‘Iftar party’, zeable force along the LOC in J&K. Pakistan followed President Musharraf announced an important declaration suit, until both nations had aligned a vast array of troops regarding settlement of the J&K acceptable to Pakistan, and weapons against one another. Armies of both countries India and people of J&K. He remarked that the solution frequently exchanged of artillery fires. The mobilization of would have to be met thee steps: troops sparked worldwide fears of a deadly military con- i) Both sides should identify the regions on both flict between India and Pakistan. sides of LOC, In order to defuse the growing tensions Bush Administra- ii) Demilitarize these regions, tion took initiatives and succeeded in getting both sides to iii) Determine their status through independence or make conciliatory move. On 12 January 2002, in his ad- joint control or UN mandate. dress to his nation, Musharraf committed Pakistan’s “polit- He opinioned that Pakistanis demand for a plebiscite was ical, diplomatic and moral” support to the struggle of impractical while India’s offer for making LOC a perma- people of Kashmir. He went on to criticize the Pakistani nent border was unacceptable. The Musharraf’s an- militant Islamic groups for i) creating violent activities, ii) nouncements drew much attention but Indian Prime Minis- aggravating internal instability, iii) harboring sectarianism ter M. Singh refused to comment describing them as “of in Pakistan politics iv) creating war like situation against the cut remarks”. India. He banned two militant groups, the Lasker-e-Toiba President Musharraf [79] proposed four point solutions re- and Jais-e- Mohammed. In the following weeks, 2,000 ac- garding the resolution of J&K disputes as follows: tivists of the banned militant groups were arrested in Pakis- Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 94 Neutrosophic Sets and Systems, Vol. 2, 2013 i) troops will be withdrawn from the region in a underworld Dawood Ibrahim and Jaish-e-Mohammad chief staggered manner Maulana Masood Azhar staying in Pakistan. . To defuse ii) the border will remain unchanged, however the tension between India and Pakistan, Secretary of State people will be allowed to move freely in the re- of the USA, Condoleeza Rice visited Indian subcontinent. gion Ultimately USA succeeded in defusing the tension. iii) self governance or autonomy but not indepen- dence 3.27 Is China a third party in J&K conflict? iv) a joint management mechanism will be created Indian stand on the question is contradictory, ambiguous with India, Pakistan and Kashmiri Representa- and unclear and neutrosophic in nature. India strongly ob- tives jected the border agreement between Pakistan and China On 5 December 2006, during an interview with NDTV signed in 1963 by which China gained 2700 square miles President Musharraf opinioned that Pakistan is prepared to of the Pakistan occupied Kashmir. Based on the Simla give up her claim on Kashmir, also ready to give up her old Agreement 1972 between India and Pakistan, India strong- demand for a plebiscite and forget all UN resolution if In- ly is of the opinion that J&K problem is a bilateral dispute. dia accepts the four-point resolution of Kashmir dispute of- Third party involvement is not welcomed by India to re- fered by him. He remarked that Pakistan is absolutely solve the issue. However, Pakistan wants that China would against the independence of Kashmir so is India. He stated play a definite role to resolve the J&K conflict. China that for compromise solution both countries would have to adopts a neutral role as seen in the Kargil conflict in 1999. give up their positions and step back. So, depending upon Simla Agreement, Indian point of On 31 December 2006, G. N. Azad, the Chief Minister of view and present status, J&K conflict is considered as a bi- J&K stated that ‘joint mechanism” is possible in the field lateral problem between India and Pakistan. of trade, water, tourism and culture, and this could lead the 3.28 Internal development in Indian administered J&K way for a resolution to the longstanding Kashmir problem. On 8 January 2007, he further stated that the latest four Special status was conferred on J&K under Article 370 of point-solution offered by President Musharraf should not Indian Constitution [80]. Constituent Assembly was be put aside without discussing positively. elected by J&K on 31 October, 1951. The accession of On 19 January 2007, following the meeting with Indian J&K to India was ratified by the State’s Constituent As- External Affairs Minister, Chief Minister, G. N. Azad of sembly in 1954. The Constituent Assembly also ratified the J&K said “The date for the composite dialogue has been Maharaja’s accession of 1947 in 17 November 1956. In fixed for 13-14 March 2007 and I am sure all outstanding 1956, the category of part B was abolished and J&K was issues and proposals floated from time to time by President included as one of the States of India under article 1. On Musharraf will be discussed.” On the same day APHC January 26, 1957 Constituent Assembly dissolves itself. leader Mirwaiz Umar Farooq told the BBC that the next On 30 March 1965, article 249 of Indian Constitution ex- three months would be crucial. tended to J&K whereby the Central Government at New On 2 February 2007, President Musharraf said: “We can- Delhi could legislate on any matter enumerated in state list. not take people on board who believe in confrontation and Designation like Prime Minister and Sarder-i-Riyasat are who think that only militancy solves the problem”. On 3 replaced by Chief Minster and Governor respectively. In February 2007, Indian Prime Minister M. Singh welcomed 1964, decision to extend Articles 356 and 357 of the Union President Musharraf’s statement that militancy or violence Constitution of India to J&K announced. On 12 February cannot resolve the Kashmir issue. On 4 February 2007, In- 1975, Chief Minister Abdullah signed Kashmir Accord dian External Affairs Minister, Pranab Mukherjee com- that affirmed its status as a constituent unit of the India and mented on Musharraf’s proposals: “It is good. Everybody the State J&K will be governed by Article 370 of the Con- should advise terrorists to give up violence and join the stitution of India. process of dialogue.” The idea of four point resolution was purely personal that did not have the mandate. However, 3.29 Development of Pakistan administered Musharraf had to resign from his post for internal problem. Jammu and Kashmir His endeavor to resolve the J&K problem failed due to no Azad Kashmir was established on 24 October 1947. The response from India. UNCIP resolution depicts the status of Azad Kashmir as neither a sovereign state nor a province of Pakistan, but 3.26 Terror attack at Mumbai rather a "local authority" with responsibility over the area On 26 November 2008, Mumbai, the capital of Maharastra assigned to it under the ceasefire agreement. The "local and financial capital of India witnessed deadly terror at- authority" or the provisional government of Azad Kashmir tacks. India adopted a tougher-than –usual stand against had handed over matters related to defense, foreign affairs, Pakistan in the wake of the Mumbai terror attack and de- negotiations with the UNCIP and coordination of all affairs manded to hand over 20 terrorists including Pakistan-based relating to Gilgit and Baltistan to Pakistan under the Karachi Agreement [81] of April 28, 1949. Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir Neutrosophic Sets and Systems, Vol. 2, 2013 95 Zulfiqar Ali Bhutto’s government virtually annexed the Table 1: Payoff matrix POK by promulgating the Azad Jammu and Kashmir Act Pakistan of 1974. Azad Kashmir adopts Islam as the state religion I II vide Article 3. The constitution prohibits activities I 1 -1 prejudicial or detrimental to the ideology of the State's II 0 -1 accession to Pakistan (Article 7). It disqualifies non- India III 0 -1 Muslims from election to the Presidency and prescribed in IV 1 0 the oath of office the pledge 'to remain loyal to the country and the cause of accession of the State of Jammu and Pakistan strategy vector: Kashmir to Pakistan'. It provides two executive forums, i) Full compliance with Simla Agreement 1972, namely the Azad Kashmir government in Muzaffarabad ii) Partial or non-compliance with Simla agreement 1972 and the Azad Kashmir council in Islamabad. The Pakistan India’s strategy vector: government can dismiss any elected government in Azad i) Make territorial concessions, Kasmir irrespective of the support it may enjoy in the Azad ii) Accept the third party mediation ( USA), Kasmir Legislative Assembly [82] by applying the Section iii) Apply the strategy of all-out military campaign, 56 of the constitution,. The Northern Areas have the status iv) Continue fencing along the LOC (see Fig. 1). of a Federally Administered Area. 3.30 The Indo-Pak conflict over Jammu and Kashmir The conflict was based on the neutrosophic explanation and understanding of the neutrosophic situation in India and Pakistan. From Pakistani point of view, she hoped J&K was going to accede to Pakistan as the majority of the population being Muslims. If Junagadh, despite its Muslim rulers’ accession to Pakistan, belonged to India because of Fig.1 Photograph of fence along LOC its Hindu majority, then Kashmir surely belonged to Pakis- The above payoff matrix has been constructed with tan. Princely state Hydrabad became Independent on 15 reference to the row player i.e. India. In the process of August 1947 like J&K. But India invaded it because of formulating the payoff matrix, it is assumed that the Hindu majority. Pakistan regarded the Accession of J&K combination (I, I) will hopefully resolve the conflict while as a coerced attempt to force the hand of Maharaja. Popu- the combination (IV, II) will basically imply a status quo lar outbursts took place but J&K had acceded to India, be- with continuing conflict. If Pakistan can get India to either cause the ruler was a Hindu. From Indian point of view, make territorial concessions (Muslims dominated Kashmir J&K had acceded to her. Armed Pakistani raiders having valley or other important stragic areas of J&K such Kargil) Pakistani complicity and support invaded some portion of or accept the third party mediation like USA without fully J&K. Both the countries failed to implement the UN reso- complying with the Simla Agreement i.e. strategy lutions. The plebiscite has never been held. India viewed combinations (I, II) and (II, II), then it reflects that that that the time has changed. India strongly argues that Pakistan will be benefited but India will be loser. If India legislative measures subsequently legitimized the question accepts the third party mediation and Pakistan agrees to of accession. After Simla Agreement, J&K problem be- comply fully with the Simla Agreement, then though it came bilateral issue and its solution should be based on potentially ends the conflict, there should be a political Simla Agreement 1972. jeopardy in India as a result of lack of strategic and political concensus among the political parties and so the 4 Neutrosophic game theoretic model formulation strategy combination (II, I) is not a favorable payoff for of the Indo-Pak conflict over Jammu and Kashmir India. If India employs an all-out military campaign, an Following the above discussion and based on Simla devastating war seems to occur as both the countries are Agreement 1972 , game theoretic model is formulated. capable of using nuclear powers i.e. strategy combination The problem is modeled as a standard two person (2×2) (III, I) would not produce a positive payoff for either side. zero-sum game. If there occurs a unexpected and sudden change of mind set up for J&K within the Pakistani leaderships (from inside or outside pressure) and Pakistan chooses to fully abide by the Simla Agreement 1972 considering LOC as the permanent international border i.e. strategy combination (IV, I) will bring a potential end to the conflict as both countries may think that they will be loser Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 96 Neutrosophic Sets and Systems, Vol. 2, 2013 and winner at the same time (neutrosophically true) in the they were wholeheartedly seeking strategies to put an end sense that they will not get the whole J&K but Pakistan the conflict. President Musharraf played very clever and can cansole her saying that she has gained one third of diplomatic role by using the media very cautiously and J&K while India may think she gained two third of J&K. cleverly to make the international community and his In the payoff matrix (see Table 1), all the elements of the country understand that he tries his level best to reach a first, second and third rows are less than or equal to the meaningful, desicive and effective agreement but fails due corresponding elements of the forth row, therefore from to Indian rigid position regarding J&K issue. He left Agra the game theory [83] point of view, forth row dominates and thereby tried to obtain his acceptance to his own nation the first three rows. On the other hand, every element of and international community. This immediately shows why the first column is greater than the corresponding elements such a negotiation would break down at early stage. of the second column, therefore, first column is dominated The government of India and Pakistan are dealing with by second column. It shows that the above game has a militancy and terrorism in own land. But main issue saddle point having the strategy combination (IV, II), remains the J&K conflict. which reflects that in their very attempt to out-bargain each other both countries actually end up continuing the conflict 4.1 Case for applying neutrosophic game theory indefinitely! Thus the model model offers an equilibrium It is experienced that none of the strategy vectors available solution. to either side will remain temporarily stationary because In the subcontinent political arena, Pakistani leadership’s crucial events come into light on the global political arena, best interest was to transform the conflict more complex in general, and the south Asian subcontinent in particular. and keep the conflict more alive with full strenght to gain Moreover, there is a broad variety of ambiguities about the political support from inside and outside and ultimately motives behind Pakistan authority’s primary goals about compelled India to make territorial or other concessions. the driving force behind Pakistan authority’s primary goal However, for international pressure mainly from USA, and the strategies it adopts to achieve that goal. Pakistan had to state some overt declaration that negotiated Pakistan’s principal ally USA is also a great factor. The settlement over J&K based on Simla Agreement 1972 is influence of USA has a great impact on forming strategies. possible. However, Pakistan covertly continues her sincere The terrorist activities by Pakistan baked terrorist groups help to separatist groups by means of monetary, logical, are sometimes monitored by the wishes of USA. Although psychological and military equipments. By doing so Pakistan has not hundred percent control over foreign Pakistan is now in deep troubal with various militant mercenaries coming from different part of world namely, groups and Jihadi groups. She has to deal internal security Saudi Arabia, Afghanistan, Chechnia, Sudan etc. Pakistan probles caused by Pakistani Taliban groups. Under such is constantly trying to bring India under pressure by volatile circumstances, it would be quite impossible for harboring terrorist attacks on Indian ruled Kashmir and Pakistan to chalk out a distinct governing strategy to meet destabilizing the normalcy in the state in order to with counter strategy. understand the international community that international Both the countries, in general, played their games under intervention is requierd to resolve the J&K conflict. It is international pressure. Although Pakistan signed Lahore also difficult to state apart a true bargaining strategy from declaration with India by the then Prime Minister Nawaj one just meant to be a political decay. In the horizon of Sarif, Pakistani military boss Mr. Musharraf occupied continuous conflict, we believe and advocate an some heights of Kargil in 1999 to derail the peace process application of the conceptual framework of the and draw international attention to the J&K conflict. An neutrosophic game theory as a generalization of the important lesson of the Kargil conflict seems to be that no dynamic fuzzy game paradigm. military expedition could be a success if it is pursued without paying to serious attention to the totality of the It generalized terms, a well-specified dynamic game at a scenarios having domestic, political, economic, particular time t is a particular interaction between decision geographical, international opinions and sensitivities. makers with well-defined rules and regulations and roles Another important of Kargil conflict seems that national for the decision makers, which remain in place at time t but community does not want to military solution relating to are free to change over time. However, it is likely that the J&K problem. However, one positive aspect of the Lahore decision makers may suffer from the role of ambiguity i.e. declaration reflects that both the countries are capable of a typical situation where none of the decision makers are transforming the game scenario an open one in the sense exactly sure what to expect from others or what the other that the conflicting countries are capable of dynamically decision makers expect from them. In the context of indo- constructing and formulating objectives and strategies in Pak continuing conflict, for example, Pakistan leadership the course of their peaceful, mutual interaction within a would probably not have been sure of its role when Mr. formally defined socio- political set up. Musharraf met with Indian prime minister Atal Behari During the Agra summit in 2001, when probably President Vajpayee at the Agra summit to chalk out a peace Musharraf was thinking to make the conflict very alive agreement. Mr. Musharuff went to that summit under the while offering the impression to the other side (India) that international pressure or to prove himself to be an efficient Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 97 Neutrosophic Sets and Systems, Vol. 2, 2013 leader of Pakistan or against his free will and would have conflicting parties can effectively read and precisely liked to avoid Agra if he could because he did not want to understand each other’s nature of expectations. sign any final agreement on J&K which can be against This was reflected in Agra summit when probably common feeling of people of Pakistan. Mushauff Musharraf (Pakistan) was thinking in terms of keeping the demanded for declaration of J&K to be a disputed territory conflict alive while offering the impression to the other at least. Having no such capitulation forthcoming from side (India) that they were wholeheartedly seeking India, Musharuff left Agra without signing any joint strategies to put an end the conflict. Mr. Musharraf played statement. very diplomatic role by using the media very cautiously In this political context, Pakistani leadership’s best interest and cleverly to make the international community and his was to keep the conflict alive with full strength ultimately country understand he tried his level best to reach a fruitful compelled India to make territorial or other concession. agreement but failed and left at midnight and thereby tried However, for international pressure mainly from USA, to obtain his acceptance to his own nation and international Pakistan had to offer some overt declaration that community. This immediately comes to light why such a negotiated settlement over Jammu and Kashmir based on negotiation would break down at early stage. Simla Agreement 1972 is possible. Pakistan covertly If the Indo-Pak conflict over Jammu and Kashmir is continues her sincere help separatist groups by means of constituted as fuzzy dynamic fuzzy bargaining game, the monetary, logical, psychological and military equipments. players’ fuzzy set judgment functions over expected Under such volatile circumstances, it would be quite outcomes can be formulated as follows according to the impossible to chalk out a distinct governing strategy to rules of well-developed fuzzy set theory due to Zadeh [85] meet with counter strategy. For Pakistan, the fuzzy evaluative judgment function at However, both the countries, in general, played their time t, μ J () tP, (P, t) will be represented by fuzzy set games under international pressure. Although Pakistan membership function as follows: signed Lahore declaration with India by the then Prime Minister Nawaj Sarif, Pakistani military boss Mr. ∈(),1,5.0 for ΟWorst Musharraf occupied some heights of Kargil in 1999 to μ J()Ρ t, = ,5.0 x =for ΟWorst derail the peace process. An important lesson of the Kargil ,0 x Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 98 Neutrosophic Sets and Systems, Vol. 2, 2013 indeterminacy would be practically encountered due to the India and Pakistan in resolving the conflict and the reasons fact any mediated, two-way negotiation process is likely to for their failure. We have looked the Indo-Pak relations become over catalyst by the subjective utility preferences and recent developments and their views on J&K situations. of the mediator. Alternative possible solutions are also considered. Most of Let T, I, F represent real subsets of the real standard unit the solutions are either impractical or unacceptable to India, interval [0,1]. Statically, T, I, F are subsets while Pakistan, and or the various militant groups. Pro Pakistani dynamically, in the context of a dynamic game, they may militant groups and Pakistan would opt for free and be considered as set-valued vector functions. If a logical impartial plebiscite. Even some militant groups would proposition is said to be t % true in T, i % indeterminate in oppose the plebiscite because the UNSC resolutions do not I and f % false in F, then T, I, F are considered as the offer them the very option of independence. In the process neutrosophic components. According to Smarandache [7] of formulating the payoff matrix for game theory model, neutrosophic probability is useful to events that are the combination (I, I) will hopefully resolve the conflict i.e. shrouded in a veil of indeterminacy like the actual implied maintaining the status quo along the LOC with some volatile of long-term options. Bhattacharya et al.y applied border adjustments favorable to Pakistan. Otherwise, the the concept of neutrosophic probability in order to the proposed model offers the solution wich state that both formulate neutrosophic game theretic model [86] to Israel the countries will continue the conflict indefinitely. The –Palestine conflict. It is worthy of mention that the application of game theoretic method to the ongoing Indo- proposed approach uses a subset-approximation for truth- Pak conflict over J&K is based on identifying and values, indeterminacy and falsity-values. It is capable of evaluating the best options that each side has and is trying providing a better approximation than classical probability to achieve chosen options. The Simla Agreement 1972 is to uncertain events. used as an instance with Pakistan being left to choose Therefore, the neutrosophic evaluative function for between two mutually exclusive options. Pakistan at time t, JN(P, t) will be represented by the The solution reflects the real facts that Pakistan does not neutrosophic set membership function μ ()x as want to ever agree to have full compliance with the Simla J N ()Ρ, t follows: Agreement 1972, as she will see always herself worse off (.5,1),for OWorst < x < OBest , x ∈T that way. Realizing that Pakistan will never actually μ ()x = .5,for x = O , x ∈T comply with Simla Agreement 1972, India will find her J N ()Ρ, t Worst 0, for x < OWorst , x ∈T best interest to continue the status quo with an ongoing conflict, as she will see herself ending up on the worse end of the bargaining if she chooses to apply any other strategy. It is experienced that none of the strategy vectors available On the other hand, the neutrosophic evaluative judgment to either side will remain temporally stationary due to function for India at time t, JN(I, t) will be represented by action and reaction of militant groups and Indain securty the neutrosophic set membership function μ (y) as forces in J&K. Moreover, there exists a broad variety of JN ()Ι,t follows: ambiguities concerning primary goals of the two countries μ (y) = and the strategies they adopt to achieve those goals. JN ()Ι,t Due to the impact of globalization, people have to interact 1, for y ≥ ΘBest , y∈F ∈()0.5,1 , for Θ < y < Θ , y∈F with people from other countries. In the days of cross- Worst Best border strategic alliances and emphasis on various groups, 0.5, for y = Θ , y∈F Worst it would really be tragic if other nations remain standstill 0, for y ≤ Θ , y∈F Worst or ignore the conflict. The government of India fails to solve its own problems of northeast states such as Relying on three-way classification of neutrosophic Nagaland, Assam, Mizoram, and Arunachal Pradesh. On semantic space, it is t %true in sub-space that bilateral the other hand, Pakistan is constantly facing with the negotiation will produce a favourable outcome within the myriad problems of democracy and absence of it. Pakistan evaluative judgment space of the Pakistan while it is f % fails to curve insurgencies in Balochistan, the largest false in the sub-space F that the outcome will be favorable province of Pakistan having resource of natural gas and within the evaluative judgment space of the Pakistan. mineral and sparsely populated. We have discussed the However, there is an i % indeterminacy in sub-space I development of the Jammu and Kashmir conflict. We have whereby the nature of the outcome may be neither examined the various efforts by Indo-Pak and other favorable nor unfavorable within the evaluative judgment countries to resolve the grave conflict. The effect of space of either competitor. nuclear power acquisition by both the countries,9/11 terror attack on USA and USA and its allies invasion in Iraq. The 5 Conclusion most beneficiary party of Jammu and Kashmir conflict is We have discussed the crisis dynamics of the continuing the republic of China. It invaded about 38000 square Indo-Pak conflict over J&K. we have briefly examined the kilometers territory from Indian ruled Jammu and Kashmir efforts made by various study groups and persons and in 1962 war and later Pakistan ceded 5180 square Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 99 Neutrosophic Sets and Systems, Vol. 2, 2013 kilometers areas to Beijing under a 1963 pact. The ongoing was a place of religious center and different people with conflict between and Pakistan reflects the fact that both the their different faiths live together peacefully. But, poeple countries are incapable of solving J&K conflict and other of J&K are at present neutrosophically divided in the core issues. At the international level, third party question of independent of J&K. The disintegration of arrangements can be established with the participation of India based on two nation theory does not provide any an intervening state, group of states, or international good for the people of Indian subcontinent. So people of organizations in order to play a crucial role in helping to Indian subcontinent can neutrosophically hope that India overcome a strategic impasse. The question remains and Pakisatn will rethink their decision of partition based whether India and Pakistan would accept third party on vaguely defined two nation theory. Rather India and mediation. Limits on international direct intervention in the Pakistancan form a union of indepent states by considering J&K grave conflict have historically had to do with India’s their origin, culteral heristage, common interest, blood insistence that the 1972 Simla Agreement between India relation. If East Germany and West Germany are able to and Pakistan is sufficient to deal with the issue bilaterally. get united, why not the subcontinent? According to It is to noted that when well conceived and pragmatic, neutrosophy nothing is impossible. So according to India has come to appreciate mediation by international neutrosophy, the resolution of J&K is neutrosophically third party specially USA mediation in Kargil crisis in possible. The present paper provides the conceptual 1999 and 2001-2002 mobilization crisis. framework of neutrosophic game theroetic model of the μ ()x or μ (y) would be interpreted as Pakistan’s complex J&K conflict hoping that neotrosophic linear J N ()Ρ t, J N ()Ι t, (or India’s) degree of satisfaction with the negotiation programming will be able to solve the problem in near based settlement. It is likely that Pakistan authority’s future. ultimate objective is to annex J&K and if that is the case then of course there will always be an unbridgeable References incongruence between μ ()x and μ (y) due to [1] M. Intriligator. Research on conflict theory, Journal of J N ()Ρ t, J N ()Ι t, mutually inconsistent evaluative judgment spaces between Conflict Resolution 26(2) (1982), 307-327. India and Pakistan to the conflict. Therefore, for any form [2] G, Anandalingam, And V. Apprey. Multilevel of negotiation in order to produce a positive result, the programming and conflict resolution, European Journal of first and foremost requirement should be to make the Operational Research, 51(1991), 233-247. evaluative judgment spaces consistent. On the other hand, [3] H.V. Stackelberg. The Theory of the Market Economy, Oxford University Press, Oxford, 1952. if the evaluative judgment spaces are inconsistent, the [4] Y.C. Ho, P.B. Luh, and R. Muralidharan. Information negotiation space will generate into a null set at the very structure, Stackelberg games, and incentive controllability. onset of the bargaining process, thereby pre-empting any IEEE Transactions on Automatic Contorl, 26(2) (1981), equilibrium solution different from the status quo. Since 454-460. these spaces are not crisp, according to Reiter [87] they are [5] P. Yakir. The conflict between Israel, and the malleable to some extent. Therefore, even while retaining Palestinians: a rational analysis. Jerusalem their core forms, subtle changes could be induced for Letters/Viewpoints, No, 448, 22 Shvat 5761, 15 February making these spaces workably consistent. Then the goal of 2001. the mediator should be such that it will allow India and [6] S. Pramanik and T.K. Roy. Game theoretic model to the Pakistan to redefine their primary objectives without Jammu-Kashmir conflict between India and Pakistan. necessarily feeling that such redefinition itself reflects a International Journal of Mathematical Archive (IJMA), 4(8) (2013), 162-170. concession. When this type of redefinition of primary objectives has been achieved, the evaluative judgment [7] F. Smarandache. A unifying field in logics. neutrosophy: spaces generate a negotiation space that will not become neutrosophic probability, set and logic.: American Re- null ab initio. However, it should be mentioned that there search Press, Rehoboth, USA, 1999, 2000, 2003, 2005. exists also an indeterminate aspect to any process of [8] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunder- mediated bilateral dialogues between the two countries raman. Single valued neutrosophic sets, Multispace and Multistructure, 4(2010), 410–413. becasue of the catalyzation effect brought about the subjective utility preferences of the mediator. [9] W.W. Lawrence.. The Valley of Kashmir, London, UK, We build on an earlier attempted justification of a game 1895. theoretic explanation of the Indo-Pak conflict over J&K by [10] A. Durand. The Making of a Frontier, London, UK, 1900, Pramanik and Roy [6] and go on to argue in favor of a pp.65-66. neutrosophic adaptation of the standard 2x2 zero-sum [11] S. Sarkar. Modern India, New Delhi, India, 1993, pp.8- 9. [12] S. Bose. The Challenge in Kashmir, New Delhi, India, game theoretic model in order to identify an optimal 1997, pp.24-25. outcome. We hope that the concept of the interval [13] B. Puri. Kashmir Towards Insurgency, New Delhi, India neutrosophic set [88] can be used to formulate interval 1993. neutrosophic game theoretic model of the Indo-Pak [14] Ghulam Abbas speech, Dawn 10 August 1946. conflict in more general way. From ancient period J&K Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 100 Neutrosophic Sets and Systems, Vol. 2, 2013 [15] White Paper on Indian States (Ministry of States [52] Indo-Pakistan Ceasefire Agreement on Kutch, Asian Government. of India 1950), New Delhi, India, 1950, p.28. Recorder, Vol. XI 29), 6555. [16] Mountbatten, Time to Look Forward, Speeches 1947-48, ( (London, 1949), pp.13-18. [53] The Dawn, 11 August 1965. [17] M.A.K. Azad.. India wins freedom. Orient Longman [54] S/Res/209, (1965), 4 September 1965. Limited, Hyderabad, 1988, pp.270-276. [55] S/Res./210 (1965), 5 September 1965. [18] S. Gupta. Kashmir: a study in India-Pak relations, Asia [56] H.R. Gupta. India-Pakistan war 1965, Vol.II, Delhi, India, Publishing House, Calcutta, 1966. 1968, pp. 226-227. [19] Government of India Act, 1935, 2 Aug.1935, reprinted in [57] S/6683, 16 September 1965. Philips, C.H., H.L. Singh and B.N. Pandey, ‘The [58] S. Chopra. UN Mediation in Kashmir- A Study in Power evolution of India and Pakistan 1858-1947: Selective Politics, Kurukshetra, India, 1971,pp. 225-233. Documents’, Oxford University Press, London, p.321. [59] A. Raju. Third-Generation Indian Perceptions of the [20] Ranbir, 20 February 1947. Kashmir Issue, Regional Centre for Strategic Studies, [21] Indian Annual Register, vol.1, June 15, 1947, p.214. Colombo, Srilanka, 2001, p.15. [22] Khidmat, Srinagar, 2 July 1947. [60] S/10410/3 Dec. 1971, also S/10410/ADD.4. Dec.1971. [23] M.C. Mahajan. Looking Back, Bombay, India, 1963, [61] S.C.O.R., 1606 meeting, S/PV, 4 December 1971, pp. 42, p.265. 74-75. [24] V.K. Menon. Kashmir: speeches in the Security Council, [62] US Seventh Fleet Ordered oft Bay of Bengal, Asian January-February 1957, Publications Division,New Delhi, Recorder, XVIII ( 3),, p. 10573. India, 1957,pp.8-9. [63] V.S. Budhraj. Soviet Union and the Hindustan Sub- [25] C. Das Gupta. War and diplomacy in Kashmir, 1947-48, continent, Bombay, India, 1973, p. 234. Sage Publication, New Delhi, India, 2002. [64] Asian Recorder, XVIII(3), P. 1671. [26] L. Birwood. Two nations and Kashmir, Robert Hale, [65] A. Appadorai, and M. S. Rajan. India’s foreign policy and London,UK, 1956, p.77. relations, South Asian Publishers, New Delhi, India, [27] S. L. Poplai (Ed), Selected documents on Asian affairs: 1985,p.87. India 1947-50, Oxford University Press, Bombay, 1959, [66] Document -S/PV /21 Dec. 1971. Vol. I, p. 75. [67] http://www.jammu-kashmir.com/documents/simla.html. [28] P.L. Lakhanpal. Essential documents: notes [68] Simla Agreement Comes into Force ,Asian Recorder, on the Kashmir Dispute, Council on World XVIII(34), 1972, p.10936. Affairs, New Delhi, India,1965,p.57. [69] Delineating Maps Exchanged with Pakistan , Asian [29] M. Cohen, Thunder over Kashmir, Orient Longmans, Recorder, XIX(1), p. 11157. Calcutta, India, 1955, p.2. [70] Asian Recorder, Vol. XIX, No. 2, “Troops Withdrawal by [30] C. Johnson. Mission with Mountbatten, Robert Hale, India and Pakistan”, p. 11172. London, UK, 1951, pp.225-229. [71] Recognition of Bangladesh, Asian Recorder, XX, p. [31] H.V. Hodson. The great divide: Britain, India, Pakistan, 11903. London, UK, 1969. [72] Repatriation of POW’s Completed , Asian Recorder, [32] S/628, 2 January 1948. XX(21), p. 12011. [33] U.N. Doc. S/646/Corr. 1 (1948), Letter from the Minister [73] V.R. Trivedi, Facts about Kashmir, New Delhi, India, of Foreign Affairs of Pakistan to the Secretary General 1990, (Jan. 15, 1948). [74] Events of the Year, Kashmir Times 31 December 1990. [34] S/726, 21 April 1948. [75] U. S. State Department, Country, Report on Human rights [35] S/819, 3 June, 1948. Practices for 1981-2001, Washington D.C.: U. S. [36] S/1100, Para 75, 9 November, 1948. Government Printing Press, Amnesty International [37] S/1196, 10 January 1949. “ Arms and Abuses in Indian Punjab and Kashmir,” [38] S.C. Res. 91, U.N. SCOR, 6th Sess., 539th Mtg., U.N. September 1994. Doc S/2017/Rev.1 (1951). [76] India Abroad, 26 February, 1999, p. 4. [39] Annex 26 of UNCIP Third Report-S/1430 Add 1 to 3. [77] S. Ganguly, Conflict Unending, Columbia University Press, New York, USA, pp. 114-133. [40] S/1430/Add. 1, Annex 17. ~ [41] S/1430/Add. 1, Annex 20. [78] http://www1.appstate.edu/ stefanov/kashmir%20 [42] Minority Report of the Czech delegation, S/1430/Add. 3. Between%20and%Pakistan.pdf. [43] S.C.O.R. 457th meeting, on 22 December, 1949. [79] P. Musharraf. In the line of fire: a memoir. Free Press, p. [44] S/1453, Letter to General McNaughtan from Zafrulla 303. Khan, 13 January 1950. [80] The Constitution of India, Publication Division, [45] S/1469, 14 March 1950. Government of India, New Delh, India, 1990, pp.111-112. [46] S/1791. [81] http://kashmir.ahrchk.net/mainfile.php/after1947/43/. [47] U.N. Doc. S/3922 (1957). [82] http://www.hrw.org/reports/2006/pakistan0906/4.htm#_ft [48] S/2967, 27 March 1953. n38. [49] A. Sayed. Sino-Pakistan agreement: an overview, [83] E.N. Barron, Game theory an introduction, John Wiley Pakistan Horizon, XXII( 3) (1969), 11. Sons, Inc, Hoboken, New Jersy, 2007. [50] S/6291, 20 April 1965. [84] T.R. Burns,and E. Rowzkowska. Fuzzy games and [51] R.P. Anand. The Kutch award. India Quarterly 24 (1963), equilibria: the perspective of the general theory of 184-185. games on Nash and normative equilibria”, In: S. K. Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir 101 Neutrosophic Sets and Systems, Vol. 2, 2013 Pal, L. Polkowski, and A. Skowron, (eds.) Rough- [87] R. Reiter. A logic for default reasoning, Artificial Neuro Computing: Techniques for Computing with Intelligence, 13(1980), 81-132. Words, Springer-Verlag, Berlin/London, 2002. [88] H. Wang, F. Smarandache, Y. Q. Zhang, and R. [85] L.A. Zadeh. Fuzzy Sets, Information and Control 8 Sunderraman, Interval neutrosophic sets and logic: (1965),338-353. theory and applications in computing. Hexix Book [86] S. Bhattacharya, F. Smarandache, and M. Khoshne- Series No.5, 2005. visan. The Israel-Palestine question – a case for ap- plication of neutrosophic game theory. http://fs.gallup.unm.edu/ScArt2/Political.pdf. Received: December 21st, 2013. Accepted: January 19th, 2014 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir Neutrosophic Sets and Systems, Vol. 2, 2014 102 Entropy Based Grey Relational Analysis Method for Multi- Attribute Decision Making under Single Valued Neutrosophic Assessments Pranab Biswas1*, Surapati Pramanik2, and Bibhas C. Giri3 1*Department of Mathematics, Jadavpur University, Kolkata,700032, India. E-mail: [email protected] 2 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, 743126, India. Email: [email protected] 3Department of Mathematics, Jadavpur University, Kolkata,700032, India. Email:[email protected] Abstract. In this paper we investigate multi-attribute decision relational analysis method to neutrosophic environment and making problem with single-valued neutrosophic attribute values. apply it to multi-attribute decision making problem. Information Crisp values are inadequate to model real life situation due to entropy method is used to determine the unknown attribute imprecise information frequently used in decision making weights. Neutrosophic grey relational coefficient is determined process. Neutrosophic set is one such tool that can handle these by using Hamming distance between each alternative to ideal situations. The rating of all alternatives is expressed with neutrosophic estimates reliability solution and the ideal single-valued neutrosophic set which is characterised by neutrosophic estimates un-reliability solution. Then neutrosophic truth-membership degree, indeterminacy-membership degree, relational degree is defined to determine the ranking order of all and falsity-membership degree. Weight of each attribute is alternatives. Finally, an example is provided to illustrate the completely unknown to decision maker. We extend the grey application of the proposed method. Keywords: Neutrosophic set; Single-valued neutrosophic set; Grey relational analysis; Information Entropy; Multi-attribute decision making. 1 Introduction such tool that utilizes this impreciseness in a mathematical form. MADM with imprecise information can be modelled Multiple attribute decision making (MADM) problems in quite well by using fuzzy set theory into the field of the area of operation research, management science, decision making. economics, systemic optimization, urban planning and many other fields have achieved very much attention to the Bellman and Zadeh [7] first investigated decision making researchers during the last several decades. It is often used problem in fuzzy environment. Chen [8] extended one of to solve various decision making and/or selection problems. known classical MADM method, technique for order These problems generally consist of choosing the most preference by similarity to ideal solution (TOPSIS). He desirable alternative that has the highest degree of developed a methodology for solving multi-criteria satisfaction from a set of alternatives with respect to their decision making problems in fuzzy environment. Zeng [9] attributes. In this approach, the decision makers have to solved fuzzy MADM problem with known attribute weight provide qualitative and/ or quantitative assessments for by using expected value operator of fuzzy variables. determining the performance of each alternative with However, fuzzy set can only focus on the membership respect to each attribute, and the relative importance of grade of vague parameters or events. It fails to handle non- evaluation attribute. membership degree and indeterminacy degree of imprecise In classical MADM methods, such as TOPSIS (Hwang & parameters. Yoon [1]), PROMETHEE (Brans et al. [2]), VIKOR (Op- Atanassov [10] introduced intuitionistic fuzzy set (IFS). It ricovic [3-4]), ELECTRE (Roy [5]) the weight of each at- is characterized by the membership degree, tributes and rating of each alternative are naturally con- non-membership degree simultaneously. Impreciseness of sidered with crisp numbers. However, in real complex the objectives can be well expressed by using IFS than situation, decision maker may prefer to evaluate the fuzzy sets (Atanassov [11]). Therefore it has gained more attributes by using linguistic variables rather than exact and more attention to the researchers. Boran et.al [12] values due to his time pressure, lack of knowledge and lack extended the TOPSIS method for multi-criteria of information processing capabilities about the problem intuitionistic decision making problem. Z.S. Xu[13] domain. In such situations, the preference information of studied fuzzy multiple attribute decision making problems, alternatives provided by the decision maker may be vague, in which all attribute values are given as intuitionistic imprecise or incomplete. Fuzzy set (Zadeh [6]) is one of fuzzy numbers and the preference information on Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 103 alternatives can be provided by the decision maker. Z. Xu completely unknown to decision maker. Entropy method is [14] proposed a solving method for MADM problem with used for determining the unknown attribute weights. In interval-valued intuitionistic fuzzy decision making by this modified GRA method, the ideal neutrosophic using distance measure. estimates reliability solution and the ideal neutrosophic estimate un-reliability solution has been developed. In IFSs, sum of membership degree and non-membership Neutrosophic grey relational coefficient of each alternative degree of a vague parameter is less than unity. Therefore, a is determined to rank the alternatives. certain amount of incomplete information or indeterminacy arises in an intuitionistic fuzzy set. It cannot handle all In order to do so, the remaining of this paper is organized types of uncertainties successfully in different real physical as follows: Section 2 briefly introduce some preliminaries problems such as problems involving incomplete relating to neutrosophic set and the basics of single-valued information. Hence further generalizations of fuzzy as well neutrosophic set. In Section 3, Hamming distance between as intuitionistic fuzzy sets are required. two single-valued neutrosophic sets is defined. Section 4 represents the model of MADM with SVNSs and Florentin Smarandache [15] introduced neutrosophic set discussion about modified GRA method to solve MADM (NS) and neutrosophic logic. It is actually generalization of problems. In section 5, an illustrative example is provided different type of FSs and IFSs. The term “neutrosophy” to show the effectiveness of the proposed model. Finally, means “knowledge of neutral thought”. This “neutral” section 6 presents the concluding remarks. concept makes the differences between NSs and other sets like FSs, IFSs. Wang et al. [16] proposed single-valued 2 Preliminaries of Neutrosophic sets and Single neutrosophic set (SVNS) which is a sub-class of NSs. SVNS is characterized by truth membership degree (T), valued neutrosophic set indeterminacy membership degree (I) and falsity Neutrosophic set is a part of neutrosophy, which studies membership degree (F) that are independent to each other. the origin, nature, and scope of neutralities, as well as their This is the key characteristic of NSs other than IFSs or interactions with different ideational spectra (Smarandache fuzzy sets. [15]), and is a powerful general formal framework, which Such formulation is helpful for modelling MADM with generalizes the above mentioned sets from philosophical neutrosophic set information for the most general point of view. Smarandache [15] gave the following ambiguity cases, including paradox. The assessment of definition of a neutrosophic set. attribute values by the decision maker takes the form of 2.1 Definition of neutrosophic set single-valued neutrosophic set. Ye [17] studied multi- criteria decision making problem under SVNS Definition 1 Let X be a space of points (objects) with environment. He proposed a method for ranking of generic element in X denoted by x. Then a neutrosophic set alternatives by using weighted correlation coefficient. Ye A in X is characterized by a truth membership function TA, [18] also discussed single-valued neutrosophic cross an indeterminacy membership function IA and a falsity membership function FA. The functions TA, IA and FA are entropy for multi-criteria decision making problems. He - + used similarity measure for interval valued neutrosophic real standard or non-standard subsets of] 0 , 1 [ that is - + - + - + set for solving multi-criteria decision making problems. TA : X → ]0 , 1 [ ; IA : X → ]0 , 1 [; FA : X → ]0 , 1 [ Grey relational analysis (GRA) is widely used for MADM It should be noted that there is no restriction on the sum of problems. Deng [19-20] developed the GRA method that is T (x), I (x), F (x) i.e. 0- ≤ T (x) + I (x) +F (x) ≤ 3+ applied in various areas, such as economics, marketing, A A A A A A personal selection and agriculture. Zhang et al. [21] Definition 2 The complement of a neutrosophic set A is discussed GRA method for multi attribute decision making denoted by Ac and is defined by with interval numbers. An improved GRA method )x(T = + − )x(T}1{ ; + −= )x(I}1{)x(I ; proposed by Rao & Singh [22] is applied for making a Ac A Ac A decision in manufacturing situations. Wei [23] studied the + −= )x(F}1{)x(F GRA method for intuitionistic fuzzy multi-criteria decision Ac A making. Therefore, it is necessary to pay attention to this Definition 3 (Containment) A neutrosophic set A is issue for neutrosophic environment. contained in the other neutrosophic set B, A ⊆ B if and only if the following result holds. The aim of this paper is to extend the concept of GRA to ≤ )x(Tinf)x(Tinf , ≤ x(Tsup)x(Tsup ) (1) develop a methodology for solving MADM problems with A B A B single valued neutrosophic set information. The A ≥ B )x(Iinf)x(Iinf , A ≥ B )x(Isup)x(Isup (2) information taken from expert’s opinion about attribute A ≥ B )x(Finf)x(Finf , A ≥ B )x(Fsup)x(Fsup (3) values takes the form of single valued neutrosophic set. It for all x in X. is assumed that the information about attribute weights is Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 104 2.2 Some basics of single valued neutrosophic ~ Definition 7 Two single valued neutrosophic sets NA and sets (SVNSs) ~ ~ ~ ~ ~ NB are equal, i.e. NA = NB , if and only if NA ⊆NB and In this section we provide some definitions, operations and ~ ~ properties about single valued neutrosophic sets due to NA ⊇ NB . ~ Wang et al. [16]. It will be required to develop the rest of Definition 8 (Union) The union of two SVNSs NA and the paper. ~ ~ ~ ~ ~ NB is a SVNS NC , written as NC = NA ∪NB . Its truth Definition 4 (Single-valued neutrosophic set). Let X be a membership, indeterminacy-membership and falsity mem- universal space of points (objects), with a generic element ~ ~ bership functions are related to those of N and N by of X denoted by x. A single-valued neutrosophic set A B ~ ~ ~ ~ ))x(T),x(T(max=)x(T ; N ⊂X is characterized by a true membership NC NA NB function ~ )x(T , a falsity membership function ~ )x(F and ~ ~ ~ ))x(I),x(I(max=)x(I ; N N NC NA NB an indeterminacy function ~ )x(I with ~ )x(T , ~ )x(I , ~ ~ ~ ))x(F),x(F(min=)x(F for all x in X. N N N NC NA NB ~ )x(F ∈ [0, 1] for all x in X. N Definition 9 (Intersection) The intersection of two SVNSs ~ ~ ~ ~ ~ ~ ~ When X is continuous a SVNSs N can be written as NA and NB is a SVNS NC , written as NC = NA ∩NB , ~ N = ~ ~ ~ ,x)x(F),x(I),x(T ∈∀ .Xx whose truth membership, indeterminacy-membership and ∫ N N N ~ x falsity membership functions are related to those of ~ NA and when X is discrete a SVNSs N can be written as ~ and N by ~ ~ ~ ))x(T),x(T(min=)x(T ; ~ m B NC NA NB N = ∑ ~ ~ ~ x/)x(F),x(I),x(T , ∈∀ .Xx N N N ~ ~ ~ ))x(I),x(I(min=)x(I ; =1i NC NA NB Actually, SVNS is an instance of neutrosophic set which ~ ~ ~ ))x(F),x(F(max=)x(F for all x in X. can be used in real life situations like decision making, sci- NC NA NB entific and engineering applications. In case of SVNS, the 3 Distance between two neutrosophic sets. degree of the truth membership ~ )x(T , the indeterminacy N Similar to fuzzy or intuitionistic fuzzy set, the general membership ~ )x(I and the falsity membership ~ )x(F N N SVNS having the following pattern values belong to [0, 1] instead of non standard unit inter- ~ - + = ~~ ~ x:))x(F),x(I),x(T/(x{( ∈X}. For finite SVNSs val] 0 , 1 [as in the case of ordinary neutrosophic sets. N N N N ~ It should be noted that for a SVNS N , can be represented by the ordered tetrads: ~ 0 ≤ ~ + ~ + ~ )x(Fsup)x(Isup)x(Tsup ≤ 3 , ∀ ∈ .Xx (4) N = 1 ~ 1 ~ 1 ~ 1)),x(F),x(I),x(T/(x{( N N N N N N , ∀x ∈X and for a neutrosophic set, the following relation holds ~ ~ ~ ))}x(F),x(I),x(T/(x..., - + m N m N m N m 0 ≤ ~ ~ ~ )x(Fsup+)x(Isup+)x(Tsup ≤ 3 , ∀ ∈ .Xx (5) N N N Definition 10 Let For example, suppose ten members of a political party will ~ N = ~ ~ ~ )),x(F),x(I),x(T/(x{( critically review their specific agenda. Five of them agree A 1 NA 1 NA 1 NA 1 with this agenda, three of them disagree and rest of two ~ ~ ~ ))}x(F),x(I),x(T/(x..., members remain undecided. Then by neutrosophic notation n NA n NA n NA n ~ it can be expressed as 3.0,2.0,5.0x . N = ~ ~ ~ )),x(F),x(I),x(T/(x{( B 1 NB 1 NB 1 NB 1 ~ and (6) Definition 5 The complement of a neutrosophic set N is ~ ~ ~ ))}x(F),x(I),x(T/(x..., ~ n NB n NB n NB n denoted by N c and is defined by be two single-valued neutrosophic sets (SVNSs) in X= {x1, ~ )x(T = ~ )x(F ; ~ =)x(I 1 − ~ )x(I ; ~ =)x(F ~ )x(T c N c N c N x2,…, xn). N N N ~ ~ Then the Hamming distance between two SVNSs NA and Definition 6 A SVNS NA is contained in the other SVNS ~ ~ ~ ~ NB is defined as follows: N , denoted as N ⊆ N , if and only if B A B ⎧ ⎫ ~ 1 ~ 1 ~ 1 −+− ~ 1)x(I)x(I)x(T)x(T ⎛ ~~ ⎞ n ⎪ NA NB NA NB ⎪ d ~ , = ∑ ~ )x(T ≤ ~ )x(T ; ~ ≥ ~ )x(I)x(I ; ~ )x(F ≥ ~ )x(F N ⎜ A NN B ⎟ ⎨ ⎬ (7) NA NB NA NB NA NB ⎝ ⎠ =1i ⎪ ⎪ ~ 1 −+ ~ 1)x(F)x(F ∈∀ .Xx ⎩⎪ NA NB ⎭⎪ and normalized Hamming distance between two SVNSs ~ ~ NA and NB is defined as follows: Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 105 ~ ~ N ⎛ ⎞ lational coefficients, the grey relational degree between the d ~ ⎜ ,NN ⎟ = N ⎝ A B ⎠ reference sequence and every comparability sequences is 1 n calculated. If an alternative gets the highest grey relational ∑ ~ ~ ~ ~ ~ −+−+− ~ )x(F)x(F)x(I)x(I)x(T)x(T {}N 1 N 1 N 1 N 1 N 1 N 1 n3 =1i A B A B A B grade with the reference sequence, it means that the (8) comparability sequence is most similar to the reference with the following two properties sequence and that alternative would be the best choice ~ ~ (Fung [24]). The steps of improved GRA under SVNS are 1. 0 ≤ d ~ ( ,NN )≤ n3 (9) N A B described below: N ~ ~ 2. 0 ≤ d ~ ( ,NN )≤ 1 (10) N A B Step 1 Determine the most important criteria. Proof: The proofs are obvious from the basic definition Generally, there are many criteria or attributes in decision of SVNS. making problems where some of them are important and others may not be so important. So it is crucial, to select 4 GRA method for multiple attribute decision the proper criteria or attribute for decision making making problem with single valued neutrosophic situations. The most important criteria may be chosen with information help of experts’ opinions or by some others method that Consider a multi-attribute decision making problem with m are technically sound. alternatives and n attributes. Let A1, A2, ..., Am and C1, Step 2 Data pre-processing C2, ..., Cn denote the alternatives and attributes respectively. The rating describes the performance of alternative A Assuming for a multiple attribute decision making problem i having m alternatives and n attributes, the general form of against attribute Cj. For MADM weight vector W = {w1, w ,...,w } is assigned to the attributes. The weight > 0w decision matrix can be presented as shown in Table-1. It 2 n j may be mentioned here that the original GRA method can ( j = 1, 2, ..., n) reflects the relative importance of attributes effectively deal mainly with quantitative attributes. Cj ( j = 1, 2, ..., m) to the decision making process. The However, there exists some difficulty in the case of weights of the attributes are usually determined on qualitative attributes. In the case of a qualitative attribute subjective basis. They represent the opinion of a single (i.e. quantitative value is not available); an assessment decision maker or synthesize the opinions of a group of value is taken as SVNSs. experts using a group decision technique, as well. The values associated with the alternatives for MADM Step 3 Construct the decision matrix with SVNSs problems presented in the decision table. For multi-attribute decision making problem, the rating of alternative Ai (i = 1, 2,…m ) with respect to attribute Cj Table 1 Decision table of attribute values (j = 1, 2,…n) is assumed as SVNS. It can be represented with the following looks C1 C2 ... Cn ⎧C C C ⎫ A d...dd A = 1 , 2 ,..., n C: ∈C . 1 ⎡ 11 12 n1 ⎤ i ⎨ F,I,T F,I,T F,I,T j ⎬ ⎢ ⎥ ⎩ 1i1i1i 2i2i2i ininin ⎭ A2 ⎢ 21 22 d...dd n2 ⎥ Cj ⎢ ⎥ = j ∈ CC: for j = 1, 2,…, n. = dD = ...... (11) F,I,T ijijij ij ×nm ⎢ ⎥ ⎢ ⎥ ...... ⎢ ⎥ Tij, Iij, Fij are the degrees of truth membership, degree of ⎢ ⎥ indeterminacy and degree of falsity membership of the Am ⎣ 1m 2m d...dd mn ⎦ alternative Ai satisfying the attribute Cj, respectively where ≤≤ 1T0 , ≤≤ 1I0 , ≤≤ 1F0 and ≤++≤ 3FIT0 . GRA is one of the derived evaluation methods for MADM ij ij ij ijijij based on the concept of grey relational space. The main The decision matrix can be taken in the form: procedure of GRA method is firstly translating the performance of all alternatives into a comparability sequence. This step is called data pre-processing. According to these sequences, a reference sequence (ideal target sequence) is defined. Then, the grey relational coefficient between all comparability sequences and the reference sequence for different values of distinguishing coefficient are calculated. Finally, based on these grey re- Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments 1 106 Neutrosophic Sets and Systems, Vol. 2, 2014 Table 2 Decision table with SVNSs ~ 2. (E NAi =1) if ~ = F,I,TD ijijij N ×nm ~ 1 ~ 1 ~ 1 = 5.0,5.0,5.0)x(F),x(I),x(T ∀ ∈Xx . C1 C2 Cn NA NA NA ⎡ ⎤ ~ ~ ~ ~ A 111111 121212 F,I,T...F,I,TF,I,T n1n1n1 1 ⎢ ⎥ 3. (E ≥ (E) NN BiAi ) if NA is more uncertain than NB i.e. ⎢ F,I,T...F,I,TF,I,T ⎥ A2 212121 222222 n2n2n2 ~ + ~ ≤ ~ + ~ )x(F)x(T)x(F)x(T and ⎢ ⎥ NA 1 NA 1 NB 1 NB 1 = . ⎢ ...... ⎥ (12) ⎢ ⎥ ~ ~ ~ −≤− ~ )x(I)x(I)x(I)x(I . ⎢ ...... ⎥ N i N i N i N i ⎢ ⎥ A Ac B Bc A ⎢ F,I,T...F,I,TF,I,T ⎥ ~ ~ m ⎢ 1m1m1m 2m2m2m mnmnmn ⎥ 4. (E N = (E) N ) ∀ ∈Xx . ⎣ ⎦ iAi Ac Step 4: Determine the weights of criteria. In order to obtain the entropy value E j of the j-th attribute In the decision-making process, decision makers may often Cj ( j = 1, 2,…, n), equation (13) can be written as : face with unknown attribute weights. It may happens that m 1 C the importance of the decision makers are not equal. j 1E −= ∑()+ − ijiijiijiij i )x(I)x(I)x(F)x(T Therefore, we need to determine reasonable attribute n =1i weight for making a proper decision. Many methods are for i = 1, 2,..,m; j = 1, 2,…,n. (14) available to determine the unknown attribute weight in the It is also noticed that j ∈ ].1,0[E Due to Hwang and Yoon literature such as maximizing deviation method (Wu and [1], and Wang and Zhang [29] the entropy weight of the j- Chen [25]), entropy method ( Wei and Tang [26]; Xu and th attibute Cj is presented by Hui [27]), optimization method (Wang and Zhang [28-29]) − E1 j etc. In this paper, we propsoe information entropy method. w j = n (15) ∑ − j )E1( 4.1 Entropy method: =1j T Entropy has an important contribution for measuring We get weight vector W= ( w1, w2,…,wn) of attributes Cj uncertain information (Shannon [30-31]). Zadeh [32] n introduced the fuzzy entropy for the first time. Similarly (j = 1,2,…, n) with j ≥0w and ∑ j = 1w =1j Bustince and Burrillo [33] introduced the intuitionistic fuzzy entropy. Szmidt and Kacprzyk [34] extended the Step 5. Determine the ideal neutrosophic estimates axioms of De Luca and Termini’s [35] non-probabilistic reliability solution (INERS) and the ideal neutrosophic entropy in the setting of fuzzy set theory into intuitionistic estimates un-reliability solution (INEURS) for fuzzy information entropy. Vlachos and Sergiadis [36] also neutrosophic decision matrix. studied intuitionistic fuzzy information entropy. Majumder For a neutrosophic decision making matrix ~~ ]q[=D N Nij n×m and Samanta [37] developed some similarity and entropy measures for SVNSs. The entropy measure can be used to = F,I,T ijijij , Tij, Iij, Fij are the degrees of membership, n×m determine the attributes weights when it is unequal and degree of indeterminacy and degree of non membership of completely unknown to decision maker. Hwan and Yoon the alternative A of A satisfying the attribute C of C. The (1981) developed a method to determine the attribute i j neutrosophic estimate reliability estimation can be easily weights based on information entropy. determined from the concept of SVNS cube proposed by In this paper we propose an entropy method for Dezert [38]. determining attribute weight. According to Majumder and Definition 11 From the neutrosophic cube, the Samanta [37], the entropy measure of a SVNS ~ membership grade represents the estimates reliability. The NA = ~ 1 ~ 1 ~ 1)x(F),x(I),x(T is NA NA NA ideal neutrosophic estimates reliability solution (INERS) +++ + m ~~ ~ ~ ~ 1 C = ]q,...,q,q[Q is a solution in which every (E N 1) −= ~ + ~ ~ − ~ )x(I)x(I)x(F)x(T N 1 NN 2 Nn Ai ∑ ()N i N i N i N i (13) n =1i A A A A + + + + + component ~ j j F,I,T=q j , where j ij},T{max=T which has the following properties: Nj i ~ ~ + + 1. (E N = 0) if N is a crisp set and ~ = 0)x(I j = ij}I{minI and j = ij}F{minF in the neutrosophic Ai A NA i i i ∈∀ Xx . decision matrix ~ = F,I,TD for i = 1, 2, .., m; j = 1, N ijijij ×nm 2, …, n. Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 1 07 n Definition 12 Similarly, in the neutrosophic cube − − and i ∑ w χ=χ ijj for i = 1, 2,…, m. (19) maximum un-reliability happens when the indeterminacy =1j membership grade and the degree of falsity membership Step 8. Calculate the neutrosophic relative relational reaches maximum simultaneously. Therefore, the ideal degree. neutrosophic estimates un-reliability solution (INEURS) We calculate the neutrosophic relative relational degree of −−− − = ~~ ~ ~ ]q,...,q,q[Q can be taken as a solution in the each alternative from ITFPIS with the help of following N NN N 1 2 n equations: − −−− − form ~ = F,I,Tq , where = },T{minT Nj jjj j ij i χ+ R = i , for i = 1, 2,…, m. (20) − i χ+χ −+ j ij}I{max=I and j = ij}F{maxF in the neutrosophic ii i i decision matrix ~ = F,I,TD for i = 1, 2,..,m; j = 1, Step 9. Rank the alternatives. N ijijij ×nm 2,…,n. According to the relative relational degree, the ranking order of all alternatives can be determined. The highest Step 6 Calculate neutrosophic grey relational value of Ri yields the most important alternative. coefficient of each alternative from INERS and INEURS. 5 . Illustrative Examples Grey relational coefficient of each alternative from INERS In this section, a multi-attribute decision-making problem is: is considered to demonstrate the application as well as the effectiveness of the proposed method. We consider the de- + + minmin ij maxmax Δρ+Δ ij cision-making problem adapted from Ye [39]. Suppose + ji ji ij =χ + + , where there is an investment company, which wants to invest a ij maxmax Δρ+Δ ij ji sum of money to the best one from these four possible + + alternatives (1) A1 is a car company; (2) A2 is a food Δ ij = ( ~ q,qd ~ ), for i= 1, 2,…,m. and j=1, 2,…,n. (16) j NN ij company; (3) A3 is a computer company; and (4) A4 is an Grey relational coefficient of each alternative from arms company. The investment company must take a INEURS is: decision according to the following three criteria: (1) C1 is the risk analysis; (2) C2 is the growth analysis; and (3) C3 − − minmin ij ρ+Δ maxmax Δ ij is the environmental impact analysis. Thus, when the four − ji ji − possible alternatives with respect to the above three criteria ij =χ − − , where, Δ ij = ij ρ+Δ maxmax Δ ij are evaluated by the expert, we can obtain the following ji single-valued neutrosophic decision matrix: − ~ ~ ( q,qd ), for i= 1, 2,…,m. and j=1, 2,…,n. (17) ~ ij NN j = F,I,TD ijijij = N ×34 C C C ρ ∈ [0, 1] is the distinguishable coefficient or the 1 2 3 A ⎡ 5.0,2.0,2.03.0,2.0,4.03.0,2.0,4.0 ⎤ identification coefficient used to adjust the range of the 1 ⎢ ⎥ comparison environment, and to control level of ⎢ ⎥ A2 .0,6.0 2.0,2.0,5.02.0,1.0,6.02.0,1 differences of the relation coefficients. When ρ =1, the ⎢ ⎥ (21) A ⎢ ⎥ comparison environment is unaltered; when ρ = 0, the 3 ⎢ 2.0,3.0,5.03.0,2.0,5.03.0,2.0,3.0 ⎥ A ⎢ ⎥ comparison environment disappears. Smaller value of 4 ⎣⎢ 2.0,3.0,4.02.0,1.0,6.01.0,0.0,7.0 ⎦⎥ distinguishing coefficient will yield in large range of grey relational coefficient. Generally, ρ = 0.5 is considered for Step1: Determine the weights of attribute decision- making situation. Entropy value Ej of the j-th ( j = 1, 2, 3) attributes can be Step 7. Calculate of neutrosophic grey relational determined from SVN decision matrix D ~ (21) and N coefficient. equation (14) as: E1 = 0.50; E2 = 0.2733 and E3 = 0.5467. Calculate the degree of neutrosophic grey relational Then the corresponding entropy weights w1, w2, w3 of all coefficient of each alternative from INERS and INEURS attributes according to equation (15) are obtained by w1 = using the following equation respectively: 3 n 0.2958; w = 0.4325 and w = 0.2697 such that = 1w . + + 2 3 ∑ j =1j i ∑w χ=χ ijj (18) =1j Step1: Determine the ideal neutrosophic estimates reliability solution (INERS): Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 108 ++++ = ~~ ~ ~ ]q,q,q[Q = Step 6: The ranking order of all alternatives can be N NNN 1 2 3 determined according the value of neutrosophic relational ⎡ ⎤ 1i 1i 1i 2i 2i 2i ,}F{min},I{min},T{max,}F{min},I{min},T{max degree i.e. >>> RRRR . It is seen that the highest ⎢ i i i i i i ⎥ 1324 ⎢ ⎥ ⎢ ⎥ value of neutrosophic relational degree is Rtherefore4 3i 3i 3i }F{min},I{min},T{max ⎢ i i i ⎥ ⎣ ⎦ A4 i.e. Arms Company is the best alternative for = [ 2.0,2.0,5.0,2.0,1.0,6.0,1.0,0.0,7.0 ] investment purpose. Step 2: Determine the ideal neutrosophic estimates 6 Conclusion un-reliability solution (INEURS): In practical applications for MADM process, the −−−+ = ~~ ~ ~ ]q,q,q[Q = assessments of all attributes are convenient to use the N 1 2 NNN 3 linguistic variables rather than numerical values. In most ⎡ ⎤ 1i 1i 1i 2i 2i 2i ,}F{max},I{max},T{min,}F{max},I{max},T{min ⎢ i i i i i i ⎥ ambiguity cases, SVNS plays an important role to model ⎢ ⎥ MADM problem. In this paper, we study about SVNS ⎢ ⎥ 3i 3i 3i }F{max},I{max},T{min based MADM in which all the attribute weight information ⎣⎢ i i i ⎦⎥ is unknown. Entropy based modified GRA analysis = [ 5.0,3.0,2.0,3.0,2.0,4.0,3.0,2.0,4.0 ] method is proposed to solve this MADM problem. Step 3: Calculation of the neutrosophic grey relational co- Neutrosophic grey relation coefficient is proposed for efficient of each alternative from INERS and INEURS. solving multiple attribute decision-making problems. Finally, an illustrative example is provided to show the By using Equation (16) the neutrosophic grey relational feasibility of the developed approach. This proposed coefficient of each alternative from INERS can be obtained method can also be applied in the application of the ⎡ 4000.05000.03636.0 ⎤ multiple attribute decision-making with interval valued ⎢ ⎥ neutrosophic set and to other domains, such as decision ⎢ 0000.10000.15714.0 ⎥ + =χ making, pattern recognition, medical diagnosis and as: []ij ×34 ⎢ ⎥ (22) ⎢ 8000.05714.03333.0 ⎥ clustering analysis. ⎢ ⎥ ⎣⎢ 6666.00000.10000.1 ⎦⎥ Similarly, from Equation (17) the neutrosophic grey References relational coefficient of each alternative from INEURS is [1] C. L. Hwang, and K. Yoon. Multiple attribute decision ⎡ 7778.00000.10000.1 ⎤ ⎢ ⎥ making: methods and applications: a state-of-the-art sur- ⎢ 3333.04667.04667.0 ⎥ − =χ vey, Springer, London (1981). []ij ×34 ⎢ ⎥ (23) ⎢ 3684.07778.07778.0 ⎥ ⎢ ⎥ [2] J.P. Brans, P. Vinvke, and B. Mareschal. How to select and ⎣⎢ 4111.04667.03333.0 ⎦⎥ how to rank projects: The PROMETHEE method, Euro- Step 4: Determine the degree of neutrosophic grey pean Journal of Operation Research, 24(1986), 228–238. relational co-efficient of each alternative from INERS and INEURS. The required neutrosophic grey relational [3] S. Opricovic. Multicriteria optimization of civil engineer- co-efficient corresponding to INERS is obtained by using ing systems, Faculty of Civil Engineering, Belgrade equations (18) as: (1998). + + + 1 =χ 43243.0 ; 2 =χ 87245.0 ; 3 =χ 56222.0 ; [4] S. Opricovic, and G. H. Tzeng. Compromise solution by + 4 =χ 91004.0 (24) MCDM methods: a comparative analysis of VIKOR and TOPSIS. European Journal of Operation Research, 156 and corresponding to INEURS is obtained with the help of equation (19) as: (2004), 445–455. − − − 1 =χ 9111.0 ; 2 =χ 4133.0 ; 3 =χ 6140.0 ; [5] B. Roy. The outranking approach and the foundations of − ELECTRE methods. Theory Decision, 31(1991), 49–73. 4 =χ 3978.0 (25) [6] L. A. Zadeh. Fuzzy Sets, Information and Control, Step 5: Thus neutrosophic relative degree of each 8(1965), 338-353. alternative from INERS can be obtained with the help of [7] R. Bellman, and L. A. Zadeh. Decision making in a fuzzy equation (20) as: R1= 0.31507; R2= 0.66949; R3= 0.54275 and R4= 0.68835. environment, Management Science, 17B (4) (1970), 141- 164. Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments Neutrosophic Sets and Systems, Vol. 2, 2014 109 [8] C. T. Chen. Extensions of the TOPSIS for group decision [20] J. L. Deng. The primary methods of grey system theory, making under fuzzy environment, Fuzzy Sets and Systems, Huazhong University of Science and Technology Press, 114(2000), 1-9. Wuhan (2005). [9] L. Zeng. Expected value method for fuzzy multiple [21] J. J. Zhang, D. S. Wu, and D. L. Olson. The method of attribute decision making, Tsinghua Science and grey related analysis to multiple attribute decision making Technology, 11 (2006) 102–106. problems with interval numbers. Mathematical and [10] K. T. Atanassov. Intuitionistic fuzzy sets, Fuzzy Sets and Computer Modelling, 42(2005), 991–998. Systems, 20 (1986), 87–96. [22] R. V. Rao, and D. Singh. An improved grey relational [11] K. T. Atanassov. On Intuitionistic fuzzy set theory, studies analysis as a decision making method for manufacturing in fuzziness and soft computing, Springer- Verlag, Berlin situations, International Journal of Decision Science, Risk (2012). and Management, 2(2010), 1–23. [12] F. E. Boran, S. Genc, M. Kurt, and D. Akay. A multi- [23] G. W. Wei. Grey relational analysis method for criteria intuitionistic fuzzy group decision making for intuitionistic fuzzy multiple attribute decision making. supplier selection with TOPSIS method, Expert Systems Expert systems with Applications, 38(2011),11671- with Applications, 36(8) (2009), 11363–11368. 11677. [13] Z. S. Xu. Models for multiple attribute decision-making [24] C.P. Fung. Manufacturing process optimization for wear with intuitionistic fuzzy information. International of Un- property of fibre-reinforced polybutylene terephthalate certainty, Fuzziness and Knowledge-Based Systems. composites with gray relational analysis, Wear, 254(2003), 15(2007), 285-297. 298–306. [14] Z. Xu. A method based on distance measure for interval- [25] Z.B. Wu, and Y.H. Chen. The maximizing deviation valued intuitionistic fuzzy group decision making, Infor- method for group multiple attribute decision making under mation Sciences, 180(2010), 181-190. linguistic environment, Fuzzy Sets and Systems, [15] F. Smarandache. A unifying field in logics. Neutrosophy: 158(2007), 1608–1617. Neutrosophic probability, set and logic. Rehoboth: Ameri- [26] C. Wei, and X. Tang. An intuitionistic fuzzy group can Research Press (1999). decision making approach based on entropy and similarity [16] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunder- measures, International Journal of Information Technology raman. Single valued neutrosophic sets. Multispace and and Decision Making, 10(6) (2011), 1111-1130. Multistructure, 4(2010), 410–413. [27] Z. Xu, and H. Hui. Entropy based procedures for [17] J. Ye. Multicriteria decision-making method using the intuitionistic fuzzy multiple attribute decision making. correlation coefficient under single-valued neutrosophic Journal of Systems Engineering and Electronics. 20(5) environment, International Journal of General Systems, (2009), 1001-1011. 42(4) (2013), 386-394. [28] J. Q. Wang, and Z. H. Zhang. Programming method of [18] J. Ye. Single valued neutrosophic cross entropy for multi-criteria decision-making based on intuitionistic fuzzy multicriteria decision making problems, Applied Ma- number with incomplete certain information, Control and thematical Modeling (2013), doi: decision, 23(2008), 1145-1148. 10.1016/j.apm.2013.07.020. [29] J. Q. Wang, and Z. H. Zhang. Multi-criteria decision- [19] J. L. Deng. Introduction to grey system theory, The Journal making method with incomplete certain information based of Grey System (UK), 1(1) (1989), 1–24. on intuitionistic fuzzy number. Control and decision. 24 (2009), 226-230. Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments 110 Neutrosophic Sets and Systems, Vol. 2, 2014 [30] C. E. Shannon. A Mathematical Theory of Communica- tions, The Bell System Technical Journal. 27(1948), 379- 423. [31] C. E. Shannon, and W. Weaver. The Mathematical Theory of Communications, The University of Illinois Press, Urbana. (1947). [32] L. A. Zadeh. Fuzzy sets and systems, in: Proceedings of the Symposium on Systems, Theory polytechnic Institute of Brooklyn, Newyork (1965), 29-37. [33] H. Bustince, and P. Burillo. Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems, 78 (1996), 305-316. [34] E. Szmidt, and J, Kacprzyk. Entropy on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 118 (2001), 467-477. [35] A. S. De Luca, S. Termini. A definition of non- probabilistic entropy in the setting of fuzzy set theory. Information Control, 20 (1972), 301-312. [36] I. K. Valchos, G. D. Sergiadis. Intuitionistic fuzzy information- a pattern recognition, Pattern Recognition Letters, 28 (2007), 197-206. [37] P. Majumder, S. K. Samanta. On similarity and entropy of neutrosophic sets, Journal of Intelligent and Fuzzy Systems (2013), doi: 10.3233/IFS-130810. [38] J. Dezert. Open questions in neutrosophic inferences, Multiple-Valued Logic: An International Journal, 8(2002), 439-472. [39] J. Ye. Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment. European Journal of Operational Research, 205(2010), 202–204. Received: December 31st, 2013. Accepted: January 10th, 2014 Pranab Biswas, Surapati Pramanik, Bibhas C. Giri, Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments 114 Neutrosophic Sets and Systems, Vol. 2, 2014 Neutrosophic Sets and Systems An International Journal in Information Science and Engineering Vol. 2, 2014 CONTENTS Shawkat Alkhazaleh and Emad Marei. Mappings on Neutrosophic Soft Classes / 3 Said Broumi and Florentin Smarandache. On Neutrosophic Implications / 9 Rıdvan Şahin. Neutrosophic Hierarchical Clustering Algoritms / 18 A. A. Salama, Florentin Smarandache, Valeri Kroumov. Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces / 25 Karina Pérez-Teruel and Maikel Leyva-Vázquez. Neutrosophic Logic for Mental Model Elicitation and Analysis / 31 A.A.A. Agboola, B. Davvaz. On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings / 34 Vasantha Kandasamy and Florentin Smarandache. Neutrosophic Lattices / 42 Jun Ye, Qiansheng Zhang. Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making / 48 Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Munazza Naz. Soft Neutrosophic Bigroup and Soft Neutrosophic N-Group / 55 Surapati Pramanik and Tapan Kumar Roy. Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir / 82 Pranab Biswas, Surapati Pramanik, and Bibhash C. Giri. Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments / 102 114 39.95$