Liu s Generalized Intuitionistic Fuzzy Sets

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1 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 iu s Generalized Intuitionistic Fuzzy Sets Hsiang-Chuan iu Department of Bioinformatics & Department of Psychology, sia University bstract In this paper, an extensional generalized intuitionistic fuzzy set, called iu's generalized intuitionistic fuzzy set, is proposed. It is showed that not only tanassov's intuitionistic fuzzy set but also Mondal & Samanta's generalized intuitionistic fuzzy set is a special case of this new one. Responding the generalized intuitionistic fuzzy set, the generalized intuitionistic fuzzy value (GIFV) is also proposed. Some comparable examples are given, and some important notions and basic algebraic properties of this new generalized intuitionistic fuzzy set and its generalized intuitionistic fuzzy value are discussed. Keywords: intuitionistic fuzzy set, generalized intuitionistic fuzzy set, generalized intuitionistic fuzzy value, order-preserving order, addition-invariant order 69

2 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 I Introduction Intuitionistic fuzzy set (IFS) was proposed by tanassov (1986, 1999) characterized by a membership function and a non-membership function, which is a generalization of Zadeh s fuzzy set (Zadeh, 1965), whose basic component is only a membership function. Over the last decades, IFS has been applied to many different fields, such as decision making, logic programming, topology, medical diagnosis, pattern recognition, machine learning and market prediction, etc. (Tan & Chen, 2010; Xu, 2007) generalized intuionistic fuzzy set (GIFS) with some algebraic properties were proposed by Mondal and Samanta (2002), unfortunately, the addition operation and scalar multiplication operation of tanassov s IFS can no more be used for Mondal and Samanta s GIFS, and up to now, their GIFS do not have its own addition operation and scalar multiplication operation like which of tanassov s IFS yet. In this paper, an extension of GIFS, called iu generalized intuitionistic fuzzy set (denoted by -GIFS), is proposed, by good luck, if the extensional constant of -GIFS ~ equals to one, then the addition operation and scalar multiplication operation of tanassov s IFS can still be used for this new GIFS, with the extensional index =1, furthermore, an order-preserving partial order, called iu s order, proposed by author s previous work (iu, 2010) can still be used by this new GIFV, and it is pointed out that tanassov s order is not a totally partial order, and Xu s order is not addition-invariant, both of them can not be used to construct the generalized intuitionistic fuzzy aggregation operators, on the contrary, iu s order (iu, 2010) is not only a totally partial order but also an addition-invariant one, which is order-preserving for addition operation and can be used for constructing an ordered semi-vector space of generalized intuitionistic fuzzy values to handle intuitionistic fuzzy multi-criteria fuzzy decision making problems. II Intuitionistic Fuzzy Set The fundamental characteristic of a fuzzy set (Zadeh, 1965) is that it assigns to each of its element a membership degree and a non-membership degree under the constraint that the sum of two degrees equals one. tanassov (1986, 1999) extended Zadeh s fuzzy sets to intuitionistic fuzzy sets (IFSs), which assigns to each of their elements a membership degree and a non-membership degree, under the constraint that the sum of two degrees does not exceeds one. Some formal definitions about IFSs are defined as follows: Definition 1. Intuitionistic fuzzy sets (IFSs) (tanassov, 1986, 1999) Given a finite set X, an tanassov s intuitionistic fuzzy set (-IFS), ~, is defined as ~ x,, x X (1) which assigns to each element x a membership degree (x ) and a non-membership degree (x ), satisfying following conditions: (i) 0 1, x X (2) (ii) 0,, 1, x X (3) (iii) ( 1 ( ( (4) 70

3 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 Where ( x ) is called the intuitionistic fuzzy index of element x in -IFS ~, the value denotes a measure of non-determinacy. Obviously, if ( 0, then the intuitionistic fuzzy set, ~, is just a Zadeh s fuzzy set. et FS Z (X ) be the collection of all Zadeh s fuzzy sets on X, IFS (X ), be the collection of all -IFSs on X. Obviously, we have FSZ ( X ) IFS X ) (5) III Generalized Intuitionistic Fuzzy Set In definition 1, to replace the condition (2) with the condition (6), the generalized intuitionistic fuzzy sets were proposed by Mondal and Samanta (2002), denoted by MS-GIFSs. Definition 2. Mondal and Samanta s generalized intuitionistic fuzzy sets (MS-GIFSs) Given a finite set X, Mondal & Samanta s generalized intuitionistic fuzzy set (MS-GIFS), ~, is defined as x,, xx, which assigns to each element x a membership degree (x ) and a non-membership degree (x ), satisfying following conditions: (i) 0 0.5, x X (6) (ii) 0,, 1, x X (7) (iii) 1 (8) Where (x ) is called the Mondal and Samanta s generalized intuitionistic fuzzy index of element x, in MS-GIFS ~, the value denotes a measure of non-determinacy. The collection of all MS-GIFSs is denoted by GIFS MS (X ), if 0, then the Mondal and Samanta s intuitionistic fuzzy set, ~, is just a Zadeh s fuzzy set, and we have the following theorem: Theorem 1. IFS X ) GIFSMS ( X ) (9) ~ Proof. x,, x X IFS X ) We have 0 1, x X, then 0.5 (10) 2 ~ Therefore, GIFS ( X ), and the proof is completed. MS Example 1. et X { x1, x2, x3} and x1,0.8,0.3, x2,0.4,0.9, x3,0.5,0.4 GIFSMS ( X), but ~ IFS ( X ), in other words, GIFS ( X ) IFS ( X ) MS Note that FSZ ( X ) IFS X ) GIFSMS ( X ) (11) but GIFS ( X ) IFS ( X ) FS ( X ) (12) MS Z 71

4 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 IV n Extensional Generalized Intuitionistic Fuzzy Set In this paper, an extensional generalized intuitionistic fuzzy set, called iu s generalized intuitionistic fuzzy set, is proposed as bellows. Definition 3. iu s generalized intuitionistic fuzzy sets (-IFSs) Given a finite set X, and a constant 0,1, a iu s generalized intuitionistic fuzzy set (-GIFS), ~ ~, is defined as x,, x X, which assigns to each element x a membership degree (x ) and a non-membership degree (x ), satisfying following conditions: (i) 0 1, x X (13) (ii) 0, 1, x X (14) (iii) 1 (15) Where is called the extensional index of -GIFS ~, (x ) is called the generalized intuitionistic fuzzy index of element x in -GIFS ~, the value denotes a measure of non-determinacy. The collection of all -GIFSs is denoted by GIFS (X ). Theorem 2. (i) 0 GIFS( X ) FSZ ( X ) (16) (ii) 0 GIFS( X ) IFS X ) (17) (iii) 1 IFS ( X ) GIFS ( X ) GIFS ( X ) (18) MS Example 2. If 1, let X { x1, x2, x3} and ~ ~ x1,0.8,0.6, x2,0.6,0.9, x3,0.9,0.7 GIFS( X ), but GIFSMS ( X ), in other words, GIFS ( X ) GIFS ( X ) MS Note that FSZ ( X ) IFS X ) GIFSMS ( X ) GIFS ( X ) (19) GIFS ( X ) GIFSMS ( X ) IFS X ) FSZ ( X ) (20) and, hereafter, if there is no word mentioned the value of in GIFS (X ), then it means 1. V Basic lgebraic Operations On GIFS ( X ), Definition 4. et x,, xx B x, B(, B( xxgifs( X), then inclusion, equality, complementation, union, intersection, 0 and 1 on GIFS ( X ) are defined as follows: (i) B, B(, B(, xx (21) (ii) B, B, B (22) (iii) c x,, xx (23) (iv) B x, B(, B( xx (24) B x, ( (, ( ( xx (25) (v) B B 72

5 (vi) (vii) iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 0 x, (, ( xx, ( 0, ( 1, xx (26) x, (, ( xx, ( 1, ( 0, xx (27) , Definition 5. et x,, xx B x, B(, B( xxgifs( X), then partial order, addition operation and scalar multiplication operation are defined as follows: (i) B, B(, or B( B(, xx (28) (ii) B, B, B (29) B x, ( ( ( (, ( ( xx (30) (iii) B B B (iv) x,1 [1 ],[ ] xx if 0 0 if 0 We can obtain following properties; Theorem 3. et BC,, GIFS ( X),, 0,we have (i) B, B,,0,1, B, GIFS ( X) (closure) (32) (ii) B, B C C (transitivity) (33) (iii) 0 1, GIFS ( X) (34) (iv) 0 0,1 1, GIFS ( X) (35) (v) B B, B B (commutativity) (36) (vi) ( B) C ( B C )(associativity) (37) (vii) ( B) C ( B C )(associativity) (38) (viii) ( BC ) ( B) ( C )(distributivity) (39) (i ( BC ) ( B) ( C )(distributivity) (40) ( ( B) ( B) (absorption) (41) (xi) (idempotence) (42) (xii) ( c ) (43) (xiii) ( B) c ( B c ),( B) c ( B c )(De morgan aw) (44) (ix B B (commutativity) (45) (x ( B) C ( B C )(associativity) (46) (xxi) ( ) (distributivity) (47) (xxii) ( B) ( ) ( B ) (distributivity) (48) (xxiii) 0 0 (49) (31) 73

6 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 VI Semi-Ring Semi-Ring 1. Semi-group Definition 5. Semi-group (Golan,1999; Howie, 1976) et X be an nonempty set, ( X ; ) is called a semi-group on X with binary operations : X X X, if it satisfies the following conditions: (i) ab, Xab X(closure) (50) (ii) a, b, cx a( bc) ( ab) c(associativity) (51) Theorem 4. ( GIFS ( X ); ) and ( GIFS ( X ); ) are two semi-groups. 2. Monoids Definition 6. Monoid (Golan,1999, Howie, 1976) semi-groupo ( X ;, e) is called a monoid, if ( X ; ) is a semi-group satisfying the following condition: e X, ax eaae a(unit element) (52) Theorem 5. ( GIFS ( X );,0) and ( GIFS ( X );,1) are two monoids. 3. Semi-ring Definition 7. Semi-ring (Golan,1999; Howie, 1976) et X be an nonempty set, ( X ;,, ee, ') is called a semi-ring on X if ( X ;, e) and ( X ;, e') are two monoids on X satisfying the following conditions: (i) ab, Xabb a(commutative) (53) (ii) ex, ax aeeaa, aeea e (54) (iii) e' X, ax ae' e' a a(closure) (55) (iv) abc,, Xa( bc) ( ab) ( a c) (distribution) (56) Theorem 6. ( GIFS ( X );,,0,1) is a semi-ring. B Partial orders Definition 8. Partial order, Poset (Davey & Priestley, 2002; Dietrich & Hoffman,2003) et X be an nonempty set, a partial order,, on X is a binary relation which satisfies the following conditions: (i) aa, a X (reflexivity) (57) (ii) ab, ba, a, bx a b(antisymmetry) (58) (iii) ab, bc, a, b, cx a c(transitivity) (59) If is a partial order on X, then ( X, ) is called a partially ordered set, or poset. 74

7 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 Definition 9. Totally partial order (Davey & Priestley, 2002; Dietrich & Hoffman, 2003) et ( X, ) be a poset, the partial order,, is called a totally partial order on X, if it satisfies the following condition: ab, Xa b, or b a(reflexivity) (60) Note that the partial order proposed in this paper is a totally partial order. However, the above-mentioned inclusion order is not a totally partial order. C attices Definition 10. (Davey & Priestley, 2002) et ( X, ) be a poset, if ab, Xaba, b X (61) Then ( X, ) or ( X,, ) is called a lattice. ab ab, sup ab,, ab ab, inf ab, (62) Where Theorem 7. ( GIFS ( X ); ) or ( GIFS ( X );, ) is a lattice. B, C( X), B sup B,, B inf B, Proof. sup B,,inf B, GIFS ( X) Hence, ( GIFS ( X ); ) or ( GIFS ( X );, ) is a lattice. Definition 11. (Davey & Priestley, 2002) et ( X,, ) be a lattice, if it satisfies following two distributivity laws, then it is called a distributive lattice. (i) a( bc) ( ab) ( ac), abc,, X (63) (ii) a( bc) ( ab) ( ac), abc,, X (64) Theorem 8. ( GIFS ( X );, ) is a distributive lattice. Definition 12. (Davey & Priestley, 2002) et ( X,, ) be a lattice, if it satisfies following two distributivity laws, then it is called a completely distributive lattice. n n (i) a( b) ( ab), a, b X, n N (65) (ii) i i i i1 i1 n n a( b) ( ab), a, b X, n N (66) i i i i1 i1 Theorem 9. ( GIFS ( X );, ) is a completely distributive lattice. Definition 13. (Davey & Priestley, 2002) et ( X,, ) be a lattice, if it satisfies following bounded condition, then it is called a bounded lattice. 0,1 X, ax 0 a 1 (67) 75

8 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 Theorem 10. ( GIFS ( X );, ) is a bounded lattice. Definition 14. (Davey & Priestley, 2002) ( X,,, c) is a soft algebra, if ( X,, ) is a bounded and distributive lattice satisfying following two conditions. c c (i) ax,( a ) a (68) c c c c c c (ii) a, bx ( ab) a b,( ab) a b (69) Theorem 11. ( GIFS ( X );,, c) is a soft algebra. Definition 15. (Davey & Priestley, 2002) ( X,,, c) is a superior soft algebra, if ( X,,, c) is a completely distributive lattice satisfying following dense property. ab, X, ab,( aba, b) rxar b (70) Theorem 12. ( GIFS ( X );,, c) is a superior soft algebra. Proof., BX, B, that is B, B et x,, xx B x, B(, B( xx nd R x,0.5( B( ),0.5 ( B( ) xx (71) then R GIFS ( X ) and (i) B(, B(,or (72) (ii) B(, B(,or (73) (iii) B(, B( (74) we can obtain R B Therefore ( GIFS ( X );,, c) is a superior soft algebra. Note that ( PX ( );,, c) is not a superior soft algebra, where PX ( ) is the power set of X. VII Intuitionistic Fuzzy Values (IFVs) To be convenience, corresponding IFS, the intuitionistic fuzzy values(ifvs) is defined as follows (Xu, 2007): Definition 16. Intuitionistic fuzzy values (IFVs) (Xu, 2007) If, 0, and ( ) 1, then the order pair (, ) is called an intuitionistic fuzzy value (IFV). In this paper, corresponding GIFS, the generalized intuitionistic fuzzy values (GIFVs) is defined as follows: Definition 17. generalized intuitionistic fuzzy values (GIFVs). If 0, 1, then the order pair (, ) is called iu s generalized 76

9 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 intuitionistic fuzzy value ( GIFV ). Corresponding the operations on IFS and GIFS, we can obtain following two useful operations on IFV and GIFV. Definition 18. Two useful operations on IFV (tanassov, 1986, 1999; De, Biswas & Roy, 2000) et SIFV ( ) be the set of all IFVs, and SGIFV ( ) be the set of all GIFVs. If a ( a, a), b ( b, b) S( GIFV), then (i) a b ( a b ab, ab) (75) (1 (1 a), ( a) ) 0 (ii) a (76) (0,1) 0 Furthermore, we can get the following theorem: Theorem 13. If abc,, P,, 0 then (i) ab, a,(0,1) S( GIFV ) (77) (ii) a b ba (78) (iii) a( bc) ( ab) c (79) (iv) (0,1) a a(0,1) a (80) (v) ax, 0a a (81) (vi) ax,, 0 ( ) a aa (82) (vii) R, ab, X( ab) ( a ) ( b) (83) VIII Ordered Semi-Vector Space ddition-invariant partial orders Definition 19. ddition-invariant partial orders (Dietrich & Hoffman, 2003; iu, 2010) (i) et X be an nonempty set with an addition binary operations : X X X and a totally partial order,, on X, if it satisfies the following condition, then the algebraic structure ( X ;, ) is called an addition order-preserving structure, abc,, X, abacb c (84) (ii) If X ;, is an addition order-preserving structure, then the totally partial order,, is called an addition-invariant partial order. B Three kinds of partial orders Definition 20. tanassov s order (tanassov, 1986, 1999) et SIFV ( ) be the set of all IFVs, a ( a, a), b ( b, b) S( IFV) (85) tanassov s order,, is a binary relation which satisfies the following conditions: a b, (86) a b a b Note that tanassov s order is a partial order, but not a totally partial order, since if, or,, then a and b are incomparable. It can be only a b a b a b a b 77

10 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 used for IFS and GIFS, not for IFV and IFV. Definition 21. Xu s order (Xu, 2007) et SIFV ( ) be the set of all IFVs, a (, ), b (, ) P a a b b Sa ( ) ( a a), Ha ( ) ( a a) (87) Xu s partial order, X, is defined as follows: (i) a X b Sa ( ) Sb ( ), orsa ( ) Sb ( ), Ha ( ) Hb ( ) (88) (ii) a b S( a) S( b), H( a ) H( b) (89) (iii) a b a b, ora b (90) X X Definition 22. iu s order (iu, 2010) et SGIFV ( ) be the set of all GIFVs, a ( a, a), b ( b, b) P, iu s order,, is defined as follows: (i) a b a b, ora b, a b (91) (ii) a b a b, a b (92) (iii) a b a b, ora b (93) C Semi-vector space Definition 23. Semi-vector space (Pap, 1979) et X be an nonempty set, ( X ;, ) is called a semi-vector space on X over R with binary operations : X X X, and : R X X, if it satisfies the following conditions: (i) ( X ; ) is a semi-group (ii) ab, Xabb a(commutative) (94) (iii) R, ax a X (closure) (95) (iv) ax, R aa (commutative) (96) (v), R, ax ( ) a ( a) (association) (97) (vi) R, a, bx ( ab) ( a) ( b) (distribution) (98) (vii), R, ax ( ) a ( a) ( a) (distribution) (99) (viii) 1 R, ax 1aa X (unit) (100) Definition 24. Completed semi-vector space (Pap, 1979) et ( X ;, ) be a semi-vector space on X over R, if ex, ax eaae a, then ( X ;, ) is called a completed semi-vector space. Theorem 14. et be the set of all IFVs ( SGIFV ( );, ) be a semi-vector space on SGIFV ( ) over R and be the operations on SGIFV ( ) defined as Eqs. (75) and (76), then ( SGIFV ( ); ) is a monoid. Proof. It is trivial by Theorem 1. 78

11 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 D Ordered semi-vector space, operation-invariant Definition 25. Ordered semi-vector space semi-vector space on X over R ( X ;, ) with a totally partial order,, on X over R, denoted by ( X ;,, ), is called an ordered semi-vector space, if it satisfies the following order-preserving condition: abc,, X, abacb c (101) Definition 26. Operations-invariant (iu, 2010) If ( X ;,, ), is an ordered semi-vector space, we called that the partial order is operations-invariant. Note that. et be the operations on SGIFV ( ) defined as Eq. (75), ( SGIFV ( );, ) be a semi-vector space and X be Xu s partial order, then ( SGIFV ( );,, X) is not an odered semi-vector space, in other words, X is not operations-invariant. Here, a counterexample is given as bellows. Example 3. Counterexample (i) et abc,, SIFV ( ) SGIFV ( ) a ( a, a) (0.5,0.3), b ( b, b) (0.4,0.1) P (102) c ( c, c) (0.1,0.1), 2 (103) Then Sa ( ) a a (104) Sb ( ) b b (105) Since Sa ( ) 0.2 Sb ( ) 0.3 then a X b (ii) From (i) we can obtain ac ( a c ac, ac) (0.55,0.03) (106) b c ( b c bc, bc) (0.46,0.01) (107) and Sa ( c) Sb ( c ) (108) By Xu s order, we have ac X b c. Therefore Xu s order is not addition invariant. Theorem 15. et and be the operations on SGIFV ( ) over R defined as Eqs (75), ( SGIFV ( );, ) be a semi-vector space on X over R and be iu s order, then ( SGIFV ( );,, ) is an ordered semi-vector space. Proof. et a ( a, a), b ( b, b), c ( c, c) S( GIFV) (109) and 0 To prove that is addition-invariant, If a b, then ac ( a c ac, ac) b (110) c (, ) b c b c b c 79

12 iu s Generalized Intuitionistic Fuzzy Sets 測驗統計年刊第十八輯 If a bac bca cbc (111) a b, or, (112) If a b (i) If a b, we can obtain (1 ) (1 ) a c b c a b a b a c ac b c bc ac b c (ii) If,, we can obtain a b a b (113) ( a c ac, ac) ( b c bc, bc) (114) and ac bc), then ac b c (115) The proof is completed. IX Conclusions In this paper, an extensional generalized intuitionistic fuzzy set, called iu s generalized intuitionistic fuzzy set (-GIFS) is proposed. It is showed that not only tanassov s intuitionistic fuzzy set but also Mondal and Samanta s generalized intuitionistic fuzzy set is a special case of this new one. Responding the -GIFS, the iu s generalized intuitionistic fuzzy value ( GIFV ) is also proposed. Some comparable examples are given, and some important notions and basic algebraic properties of -GIFSs and GIFVs are discussed. Moreover, it is proved that GIFV ( X ) is a semi-ring, completely distributive lattice and a superior soft algebra, and SGIFV ( ) is a semi-vector space. X cknowledgments This paper is partially supported by the grant of National Science Council of Taiwan Government (NSC H ). References tanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, tanassov, K. (1999). Intuitionistic fuzzy sets: Theory and application. Heidelberg, Germany, Physica-Verlag. Davey, B.., & Priestley, H.. (2002). Introduction to attices and Order (2nd ed.). Cambridge University Press, Cambridge. De, S. K., Biswas, R., & Roy,. R. (2000). Some operation on intuitionisric fuzzy sets. Fuzzy Sets and Systems, 114, Dietrich, B.., & Hoffman,. J. (2003). On greedy algorithms, partially ordered sets, and submodular functions. IBM Journal of Research and Development, 47, Golan, J. S. (1999). Semiring and Their plication, Kluwer, Dordrecht. Howie, J. M. (1976). n Introduction to Semigroup Theory, cademic Press. 80

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