Interval Valued Intuitionistic Fuzzy Positive Implicative Hyper BCK - Ideals of Hyper BCK Algebras

Transcription

1 vailable online at wwwpelagiaresearchlibrarcom vances in pplie Science Research, 2016, 7(6):32-40 ISSN : CODEN (US): SRFC Interval Value Intuitionistic Fuzz Positive Implicative Hper BCK - Ieals of Hper BCK lgebras Satanaraana B 1, Vineela KVP 1, Durga Prasa R 2 Binu Mahavi U 3 1 Department of Mathematics, chara Nagarjuna Universit, Nagarjuna Nagar, Inia 2 Department of Basic Engineering, Devineni Venkata Ramana & DrHima Sekhar MIC College of Technolog, Kanchikacherla, Inia 3 Department of pplie Mathematics, Krishna universit PG Campus, Nuzvi, Inia BSTRCT In this paper, we introuce the notions of an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpes-1,2,3,,8 Then we present some theorems which characterize the above notions accoring to the intervalvalue level subsets lso we obtain certain relationships among these notions, interval-value intuitionistic fuzz (strong, weak) hper BCK-ieals interval-value intuitionistic fuzz positive implicative hper BCK-ieals of tpes-1,2,3,,8 obtain some relate results Kewors: Hper BCK-algebras; Fuzz positive implicative ieal; Interval value Intuitionistic Fuzz positive implicative hper BCK-ieals INTRODUCTION The hper structure theor was introuce in1934 b Mart [1] at the 8th science congress of Scinavian Mathematicians In the following ears, several authors have worke on this subject notabl in France, Unite States, Japan, Spain, Russia Ital Hper structures have man applications in several sectors of both pure applie sciences In Jun et al [2], applie hper structures to BCK-algebras introuce notion of Hper BCK-algebras which is generalization of a BC-algebra The notion of interval-value fuzz sets was first introuce b Zaeh [3] as an extension of fuzz sets n interval-value fuzz set is a fuzz set whose membership function is man-value form an interval in the membership scale This iea gives the simplest metho to capture the imprecision of the membership grae for a fuzz set On the other h, tanassov [4] introuce the notion of intuitionistic fuzz sets as an extension of fuzz set which not onl gives a membership egree, but also a non-membership egree tanassov Gargov [5] introuce the notion of interval-value intuitionistic fuzz sets which is a generalization of both intuitionistic fuzz sets interval-value fuzz sets In this paper we appl the concept of interval-value intuitionistic fuzz sets to positive implicative hper BCK-ieals of hper BCK-algebras obtain interval value intuitionistic positive implicative hper BCK-ieals of tpe-1,2,3,,8 also we obtain certain relationships among these notions PRELIMINRIES The B, C, K sstem is a variant of combinator logic that takes as primitive the combinators B, C, K This sstem was iscovere b Curr in his octoral thesis Grunlagen er kombinatorischen Logik, whose results are set out in Curr The combinators are efine as follows: B x z=x ( z) 32

2 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 C x z=x z K x =x Let H be a nonempt set enowe with hper operation that is, ο is a function from H H to P * ( H ) = P( H )\ For an two subsets B of H, enote b B the set aοb We shall use x instea of xο, a b, B x ο, or x ο Definition 21 B a hper BCK-algebra ( H,,0), we mean a nonempt set H enowe with a hper operation ο a constant 0 satisfing the following axioms: (HK-1) ( xz) ( z) x, (HK-2) ( x) z = ( xz), (HK-3) xh x, (HK-4) x x x= for all xz,, H We can efine a relation on H b letting x if onl if 0 x for ever B, H, Bis efine b a, b B such that a b In such case, we call the hper orer in H [6] In an hper BCK-algebra H the following hol (P1) x x x (P2) x0 x, 0 x x 00 0, (P3) ( B) C = ( C) B, B 0 0, (P4) 0 0 = 0, (P5) 0 x (P6) x x, (P7), (P8) B B, (P9) 0 x = 0, (P10) x 0 = x, (P11) 0 = 0, (P12) 0 = 0,(P13) B, (P14) x x 0, (P15) x0 x, (P16) z xz x (P17) x = 0 ( xz) ( z) = 0 xz z,(p18) 0 = 0 = 0 for all xz,, H for all non-empt sets, B C of H Let I be a non-empt subset of a hper BCK-algebra H 0 I Then I is calle a hper BCK-sub algebra of H, if x I for all x, I weak hper BCK-ieal of H if x I I implies x I, x, H hper BCK-ieal of H, if x I I implies x I for all x, H strong hper BCK-ieal of H, if x I I implies x I, x, H I is sai to be reflexive if x x I for all x H S-reflexive if it satisfies ( x ) I implies that x I, for all x, H Close, if x I implies that x I for all x H It is eas to see that ever S-reflexive sub-set of H is reflexive Definition 22 Let H be a hper BCK-algebra then H is sai to be a positive implicative hper BCK-algebra, if for all x,, z H, ( x) z = ( xz) ( z) [7] Definition 23 [7] Let I be a nonempt subset of H 0 I Then I is sai to be a positive implicative hper BCK-ieal of (i) tpe 1, if ( x) z I z I xz I, (ii) tpe 2, if ( x) z I z I xz I, (iii) tpe 3, if ( x) z I z I xz I, 33

3 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 (iv) tpe 4, if ( x) z I z I xz I, (v) tpe 5, if ( x) z I z I xz I, (vi) tpe 6, if ( x) z I z I xz I, (vii) tpe 7, if ( x) z I z I xz I, (viii) tpe 8, if ( x) z I z I xz I for all xz,, H The etermination of maximum minimum between two real numbers is ver simple, but it is not simple for two intervals Biswas [8] escribe a metho to fin max/sup min/ between two intervals set of intervals B an interval number a on [0,1], we mean an interval + a, a, where 0 a a + 1 The set of all close subintervals of [0,1] is enote b D [ 0, 1] The interval [ a, a ] is ientifie with the number a [0, 1] For an interval numbers a + i = a i, b i D[ 0,1], i I + We efine a i = min ai, minb i, + i I i I sup a i = max ai, max b i i I i I n put (i) a1 a2 = min ( a1, a 2) = min( a 1, b1 +, a 2, b2 + ) = + + min a1, a2, min b1, b a1 a2 = max a1, a 2 = max a 1, b 1, a 2, b 2 = max a1, a2, max b1, b 2 (ii) ( ) ( ) (iii) a1 + a2 = a1 + a2 a1 a2, b1 + + b2 + b1 + b 2 + (iv) a1 a2 a1 + + a2 b1 b2 (v) a1 a2 a1 + + = = a2 b1 = b2, md m[ a] m + a1, b + 1 ma1, mb 1 = = where 0 m 1 D 0,1,,, form a complete lattice with [0,0] as its least element [1,1] as its greatest Obviousl ( [ ] ) element People observe that the etermination of membership value is a ifficult task for a ecision maker In Zaeh [3] efine another tpe of fuzz set calle interval-value fuzz sets (i-v FSs) The membership value of an element of this set is not a single number, it is an interval this interval is a sub-interval of the interval [0,1] Let D[ 0,1] be the set of a subintervals of the interval [0, 1] Let L be a given nonempt set n interval-value fuzz set (briefl, i-v fuzz set) B on L is efine b B = ( X, + µ B( x), µ B( x) ) : x L, Where µ B () x µ B + () x are fuzz sets of L such that µ B () x µ B() x + L Let µ B ( x) = µ B(), x µ B() x B = x, µ B( x) : x L = ( µ, λ): L D 0, 1 D 0, 1 0 µ + () x + λ + () x 1 0 µ () x + λ () x 1 for all x L for all x, then ( ) where µ B: L D [ 0, 1] mapping [ ] [ ] is calle an interval-value intuitionistic fuzz set (i-v IF set, in short) in L if (that is, + = ( X, µ +, λ + ) = ( X, µ, λ) are intuitionistic fuzz sets), where the mappings µ ( x ) [ µ ( x ), µ + = ( x )]: L D [ 0, 1] λ() x = ( λ (), ()): L D x λ + x [ 0,1] enote the egree of membership ( namel µ ( x )) egree of non-membership ( namel λ ( x )) of each element x Lto respectivel 34

4 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 INTERVL-VLUED INTUITIONISTIC FUZZY POSITIVE IMPLICTIVE HYPER BCK-IDELS OF HYPER BCK-LGEBRS Definition 31 Let = ( µ, λ) be an interval-value intuitionistic fuzz subset of H µ (0) µ ( x ), λ(0) λ ( ) for all x, H Then is sai to be an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1, if for all t x z, µ ( t ) min µ ( a), µ ( b) ( ) a x z b z sup ( c), sup ( ) c ( x ) z z λ ( t ) max λ λ (ii) tpe 2, if for all t x z, µ ( t) min sup µ ( a), µ ( b) a ( x) z b z ( c), ( ) sup ( ) λ ( t) max λ λ c x z z (iii) tpe 3, if for all t x z, µ ( t) min sup µ ( a), sup µ ( b) ( ) ( c), c ( x ) z z ( ) a x z b z λ ( t ) max λ λ (iv) tpe 4, if for all t x z, Definition 32 µ ( a), ( ) sup µ b ( ) µ ( t) min a x z b z sup ( c), ( ) c ( x) z λ ( t) max λ λ for all x,, z H z Let = ( µ, λ ) be an interval-value intuitionistic fuzz sub-set of H Then is sai to be an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of (i) tpe 5, if there existst x z such that µ ( t ) min µ ( a), µ ( b) ( ) a x z b z sup ( c), ( ) c ( x) z λ ( t) max λ λ z (ii) tpe 6, if there existst x z such that µ ( t) min sup µ ( a), sup µ ( b) ( ) ( c), c ( x ) z z ( ) a x z b z λ ( t ) max λ λ (iii) tpe 7, if there exists t x z such that 35

5 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 µ ( a), ( ) sup µ b ( ) µ ( t) min a x z b z sup ( c), ( ) c ( x) z λ ( t) max λ λ z (iv) tpe 8, if there exists t x z such that sup µ ( a), µ ( b) a ( x) z b z µ ( t) min ( c), ( ) sup ( ) λ( t) max λ λ c x z z Theorem 33 Let = ( µ, λ) be an interval-value intuitionistic fuzz subset of H Then (i) If is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 3, then is an intervalvalue intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2, 4 6 (ii) If is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 (or) 4, then is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1 (iii) If is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 (tpe 4), then is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 8 (tpe 7) (iv) If is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1, then is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 5 Proof: (i) ssume = ( µ, λ ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 3 Since sup µ ( b) µ ( b) λ( b) sup λ( b) b z b z b z b z Now for all t x z, µ ( ) min sup ( ), sup ( ) min sup ( ), ( ) t µ a µ b µ a µ b a ( x) z b z a ( xo) oz b z λ( c), λ( ) ( c), ( ) ( ) sup c x z z ( ) z λ ( t) max max λ λ Thus = ( µ, λ ) c x z is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 Since sup µ ( a) µ ( a) λ () c sup λ() c a ( x) z a ( x) z c ( x) z c ( x) z Now for all t x z, sup µ ( ), sup ( ) ( ) min a µ b µ ( ), sup ( ) ( ) min t µ a µ b a x z b z a ( x) z b z ( ), ( ) sup ( ), ( ) ( t) max λ c λ max λ c λ c ( x ) z z c ( x) z z Thus = ( µ, λ) Clearl, there exists t x z such that sup µ ( ), sup ( ) ( ) min a µ b µ t a is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 4 ( x) z b z ( ), ( ) ( ) ( ) max t λ c x z c λ z 36

6 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 Thus = ( µ, λ ) is an interval- value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 6 (ii) ssume = ( µ, λ ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 sup µ Since ( a) µ ( a) a ( x) z a ( x) z λ() c sup λ() c c ( x) z c ( x) z Now for all t x z µ sup ( ), (b) ( ), ( ) ( t ) min µ a µ min µ a µ b a ( x ) z b z a (x ) z b z ( ), sup () sup ( ) ( ), sup c λ c ( ) c x z z c ( x ) z z λ ( t) max λ max λ λ Thus = ( µ, λ ) ssume = ( µ, λ ) Now for all t is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1 is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 4 Since sup µ ( b) µ ( b) λ () sup λ () b z b z z z x z, µ ( ) min ( ), sup ( ) ( ) min ( ), ( ) t µ a µ b µ a µ b a x z b z a ( x ) z b z sup ( c ), ( ) sup ( ) ( c ), sup ( ) c x z c ( x) z z λ ( t) max λ λ max λ λ z Thus = ( µ, λ) (iii) ssume = ( µ, λ ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1 is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 Clearl there exists t x z µ ( ) min t, ( ), sup ( ) ( t) max λ c λ c ( x) z z Thus = ( µ, λ ) that = ( µ, λ) there exists t x z, sup µ ( a), µ ( b) a ( x) z b z is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe-8 ssume is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 4 Clearl µ ( a), sup µ ( b) a ( x) z b z, sup ( ), ( ) ( t) max λ c λ c ( x) z z is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 7 µ ( t) min Thus = ( µ, λ ) (iv) ssume = ( µ, λ) exists t x z, µ ( t ) min µ ( a), µ ( b) is an intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1 Clearl there a ( x ) z b z, sup ( ), sup ( ) ( ) ( ) max t λ c λ c x z z Thus = ( µ, λ ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 5 Example 34 H = be a hper BCK-algebra with the following Cale table Let 0,1,2, ` ,3 0,2,3 37

7 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 Define an i-v IFS = ( µ, λ ) in H b µ (0) = µ (1) = [1,1], µ (2) = [05,06], µ (3) = [02,025] λ (0)= λ(1)=[0,0], λ(2)=[03,035], λ(3)=[07,075] Then = ( µ, λ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe1, while it is not tpe-2, because µ (3) = [02,025] < [05, 06] = µ (2) = min sup µ ( a), µ (b) a (32) 0 (20) b λ ( c ), sup λ ( ) (32) 0 (20) λ (3) = [07,075] > [03,035] = λ (2) = max c Example 35 H = be a hper BCK-algebra with hper operation ï the following is the Cale table Let 0,1,2 Define i-v IFS = ( µ, λ ) in H b ,1 0, ,2 0,1,2 µ (0) = µ (2) = [1,1], µ (1) = [02,025] λ(0) = λ(2) = [0,0], λ(1) = [06,065] Then = ( µ, λ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 4, which is not tpe 3, because µ (1) = [02,025] < [1,1] = µ (0) = min sup µ ( a ), sup µ ( b ) (20) 2 (02) λ ( c ), λ ( ) c (2 0) 2 (02) a b λ (1) = [06,065] > [0,0] = max Theorem 36 Let = ( µ, λ) be an i-v IFS of H Then the following statements hol: (i) If is an interval-value intuitionistic fuzz positive implicative hper BCK- ieal of tpe 1, then is an intervalvalue intuitionistic fuzz weak hper BCK-ieal of H (ii) If is an interval-value intuitionistic fuzz positive implicative hper BCK- ieal of tpe 2, then is an intervalvalue intuitionistic fuzz strong hper BCK-ieal of H Proof: (i) Suppose is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1 Put z = 0 in tpe1, for all t x 0 = x, we get µ ( ) min ( ), ( ) ( )0 0 = min ( ), ( ) x µ a µ b µ a µ a x b a x sup ( ) max ( ), sup ( ) sup ( )0 0 = max ( ), ( ) x λ c λ λ c λ c x c x Thus = ( µ, λ) is an interval-value intuitionistic fuzz weak hper BCK-ieal Suppose is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 Put z = 0 in tpe 2, for all t x 0 = x, we get µ ( ) min sup ( ), ( ) min sup ( ), ( ) x µ a µ b = µ a µ a ( x)0 b 0 a x ( c ), sup ( ) ( ), ( ) ( )0 0 c λ ( x) max λ λ = max λ λ (i) c x c x 38

8 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 We show that, for x, Hif x then µ ( x) µ ( ) λ ( x) λ ( ) For this, if x, H such that x that is 0 x since µ (0) µ ( ) λ (0) λ ( ) This is true for all ab, x sup µ ( ) (0) Then a = µ λ( b) = λ(0) a x b x so x µ (0), µ ( ) a x ( ), ( ) x λ c λ = λ(0), λ( ) = c x a µ ( ) min sup µ ( a), µ ( ) = min = µ ( ) ( ) max max λ ( )(ii) Since b (HK3) then for all a x x, a x so µ () a µ () x for all ab, x Hence µ ( a) µ ( x) a xx Combining (i) (iii), we get µ ( ) ( ) min sup ( ), ( ) a µ b x µ µ b x a xx λ ( ), λ ( ) x sup λ ( c) λ ( x) max c xx Thus = ( µ, λ ), λ ( a) λ ( x) λ λ sup ( c) ( x) (iii) c xx for all, x H is an interval-value intuitionistic fuzz strong hper BCK-ieal of H The following examples show that the converse of the Theorem 36 is not true in general Example 37 H = be a hper BCK-algebra with the following Cale table Let 0,1,2, ,2 Define an i-v IFS = ( µ, λ) in H b µ (0)= µ (1)=[1,1], µ (2)= µ (3)=[03,035] λ (0)= λ(1)=[0,0], λ(2)= λ(3)=[05,06] Then = ( µ, λ ) is an interval-value intuitionistic fuzz weak hper BCK-ieal, but it is not an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe1 Because = 2 µ ( a ), µ ( b ) (32) 2 22 µ (2) = [03,035] < [1,1] = µ (0) = min a b sup λ ( c ), sup λ ( ) c (32) 2 22 λ (2) = [05,06] >[0,0] = λ (0) = max Example 38 Let 0,1,2 H = be a hper BCK-algebra with the following Cale table Define an i-v IFS = ( µ, λ ) in H b º ,2 b 39

9 Satanaraana et al vppl Sci Res, 2016, 7(6):32-40 µ (0) = [1,1], µ (1) = [0,0], µ (2) = [04,05] λ(0) = [0,0], λ(1) = [1,1], λ(2) = [03,035] Then = ( µ, λ) is an interval-value intuitionistic fuzz strong hper BCK-ieal of H but it is not an intervalvalue intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2 Because sup µ ( a ), sup µ ( b ) (20) 2 02 ( c ), ( ) c (2 0) 2 22 µ (2) = [04,05] < [1,1] = µ (0) = min a b λ = [03,035] > [0,0] = λ (0) = max λ λ (2) Corollar 39 Let = ( µ, λ) be an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 1, if satisfies the - sup propert, then = ( µ, λ) is an interval-value intuitionistic fuzz s-weak hper BCKieal of H Corollar 310 Let = ( µ, λ) be an interval-value intuitionistic fuzz Positive implicative hper BCK-ieal of tpe 2 x, H Then (i) x impl µ (x) µ () λ( x) λ( ) (ii) µ ( x) min µ ( a), µ ( ) λ( x) max λ( b), λ( ) for all ab, x Corollar 311 Ever interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe-2, is an interval-value intuitionistic fuzz s-weak hper BCK-ieal (hence an interval-value intuitionistic fuzz weak-hper BCK-ieal) an interval-value intuitionistic fuzz hper BCK-ieal Corollar 312 If = ( µ, λ) µ ( x ) min µ ( a), µ ( ) is an interval-value intuitionistic fuzz positive implicative hper BCK-ieal of tpe 2, then a x sup ( ), ( ) bx ( ) max x λ b λ for all x, H Definition 313 n IFS = ( µ, λ) in H is sai to be an interval-value intuitionistic fuzz close if it satisfies x, H such that x, µ ( x) µ ( ) λ( x) λ( ) REFERENCES [1] Mart F Sur une generalization e la notion e groupe 8th Congress Math Scinaves, 1934, [2] Jun YB, Xin XL Scalar elements hper atoms of hper BCK-algebras Scientiae Mathematicae Japonica, 1999, 2: [3] Zaeh L The concept of a linguistic variable its application to approximate reasoning I, Information Sci Control, 1975, 8: Z [4] tanassov KT Intuitionistic fuzz sets Fuzz Sets Sstems, 1986, 20: [5] tanassov KT, Gargov G Interval value intuitionistic fuzz sets Fuzz Sets Sstems, 1989, 31: [6] Satanaraana B, Krishna L, Durga Prasa R On interval-value intuitionistic Fuzz hper BCK-ieals of hper BCK-algebras Journal of vance Resarch, 2014, 7: [7] Durga Prasa R, Satanaraana B, Ramesh D, Gnaneswara Re M On intuitionistic fuzz positive implicative hper BCK- ieals of BCK-algebras ppls J of Pure ppl Math, 2012, 6: [8] Biswas R Rosenfel s fuzz subgroups with interval value membership function Fuzz Sets Sstems, 1994, 1: