Intuitionistic Fuzzy WI-Ideals of Lattice Wajsberg Algebras

Transcription

1 Intern J Fuzzy Mathematial Arhive Vol 15, No 1, 2018, 7-17 ISSN: (P), (online) Publihed on 8 January 2018 wwwreearhmathiorg DOI: International Journal of Intuitioniti Fuzzy WI-Ideal of Lattie Wajberg Algebra A Ibrahim 1 and C Shajitha Begum 2 1 PG and Reearh Department of Mathemati, H H The Rajah College Pudukkotai, Tamilnadu, India ibrahimaadhil@yahooom; dribrahimaadhil@gmailom 2 Department of Mathemati, Sri Ramakrihna Engineering College Coimbatore, Tamilnadu, India nouheen2407@gmailom Reeived 20 Deember 2017; aepted 5 January 2018 Abtrat In the preent paper, we introdue the notion of intuitioniti fuzzy WI-ideal and intuitioniti fuzzy lattie ideal of lattie Wajberg algebra We how that every intuitioniti fuzzy WI-ideal of lattie Wajberg algebra i an intuitioniti fuzzy lattie ideal of lattie Wajberg algebra Alo, we diu it onvere part Further, we obtain every intuitioniti fuzzy lattie ideal i an intuitioniti fuzzy WI-ideal in lattie H-Wajberg algebra Moreover, we diu ome haraterization of intuitioniti fuzzy WI-ideal Keyword: Wajberg algebra; Lattie Wajberg algebra; WI-ideal; Fuzzy WI-ideal; Intuitioniti fuzzy WI-ideal; Intuitioniti fuzzy lattie ideal AMS Mathemati Subjet Claifiation (2010): 03B50, 03G10, 03G25, 03E72 1 Introdution Non laial logi inluding many-valued logi and fuzzy logi take the advantage of the laial logi to handle information with variou fat of unertainty, uh a fuzzine and randomne Therefore, nonlaial logi ha beome a formal and ueful tool for omputer iene to deal with fuzzy information and unertain information The theory of fuzzy et wa introdued by Zadeh [13] in 1965 ha everal appliation in many field The idea of fuzzy et handle unertainty and vaguene In fuzzy et theory the memberhip of an element to a fuzzy et i a ingle value between zero and one The generalization of fuzzy et wa propoed by Atanaov [1, 2] a intuitioniti fuzzy et whih inorporate the non-memberhip degree (that i, 1 minu um of memberhip degree), and i quite intereting and ha many ueful appliation in many area Fuzzy et and intuitioniti fuzzy et are two trong framework for unertainty handling The onept of Wajberg algebra propoed by Mordhaj Wajberg [12] in 1935 In [4], Chang introdued MV-algebra a kind of algebrai ounterpart of the multivalued Lukaiewiz propoitional Caluli [5] Wajberg algebra are formulated in term of the operation "impliation" and "quai omplement" Roe et al [11] publihed the proof of Wajberg algebra in 1958 In 1984, Font et al [6] introdued lattie truture of Wajberg algebra 7

2 A Ibrahim and C Shajitha Begum The author [7] introdued the notion of Wajberg impliative ideal (WI-ideal) of lattie Wajberg algebra and diued ome related propertie Moreover, the author [8] introdued the notion of fuzzy WI-ideal and normal fuzzy WI-ideal of lattie Wajberg algebra, and invetigated their propertie with uitable illutration In thi paper, the onept of intuitioniti fuzzy et i applied to WI-ideal, that i we introdue the notion of intuitioniti fuzzy WI-ideal and intuitioniti fuzzy lattie ideal of lattie Wajberg algebra We how that every intuitioniti fuzzy WI-ideal of lattie Wajberg algebra i an intuitioniti fuzzy lattie ideal of lattie Wajberg algebra Alo, we verify it onvere part Further, we diu the relationhip between intuitioniti fuzzy WI-ideal and intuitioniti fuzzy lattie ideal in lattie H-Wajberg algebra Alo, we invetigate ome propertie of intuitioniti fuzzy WI-ideal of lattie Wajberg algebra Finally, we how that olletion of WI-ideal of lattie Wajberg algebra i an intuitioniti fuzzy WI-ideal of lattie Wajberg algebra 2 Preliminarie In thi etion, we reall ome bai notion and their propertie that are neeary to develop our main reult Definition 21 [6] Let (A,, *, 1) be an algebra with quai omplement * and a binary operation i alled a Wajberg algebra if it atifie the following axiom for all x, y, z A, (i) 1 x = x (ii) ( x y) (( y z) ( x z)) = 1 (iii) ( x y) y = ( y x) x (iv) * * ( x y ) ( y x) = 1 Propoition 22 [6] A Wajberg algebra (A,, *, 1) atifie the following axiom for all x, y, z A, (i) x x = 1 (ii) If ( x y) = ( y x) = 1 then x = y (iii) x 1 = 1 (iv) ( x ( y x)) = 1 (v) If ( x y) = ( y z) = 1 then x z = 1 (vi) ( x y) (( z x) ( z y)) = 1 (vii) x ( y z) = y ( x z) (viii) (ix) x 0 = x 1 = x ) ( x = x (x) ( x y ) = y x Definition 23 [6] A Wajberg algebra A i alled a lattie Wajberg algebra if it atifie the following ondition for all x, y A, 8

3 Intuitioniti fuzzy WI-ideal of lattie Wajberg algebra (i) The partial ordering on a lattie Wajberg algebra A, uh that x y if and only if x y = 1 (ii) ( x y) y (iii) ( x y) = (( x y ) y ) Note From the definition 23 an algebra (A,,, *, 0, 1) i a lattie Wajberg algebra with lower bound 0 and upper bound 1 Propoition 24 [6] A lattie Wajberg algebra (A,, *, 1) atifie the following axiom for all x, y, z A, (i) If x y then x z y z and z x z y (ii) x y z if and only if y x z (iii) ( x y) = ( x y ) (iv) ( x y) = ( x y ) (v) ( x y) z = ( x z) ( y z) (vi) x ( y z) ( x z) (vii) ( x y) ( y x) = 1 (viii) x ( y z) ( x z) (ix) ( x y) z = ( x z) ( y z) (x) ( x y) z = ( x z) ( y z) (xi) ( x y) z ( x z) Definition 25 [7] The lattie Wajberg algebra A i alled a lattie H-Wajberg algebra if x y (( x y) z) = 1 for all x, y, z A, In a lattie H-Wajberg algebra A, the following hold (i) x ( x y) (ii) x ( y z) ( x z) Definition 26 [6] Let L be a lattie An ideal I of L i a nonempty ubet of L i alled a lattie ideal if it atifie the following axiom for all x, y I, (i) x I, y L and y x imply y I (ii) x, y I implie x y I Definition 27 [7] Let A be a lattie Wajberg algebra Let I be a nonempty ubet of A Then I i alled WI-ideal of lattie Wajberg algebra A atifie for all x, y A, (i) 0 I (ii) ( x y) I and y I imply x I Definition 28 [13] Let X be a et A funtion µ : X [0, 1] i alled a fuzzy ubet on X for eah x X, the value of µ (x) deribe a degree of memberhip of x in µ 9

4 A Ibrahim and C Shajitha Begum Definition 29 [13] Let µ be a fuzzy ubet of X then the omplement of µ i denoted by µ and defined a µ ( x) = 1 µ ( x) for all x X Definition 210 [8] Let A be a lattie Wajberg algebra A fuzzy ubet µ of A i alled a fuzzy WI-ideal of A if for any x, y A, (i) µ ( 0) µ ( x) (ii) µ ( x ) min{ µ (( x y) ), µ ( y)} Definition 211 [1] An intuitioniti fuzzy ubet S in a non-empty et X i an objet having the form S = {( x,, ) / x X} = ( µ, γ ) where the funtion : X [0, 1] and : X [0, 1] denote the degree of memberhip and the degree of non-memberhip repetively, and for any x X 3 Intuitioniti fuzzy Wajberg impliative ideal (Intuitioniti fuzzy WI-ideal) In thi etion, we introdue the onept of an intuitioniti fuzzy WI-ideal and an intuitioniti fuzzy lattie ideal of lattie Wajberg algebra Alo, we obtain ome propertie of an intuitioniti fuzzy WI-ideal Definition 31 Let A be a lattie Wajberg algebra An intuitioniti fuzzy ubet S = ( µ, γ ) of A i alled an intuitioniti fuzzy WI-ideal of A if it atifie the following inequalitie for any x, y A, (i) µ ( 0) ( x) and γ ( 0) ( x) µ γ (ii) min{ µ (( x y) ), µ ( y)} (iii) max{ γ (( x y) ), γ ( y) } Example 32 Let A = { 0, a, b,, d, r,, t, 1} be a et with Figure (1) a a partial ordering Define a quai omplement and a binary operation on A a in Table (1) and Table (2) * x x 0 1 a t b b r d d r b t a a b d r t a t 1 1 t 1 1 t 1 1 b b t 1 t 1 t 1 r r r d d r r t 1 1 t 1 1 r d r t 1 t 1 b b b r r r t a b b d r r t a b d r t 1 Figure 1: Table 1: Table 2: 10

5 Intuitioniti fuzzy WI-ideal of lattie Wajberg algebra Define and operation on A a follow, ( x y) y, ( x y) = (( x y ) y ) for all x y A Then (A,,, *, 0, 1) i a lattie Wajberg algebra, Conider an intuitioniti fuzzy ubet S = ( µ, γ ) on A a, 06 otherwie 04 otherwie Then S i an intuitioniti fuzzy WI-ideal of A In the ame Example 32, let u onider an intuitioniti fuzzy ubet S = ( µ, γ ) on A if x { a, a, 032 otherwie if x { a, 056 otherwie Then S i not an intuitioniti fuzzy WI-ideal of A for µ ( t) < min{ µ (( t b) ), µ ( b) }, and γ ( t) > max{ γ (( t b) ), γ ( b) } Example 33 Let A be a lattie Wajberg algebra defined in example 32, define an intuitioniti fuzzy ubet S = ( µ, γ ) of A a follow, (i) µ ( 0) = µ ( ) = 1 (ii) = m for any x { a, b,, d, r,, t, 1} (iii) γ ( 0) = γ ( ) = 0 (iv) = n for any x { a, b,, d, r,, t, 1} where m, n [0, 1] and m + n 1 Then S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A Example 34 Let A = { 0, a, b, p, q,, d, 1} be a et with Figure (2) a a partial ordering Define a quai omplement and a binary operation on A a in Table (3) and Table (4) Define and operation on A a follow, ( x y) y, ( y) = (( x y ) y ) x y Then (A,,, *, 0, 1) i a lattie Wajberg algebra x for all, A 11

6 A Ibrahim and C Shajitha Begum * x x 0 1 a b b a p 0 q 0 0 d a b p q d a b 1 b b a a p 0 a b q 0 a b p a b p d 1 d 1 d 0 a b p a b p q d 1 Figure 2: Table 3: Table 4: Conider an intuitioniti fuzzy ubet S = ( µ, γ ) on A a, p} 056 otherwie p} 038 otherwie Then S i an intuitioniti fuzzy WI-ideal of A In the ame Example 34, let u onider an intuitioniti fuzzy ubet S = ( µ, γ ) on A p, a, 045 otherwie p, 032 otherwie Then S i not an intuitioniti fuzzy WI-ideal of A for µ ( d) < min{ µ (( d p) ), µ ( p)}, and γ ( d) > max{ γ (( d p) ), γ ( p) } Propoition 35 Every intuitioniti fuzzy WI-ideal S = ( µ, γ ) of lattie Wajberg algebra A i an intuitioniti monotoni, that i, if x y, then µ ( y) and γ ( y) Proof: Let x, y A, x y Then ( x y) = 1 = 0 min{ µ (( x y) ), µ ( y)} = min{ µ (0), µ ( y)} = µ (y) Therefore µ ( y) Now, max{ γ (( x y) ), γ ( y) } 12

7 Intuitioniti fuzzy WI-ideal of lattie Wajberg algebra = max{ γ (0), γ ( y)} = γ (y) Therefore γ ( y) Propoition 36 Let an intuitioniti fuzzy ubet S = ( µ, γ ) be an intuitioniti fuzzy WI-ideal of lattie Wajberg algebra A For any x, y, z A whih atifie x y z then min{ µ ( y), µ ( z) } and max{ γ ( y), γ ( z) } Proof: Let x, y, z A, x y z Then, we have 1 = x ( y z) = z ( x y) z [from (x) of propoition 22] and o, (( x y) z) ) = 0 It follow from definition 31 that, min{ µ (( x y) ), µ ( y)} min{ min{ µ (( x y) z) ), µ ( z)}, µ ( y)} = min{ min{ µ (0), µ ( z)}, µ ( y)} = min{ µ ( y), µ ( z)} Hene min{ µ ( y), µ ( z)} max{ γ (( x y) ), γ ( y)} max{ max{ γ (( x y) z) ), γ ( z)}, γ ( y)} = max{ max{ γ ((0), γ ( z)}, γ ( y)} = max{ γ ( y), γ ( z)} Hene max{ γ ( y), γ ( z)} Definition 37 An intuitioniti fuzzy ubet S = ( µ, γ ) of lattie A i alled an intuitioniti fuzzy lattie ideal of A if it atifie the following for all x, y A, (i) S = ( µ, γ ) i intuitioniti monotoni (ii) µ ( x y) min{, µ ( y) } (iii) γ ( x y) max{ γ ( x), γ ( y) } for all x, y A Remark 38 In the definition 37 (ii) and (iii) an be equivalently replaed by µ ( x y) = min{, µ ( y)} and γ ( x y) = max{, γ ( y) } repetively by γ Example 39 Let A be a lattie Wajberg algebra defined in the Example 32 and S = ( µ, γ ) be an intuitioniti fuzzy ubet of A defined by d} m otherwie 13

8 A Ibrahim and C Shajitha Begum d} n otherwie where m, n [0, 1] and m + n 1 Then S = ( µ, γ ) i an intuitioniti fuzzy lattie ideal of A Propoition 310 Let A be a lattie Wajberg algebra Every intuitioniti fuzzy WI-ideal of A i an intuitioniti fuzzy lattie ideal of A Proof: Let S = ( µ, γ ) be an intuitioniti fuzzy lattie ideal of A Then, from the propoition 35 how that S = µ, γ ) i intuitioniti monotoni ( Now (( x y) y) = ((( x y) y)) y) ( x ) = x for any x, y A, it follow from definition 31and definition 37 that, µ ( x y) min{ µ (( x y) y) ), µ ( y)} min{, µ ( y)} max{ γ (( x y) y) ), γ ( y)} max{, γ ( y)} Hene, we have S = ( µ, γ ) i an intuitioniti fuzzy lattie ideal of A The following example how that the onvere of propoition 310 i not true Example 311 Let A be a lattie Wajberg algebra defined in the Example 33 and S = ( µ, γ ) be an intuitioniti fuzzy ubet of A defined by, b, d} 07 otherwie b, d} 03 otherwie Then, we have S = ( µ, γ ) i an intuitioniti fuzzy lattie ideal of A, but not an intuitioniti fuzzy WI-ideal of A for µ ( p) < min{ µ (( p d) ), µ ( d) } and γ ( p) > max{ γ (( p d) ), γ ( d)} Propoition 312 In a lattie H-Wajberg algebra A, every intuitioniti fuzzy lattie ideal of A i an intuitioniti fuzzy WI-ideal of A Proof: Let S = ( µ, γ ) be an intuitioniti fuzzy lattie ideal of A Clearly µ ( 0) and γ ( 0) for any x A Now, x x y for all x, y A It follow from definition 37 that, µ ( x y) = µ ( y ( x y ) ) = µ ( y ( x y) ) min{ µ ( y), µ ( x y) }, 14

9 Intuitioniti fuzzy WI-ideal of lattie Wajberg algebra and γ ( x y) = γ ( y ( x y ) ) = γ ( y ( x y) ) max{ γ ( y), γ ( x y) )} Thu S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A Propoition 313 Let A be a lattie Wajberg algebra An intuitioniti fuzzy ubet S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A if and only if the fuzzy ubet µ and γ are fuzzy WI-ideal of A, where = 1 γ ( x) for any x A Proof: Let S = ( µ, γ ) be an intuitioniti fuzzy WI-ideal of A Clearly, µ i a fuzzy WI-ideal of A For any x, y A, we have γ ( 0) = 1 γ (0) 1 γ ( 0) = and = 1 1 max{ γ (( x y) ), γ ( y)} = min{ 1 γ (( x y) ), 1 γ ( y)} = min{ γ (( x y) ), γ ( y)} Hene, we have γ i a fuzzy WI-ideal of A Converely, aume that µ and γ are fuzzy WI-ideal of A For any x, y A, we get µ ( 0) and 1 γ (0) = γ (0) γ (x) = 1 γ ( 0) min{ µ ( x y), µ ( y)}, and 1 = min{ µ (( x y) ), µ ( y)} = min{ 1 µ (( x y) ), 1 µ ( y)} = 1 max{ µ (( x y) ), µ ( y)} max{ γ (( x y) ), γ ( y)} Hene, we have S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A 15

10 A Ibrahim and C Shajitha Begum Propoition 314 Let A be a lattie Wajberg algebra and S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A Then S = µ, γ ) i an intuitioniti fuzzy WI-ideal ( of A if and only if ( µ, µ ) and ( γ, γ ) are intuitioniti fuzzy WI-ideal of A Proof: Let S = ( µ, γ ) be an intuitioniti fuzzy WI-ideal of A, then µ and γ are fuzzy WI-ideal of A from propoition 313 Hene, we have ( µ, µ ) and ( γ, γ ) are intuitioniti fuzzy WI-ideal of A Converely, if ( µ, µ ) and ( γ, γ ) are intuitioniti fuzzy WI-ideal of A, then the fuzzy ubet µ andγ are fuzzy WI-ideal of A, hene S = ( µ, γ ) i an intuitioniti fuzzy WI-ideal of A Propoition 315 Let A be a lattie Wajberg algebra, V a non-empty ubet of [0, 1] and { I t / t V} a olletion of WI-ideal of A uh that (i) A = U I t v t (ii) r > t if and only if I r It for any r, t V then the intuitioniti fuzzy ubet S = ( µ, γ ) of A defined by µ = Sup{ t V / x It } and γ = Inf { t V / x It } for any x A i an intuitioniti fuzzy WI-ideal of A Proof: Aording to propoition 313, it i uffiient to how that µ and γ are fuzzy WI-ideal of A for any x A µ ( 0) = Sup{ t V / 0 It} = SupV µ (x) If there exit x, y A uh that < min{ µ (( x y) ), µ ( y) } Then there exit t 1 uh that < t1 < min{ µ (( x y) ), µ ( y) } It follow that t1 < µ (( x y) ), t1 < µ ( y), and hene there exit t2, t3 V, t2 > t1, t3 > t1, ( x y) It It follow that ( x y) I t 2 t3 and y It 2 t, hene x I 3 t 2 t 3 That i, = Sup{ t V / x It} t2 t3 > t1 Therefore, > t1 2 and y I t3 Thi i a ontradition Hene, we have µ i a fuzzy WI-ideal of A γ i a fuzzy WI-ideal, whih an be proved by imilar method 4 Conluion In thi paper, we have introdued the definition of intuitioniti fuzzy WI-ideal and intuitioniti fuzzy lattie ideal of lattie Wajberg algebra We have diued ome of their propertie with illutration Alo, we have hown that every intuitioniti fuzzy WI-ideal of lattie Wajberg algebra i an intuitioniti fuzzy lattie ideal of lattie Wajberg algebra But, the onvere part i true only in the lattie H-Wajberg algebra Finally, we have hown that olletion of WI-ideal of lattie Wajberg algebra i an 16

11 Intuitioniti fuzzy WI-ideal of lattie Wajberg algebra intuitioniti fuzzy WI-ideal of lattie Wajberg algebra We hope that more link of intuitioniti fuzzy ubet and logi emerge by the tipulating of thi work REFERENCES 1 KTAtanaov, Intuitioniti fuzzy et, Fuzzy Set and Sytem, 20(1) (1986) KTAtanaov, New operation defined over the intuitioniti fuzzy et, Fuzzy Set and Sytem, 61(2) (1994) MBaheer Ahamed and AIbrahim, Anti fuzzy impliative filter in lattie W-algebra, International Journal of Computational Siene and Mathemati, 4(1) (2012) CCChang, Algebrai analyi of many-valued logi, Tranation of the Am Math So, 88 (1958) CCChang, A new proof of the ompletene of the Lukaiewiz axiom, Tranation of the Am Math So, 93 (1959) JMFont, AJRodriguez and ATorren, Wajberg algebra, STOCHASTICA, 8(1) (1984) AIbrahim and CShajitha Begum, On WI-Ideal Of lattie Wajberg algebra, Global Journal of Pure and Applied Mathemati, 13(10) (2017) AIbrahim and CShajitha Begum, Fuzzy and normal fuzzy WI-ideal of lattie Wajberg algebra, International Journal of Mathematial Arhive, 8(11) (2017) YBJun and KHKim, Intuitioniti fuzzy ideal in BCK-algebra, Internat J Math and Math Si, 24(12) (2000) YBJun and CHPark, Ideal of PSEUDO MV- algebra baed on vague et theory, Iranian Journal Of Fuzzy Sytem, 6(2) (2009) ARoe and JBRoer, Fragment of many valued tatement aluli, Tranation of Amerian Mathematial Soiety, 87 (1958) MWajberg, Beitrage zum Metaauagenkalkul, Monat Mat Phy, 42 (1935) LAZadeh, Fuzzy Set, Information and Control, 8(3) (1965)