Mathematical Theory and Modeling                                                           www.iiste.orgISSN 2224-5804 (Pa...
Mathematical Theory and Modeling                                                 www.iiste.orgISSN 2224-5804 (Paper)     I...
Mathematical Theory and Modeling                                                                  www.iiste.orgISSN 2224-5...
Mathematical Theory and Modeling                                                              www.iiste.orgISSN 2224-5804 ...
Mathematical Theory and Modeling                                                                   www.iiste.orgISSN 2224-...
Mathematical Theory and Modeling                                                            www.iiste.orgISSN 2224-5804 (P...
Mathematical Theory and Modeling                                                              www.iiste.orgISSN 2224-5804 ...
Mathematical Theory and Modeling                                                          www.iiste.orgISSN 2224-5804 (Pap...
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11.fully invariant and characteristic interval valued intuitionistic fuzzy dual ideals of bf

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  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 Fully Invariant and Characteristic Interval-Valued Intuitionistic Fuzzy Dual Ideals of BF-algebras B. Satyanarayana, D. Ramesh and R. Durga Prasad Department of Applied Mathematics Acharya Nagarjuna University Campus Nuzvid-521201, Krishna (District) Andhara Pradesh, INDIA. E. Mail: *drbsn63@yahoo.co.in ram.fuzzy@gmail.com durgaprasad.fuzzy@gmail.comAbstractThe notion of interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov asa generalization of both interval-valued fuzzy sets and intuitionistic fuzzy sets. Satyanarayana et. al.,applied the concept of interval-valued intuitionistic fuzzy ideals and interval-valued intuitionisticfuzzy dual ideals to BF-algebras. In this paper, we introduce the notion of fully invariant andcharacteristic interval-valued intuitionistic fuzzy dual ideals of BF-algebras and investigate some of itsproperties.1. Introduction and preliminariesFor the first time Zadeh (1965) introduced the concept of fuzzy sets and also Zadeh (1975) introducedthe concept of an interval-valued fuzzy sets, which is an extension of the concept of fuzzy set.Atanassov and Gargov, 1989) introduced the notion of interval-valued intuitionistic fuzzy sets, whichis a generalization of both intuitionistic fuzzy sets and interval-valued fuzzy sets. Meng and Jun (1993)introduced fuzzy dual ideals in BCK-algebras. On other hand, Satyanarayana et al., (2010) and (2011)applied the concept of interval-valued intuitionistic fuzzy ideals and interval-valued intuitionisticfuzzy dual ideals to BF-algebras. In this paper we introduce the notion of fully invariant andcharacteristic interval-valued intuitionistic fuzzy dual ideals of BF-algebras and investigate some of itsproperties. By a BF-algebra we mean an algebra satisfying the axioms:(1). x ∗ x = 0 ,(2). x ∗ 0 = x ,(3). 0 ∗ (x ∗ y) = y ∗ x , for all x, y ∈ XThroughout this paper, X is a BF-algebra. If there is an element 1 of X satisfying x ≤ 1 , for allx ∈ X , then the element 1 is called unit of X. A BF-algebra with unit is called bounded. In a bounded 52
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012BF-algebra, we denote 1 * x by Nx for brief. A bounded BF-algebra X called involutory ifNNx=x, for all x ∈ X .Definition 1.1 (Satyanarayana et al., 2011) A nonempty subset D in a BF-algebra X is said to be adual ideal of X if it satisfies:(D1 ) 1 ∈ D ,(D 2 ) N(Nx ∗ Ny) ∈ D and y ∈ D imply x ∈ D , for any x, y ∈ X.Definition 1.2 Let X be a set. A fuzzy set in X is a function µ : X → [0,1].Definition 1.3 (Meng and Jun (1993) A fuzzy subset  of X is said to be a fuzzy dual ideal of X if itsatisfies(FDI 1) µ(1) ≥ µ(x)(FDI 2) µ(x) ≥ min{ µ(N(Nx ∗ Ny)), µ(y)} for all x, y in X.We now review some fuzzy logic concepts. For fuzzy sets µ and λ of X and s, t ∈ [0, 1] , the setU (µ; t ) = {x ∈ X : µ(x) ≥ t} is called upper t-level cut of µ and the setL(µ; t ) = {x ∈ X : λ(x) ≤ s} is called lower s-level cut of λ . The fuzzy set µ in X is called afuzzy dual sub algebra of X , if µ(N(Nx ∗ Ny)) ≥ min{µ(x), µ(y)} , for all x, y ∈ X . Intuitionistic fuzzy sets: (Fatemi 2011 and Wang et. al., 2011) An intuitionistic fuzzy set (shortlyIFS) in a non-empty set X is an object having the form A = {(x, µ A (x), λ A (x) ) : x ∈ X} , where thefunctions µ A : X → [0,1] and λ A : X → [0, 1] denote the degree of membership (namely µ A (x) )and the degree of non membership (namely λ A (x) ) of each element x ∈ X to the set Arespectively such that for any x ∈ X . 0 ≤ µ A (x) + λ A (x) ≤ 1 . For the sake of simplicity we usethe symbol form A = ( µ A , λ A ) or A = (X, µ A , λ A ) . [ By an interval number D on [0, 1] we mean an interval a − , a + ] where 0 ≤ a − ≤ a + ≤ 1 . The setof all closed subintervals of [0, 1] is denoted by D[0, 1] . [− + For interval numbers D1 = a 1 , b1 ], 53
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 [ ]D 2 = a − , b + . We define 2 2 − [ + − ][ • D1 ∩ D 2 = min(D1 , D 2 ) = min ( a 1 , b1 , a 2 , b 2 + ]) = [min{a , a }, min{b , b }] − 1 − 2 + 1 + 2 • D1 ∪ D 2 = max(D1 , D 2 ) = max ([a − 1 , b ], [a + 1 − 2 , b ]) = [max{a , a }, max{b , b }] + 2 − 1 − 2 + 1 + 2 D 1 + D 2 = [a 1 + a − − a 1 .a − , b1 + b + − b1 .b + ] − 2 − 2 + 2 + 2and put • D1 ≤ D 2 ⇔ a 1 ≤ a − and b1 ≤ b + − 2 + 2 • D1 = D 2 ⇔ a 1 = a − and b1 = b + , − 2 + 2 • D1 < D 2 ⇔ D1 ≤ D 2 and D1 ≠ D 2 − + − + • mD = m[a 1 , b1 ] = [ma 1 , mb1 ] , where 0 ≤ m ≤ 1 .Obviously (D[0,1], ≤, ∨, ∧ ) form a complete lattice with [0, 0] as its least element and [1, 1] asits greatest element. We now use D[0, 1] to denote the set of all closed subintervals of the interval[0, 1]. Let L be a given nonempty set. An interval-valued fuzzy set (briefly, i-v fuzzy set) B on L is B{( B ) }defined by B = x, [µ − (x), µ + (x)] : x ∈ L , where µ − (x) and µ + (x) are fuzzy sets of B B B B B B [L such that µ − (x) ≤ µ + (x) for all x ∈ L . Let µ B (x) = µ − (x), µ + (x) , then ~ ]B = {(x, µ B (x)) : x ∈ L} where µ B : L → D[0, 1] . A mapping A = (µ A , λ A ) ~ ~ ~ ~: L → D[0, 1] × D[0, 1] is called an interval-valued intuitionistic fuzzy set (i-v IF set, in short) inL if 0 ≤ µ + (x) + λ + (x) ≤ 1 and 0 ≤ µ − (x) + λ − (x) ≤ 1 for all x ∈ L (that is, A A A AA + = (X, µ + , λ + ) and A − = (X, µ − , λ − ) are intuitionistic fuzzy sets), where the mappings A A A A ~µ A (x) = [µ − (x), µ + (x)] : L → D[0, 1] and λ A (x) = [λ − (x), λ + (x)] : L → D[0, 1] denote~ A A A A 54
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 ~the degree of membership (namely µ A (x)) and degree of non-membership (namely~λ A (x)) of each element x ∈ L to A respectively.2 Main ResultIn this section we introduce fully invariant and characteristic interval–valued intuitionistic fuzzy dualideals and prove some of its properties. Definition 2.1 A dual ideal F of BF-algebra X is said to be a fully invariant dual ideal if f(F) ⊆ Ffor all f ∈ End(X) where End(X) is set of all endomorphisms of BF-algebras X. ~ ~Definition 2.2 An interval-valued IFS A = (X, µ A , λ A ) is called interval-valued intuitionisticfuzzy dual ideal (shortly i-v IF dual ideal) of BF-algebra X if satisfies the following inequality ~ ~ ~(i-v IF1) µ ~ (1) ≥ µ (x) and λ (1) ≤ λ (x) A A A A ~(i-v IF2) µ (x) ≥ min { µ A (N (Nx ∗ Ny )), µ A (y)} ~ ~ A(i-v IF2) λ ~ A ~ { ~ } (x) ≤ max λ A (N(Nx ∗ Ny)), λ A (y) , for all x, y, z ∈ X .Example 2.3 Consider a BF-algebra X = {0, 1, 2, 3} with following table. ∗ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0Let A be an interval valued fuzzy set in X defined by µ A (0) = µ A (1) = [0.6, 0.7 ] , ~ ~µ A (2) = µ A (3) = [0.2, 0.3] , λ A (0) = λ A (1) = [0.1, 0.2] and λ A (2) = λ A (3) = [0.5, 0.7 ] .~ ~ ~ ~ ~ ~It is easy to verify that A is an interval valued intuitionistic fuzzy dual ideal of X. ~ ~Definition 2.4 An interval-valued intuitionistic fuzzy dual ideal A = (µ A , λ A ) of X is called a fully ~f ~ ~ ~f ~ ~invariant if µ A (x) = µ A (f(x)) ≤ µ A (x) and λ A (x) = λ A (f(x)) ≤ λ A (x) for all x ∈ X and 55
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012f ∈ End(X) .Theorem 2.5 If {A i i ∈ I} is a family of i-v intuitionistic fuzzy fully invariant dual ideals of X , ~ ~then I i∈I A i = (∧ i∈I µ A , ∨ i∈I λ A ) is an interval-valued intuitionistic fully invariant dual i iideal of X , where ~ ~ ~ ~ ∧ µ (x) = inf{µ (x) i ∈ I, x ∈ X} and ∨ λ (x) = sup{ λ (x) i ∈ I, x ∈ X}. i∈I A i Ai i∈I A i Ai ~ ~Theorem 2.6 Let S be nonempty subsets of BF-algebra X and A = (µ A , λ A ) an i-v intuitionisticfuzzy dual ideals defined by ~ [s , t ] if x ∈ S ~ [α , β ] if x ∈ S µ A (x) =  2 2 and λ A (x) =  2 2  [s1 , t 1 ] otherwise,  [α1 , β1 ] otherwise,Where [0, 0] ≤ [s1 , t 1 ] < [s 2 , t 2 ] ≤ [1, 1], [0, 0] ≤ [α 2 , β 2 ] < [α1 , β1 ] ≤ [1, 1],[0, 0] ≤ [s i , t i ] + [α i , β i ] ≤ [1, 1] for i = 1, 2 . If S is an interval-valued intuitionistic fully ~ ~invariant dual ideal of X , then A = (µ A , λ A ) is an interval-valued intuitionistic fully invariantdual ideal of X . ~ ~Proof: We can easily see that A = (µ A , λ A ) is an i-v intuitionistic fuzzy dual ideal of X . Letx ∈ X and f ∈ End(X) . If x ∈ S, then f(x)∈ f(S) ⊆ S . Thus we have ~f ~ ~ ~f ~ ~ µ A (x) = µ A (f(x)) ≤ µ A (x) = [s 2 , t 2 ] and λ A (x) = λ A (f(x)) ≤ λ A (x) = [α 2 , β 2 ] . For ifotherwise, we have~f ~ ~ ~f ~ ~µ A (x) = µ A (f(x)) ≤ µ A (x) = [s1 , t 1 ] and λ A (x) = λ A (f(x)) ≤ λ A (x) = [α1 , β1 ] . ~ ~Thus, we have verified that A = (µ A , λ A ) is an interval-valued intuitionistic fully invariant dualideal of X . ~ ~Definition 2.7 An i-v intuitionistic fuzzy dual ideal A = (µ A , λ A ) of X has the same type as an ~ ~i-v intuitionistic fuzzy dual ideal B = (µ B , λ B ) of X if there exist f ∈ End(X) such that 56
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 ~ ~ ~ ~A = B o f i. e µ A (x) ≥ µ B (f(x)), λ A (x) ≥ λ B (f(x)) for all x ∈ X . Theorem 2.8 Interval-valued intuitionistic fuzzy dual ideals of X have same type if and only if theyare isomorphic.Proof: We only need to prove the necessity part because the sufficiency part is obvious. If an i-v ~ ~ ~ ~intuitionistic fuzzy dual ideal A = (µ A , λ A ) of X has the same type as B = (µ B , λ B ) , then there ~ ~exist φ ∈ End(X) such that µ A (x) ≥ µ B (φ (x)), λ A (x) ≥ λ B (φ (x)) for all x ∈ X . Let ~ ~f : A(X) → B(X) be a mapping defined by f(A(x) = B(φ (x)) for all x ∈ X , i.e, ~ ~ ~ ~f(µ A (x)) = µ B (φ (x)), f( λ A (x)) = λ B (φ (x)) for x ∈ X . Then, it is clear that f is a ~ ~surjective homomorphism. Also, f is injective because f(µ A (x)) = f(µ A (y)) for all x, y ∈ X ~ ~ ~ ~implies µ B (φ (x)) = µ B (φ (y)) . Hence µ A (x) = µ B (y) . Likewise, from ~ ~ ~ ~ ~ ~f( λ A (x)) = f( λ A (y)) we conclude λ A (x) = λ B (y) for all x ∈ X . Hence A = (µ A , λ A ) is ~ ~isomorphic to B = (µ B , λ B ) . This completes the proof. Definition 2.9 An ideal C of X is said to be characteristic if f(C) = C for all f ∈ Aut(X)where Aut(x) is the set of all automorphisms of X . An i-v intuitionistic fuzzy dual ideal ~ ~ ~ ~ ~ ~A = (µ A , λ A ) of X is called characteristic if µ A (f(x) = µ A (x) and λ A (f(x)) = λ A (x) for allx ∈ X and f ∈ Aut(X) . ~ ~ Lemma 2.10 Let A = (µ A , λ A ) be an i-v intuitionistic fuzzy dual ideal of X and let x ∈ X . ~ ~ ~ ~Then µ A (x) = t , λ A (x) = ~ if and only if x ∈ U(µ A ; t ), x ∉ U(µ A ; ~) and x ∈ L(λ A ; ~) , ~ s ~ ~ s s ~ ~ ~x ∉ L(λ A ; t ) for all ~ > t . sProof: Straight forward Theorem 2.11 An i-v intuitionistic fuzzy dual ideal is characteristic if and only if each its level set isa characteristic dual ideal. 57
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 ~ ~Proof: Let an i-v intuitionistic fuzzy dual ideal A = (µ A , λ A ) be characteristic,~ ~ ~ ~ ~ ~ ~t ∈ Im(µ A ), f ∈ Aut(X),x ∈ U(µ A ; t ) . Then µ A (f(x)) = µ A (x) ≥ t , which means that ~ ~ ~ ~ ~ ~ ~ ~f(x) ∈ U(µ A ; t ) . Thus f(U(µ A ; t )) ⊆ U(µ A ; t ) . Since for each x ∈ U(µ A ; t ) there exist ~ ~ ~ ~y ∈ X such that f(y) = x we have µ A (y) = µ A (f(y)) = µ A (x) ≥ t , hence we conclude ~ ~y ∈ U(µ A ; t ). ~ ~ ~ ~ ~ ~Consequently, x = f(y) ∈ f(U(µ A ; t ) . Hence f(U(µ A ; t ) = U(µ A ; t ) . ~ ~ ~ ~ ~Similarly, f(L( λ A ; ~)) = L( λ A ; ~ ) . This proves that U(µ A ; t ) and L( λ A ; ~ ) are characteristic. s s s ~ ~Conversely, if all levels of A = (µ A , λ A ) are characteristic dual ideals of X, then for x ∈ X , ~ ~ ~ ~f ∈ Aut(X) and µ A (x) = t < ~ = λ A (x) , by lemma 2.10, we have x ∈ U(µ A ; t ), x ∉ U(µ A ; ~) ~ s ~ s ~ ~ ~ ~ ~ ~ ~and x ∈ L( λ A ; ~ ), x ∉ L( λ A ; t ) . Thus f(x) ∈ f(U(µ A ; t )) = U(µ A ; t ) and s ~ ~ ~ ~ ~ ~ ~ ~f(x) ∈ f(L( λ A ; ~ ) = L( λ A ; ~ ) , i.e. µ A (f(x)) ≥ t and λ A (f(x)) ≤ ~ . For µ A (f(x)) = t1 > t , s s s~ ~ ~ ~ ~ ~ ~λ A (f(x)) = ~1 < ~ . We have f(x) ∈ U(µ A ; t1 ) = f(U(µ A ; t1 ) , f(x) ∈ L( λ A ; ~1 ) = f(L( λ A ; ~1 )) . s s s s ~ ~ ~Hence x ∈ U(µ A ; t1 ) , x ∈ L(λ A ; ~ ) this is a contradiction. Thus µ A (f(x)) = µ A (x) and s1 ~ ~~ ~ ~ ~λ A (f(x)) = λ A (x) .So, A = (µ A , λ A ) is characteristic.ReferencesAtanassov, K.T. and G. Gargov, 1989. Interval valued intuitionistic fuzzy sets, Fuzzy sets andsystems, 31: 343-349.Fatemi A., 2011. Entropy of stochastic intuitionistic fuzzy sets, Journal of Applied Sci. 11: 748-751.Meng, J. and Y.B. Jun, 1993. Fuzzy dual ideals in BCK-algebras, Comm. Korean Math. Soc., 8:225-231.Satyanarayana, B., D. Ramesh, M.V. Vijayakumar and R. Durga Prasad, 2010. On fuzzy ideal inBF-algebras. Int. J. Math. Sci. Engg. Apple., 4: 263-274.Satyanarayana, B., D. Ramesh and R. Durga Prasad, 2011. On interval-valued intuitionistic fuzzy dualideals of bf-algebras, annals of the constantin brancusi university of targu jiu. Eng. Series, 3: 116-126.Wang, H., G. Qian and X. Feng, 2011. An intuitionistic fuzzy AHP based on synthesis of eigenvectorsand its application. Inf. Technol. J., 10: 1850-1866. 58
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012Zadeh, L.A., 1965. Fuzzy sets, Informatiuon Control, 8: 338-353.Zadeh, L.A., 1975. The concept of a linguistic variable and its application to aproximate reasoning,Information Sciences, 8: 199-249. 59
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