The notion of interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov as a 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 intuitionistic fuzzy dual ideals to BF-algebras. In this paper, we introduce the notion of fully invariant and characteristic interval-valued intuitionistic fuzzy dual ideals of BF-algebras and investigate some of its properties

Prasad, R. Durga

Ramesh, D.

Satyanarayana, B.

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 52  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.com Abstract The notion of interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov as a 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 intuitionistic fuzzy dual ideals to BF-algebras. In this paper, we introduce the notion of fully invariant and characteristic interval-valued intuitionistic fuzzy dual ideals of BF-algebras and investigate some of its properties. 1. Introduction and preliminaries  For the first time Zadeh (1965) introduced the concept of fuzzy sets and also Zadeh (1975) introduced the 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, which is 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 intuitionistic fuzzy dual ideals to BF-algebras. In this paper we introduce the notion of fully invariant and characteristic interval-valued intuitionistic fuzzy dual ideals of BF-algebras and investigate some of its properties.   By a BF-algebra we mean an algebra satisfying the axioms:  0x x(1). =∗ , x0 x(2). =∗ , xyy)(x0 (3). ∗=∗∗ , for all Xyx, ∈  Throughout this paper, X  is a BF-algebra. If there is an element 1 of X satisfying 1x ≤ , for all Xx ∈ , then the element 1 is called unit of X. A BF-algebra with unit is called bounded. In a bounded Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 53 BF-algebra, we denote x*1 by Nx for brief. A bounded BF-algebra X called involutory if NNx=x, for all Xx ∈ .  Definition 1.1 (Satyanarayana et al., 2011) A nonempty subset D in a BF-algebra X is said to be a dual ideal of X if it satisfies: D1 )(D1 ∈ , DNy)N(Nx  )(D2 ∈∗ and Dy ∈ imply Dx ∈ , for any X.y x, ∈ Definition 1.2 Let X be a set. A fuzzy set in X is a function .[0,1]X:µ →   Definition 1.3 (Meng and Jun (1993) A fuzzy subset  of X is said to be a fuzzy dual ideal of X if it satisfies (FDI 1) µ(x)µ(1) ≥  (FDI 2) µ(y)} Ny)),µ(N(Nx min{µ(x) ∗≥ for all x, y in X. We now review some fuzzy logic concepts. For fuzzy sets µ and λ of X and 1] [0, ts, ∈ , the set ( ) { }tµ(x):Xx tµ;U ≥∈=  is called upper t-level cut of µ and the set ( ) { }sλ(x):Xx tµ;L ≤∈= is called lower s-level cut of λ . The fuzzy set µ  in X is called a fuzzy dual sub algebra of X , if { }µ(y) µ(x),minNy))µ(N(Nx ≥∗ , for all Xyx, ∈ .  Intuitionistic fuzzy sets: (Fatemi 2011 and Wang et. al., 2011) An intuitionistic fuzzy set (shortly IFS) in a non-empty set X is an object having the form ( ){ }Xx:(x)λ (x),µ x,A AA ∈= , where the functions 1] [0,X:µA → and 1] [0,X:λA →  denote the degree of membership (namely (x)µA ) and the degree of non membership (namely (x)λA ) of each element Xx ∈  to the set A  respectively such that for any Xx ∈ . 1(x)λ(x)µ0 AA ≤+≤ . For the sake of simplicity we use the symbol form )λ ,µ (A AA=  or )λ ,µ (X,A AA= .  By an interval number D on 1] [0,  we mean an interval [ ]+− a ,a  where 1aa0 ≤≤≤ +− . The set of all closed subintervals of 1] [0,  is denoted by 1] D[0, .  For interval numbers [ ]+−= 111 b ,aD  , Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 54 [ ]+−= 222 b ,aD . We define   • )D ,min(DDD 2121 =∩ = min [ ] [ ]( )+−+− 2211 b  ,a  ,b  ,a  = { } { }[ ]++−− 2121 b,bmin ,a,amin   • )D,max(DDD 2121 =∪ = max [ ] [ ]( )+−+− 2211 b  ,a  ,b  ,a  = { } { }[ ]++−− 2121 b,bmax ,a,amax   ].bbbb ,.aaa[aDD 2121212121++++−−−−−+−+=+ and put  • −− ≤⇔≤ 2121 aaDD  and ++ ≤ 21 bb   • −−=⇔= 2121 aaDD and ++= 21 bb ,  • 2121 DDDD ≤⇔< and 21 DD ≠   • ]mb ,[ma]b ,m[amD 1111 +−+− == , where 1m0 ≤≤ .  Obviously ( )∧∨≤  , ,  D[0,1],  form a complete lattice with 0] [0,  as its least element and 1] [1,  as its 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 defined by ( ){ }Lx:(x)]µ (x),[µ x,B BB ∈= +− , where (x)µB−  and (x)µB+  are fuzzy sets of L such that (x)µB− ≤ (x)µB+  for all Lx ∈ . Let (x)µ~B  = [ ](x)µ (x),µ BB +− , then { }Lx:(x))µ~ (x,B B ∈=  where 1] D[0,L:µ~B → .  A mapping )λ~ ,µ~(A AA= 1] D[0,1] D[0,L: ×→ is called an interval-valued intuitionistic fuzzy set (i-v IF set, in short) in L  if 1(x)λ(x)µ0 AA ≤+≤ ++  and 1(x)λ(x)µ0 AA ≤+≤ −−  for all Lx ∈  (that is, )λ ,µ (X,A AA +++ =  and )λ,µ(X,A AA −−− = are intuitionistic fuzzy sets), where the mappings 1] D[0,L:(x)]µ (x),[µ(x)µ~ AAA →= +− and =(x)λ~A :(x)]λ (x),[λ AA +−  1] D[0,L →  denote Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 55 the degree of membership (namely (x))µ~A  and degree of non-membership (namely (x))λ~A  of each element Lx ∈  to A  respectively.  2 Main Result In this section we introduce fully invariant and characteristic interval–valued intuitionistic fuzzy dual ideals and prove some of its properties.  Definition 2.1 A dual ideal Fof BF-algebra X is said to be a fully invariant dual ideal if Ff(F) ⊆  for all End(X)f ∈  where End(X) is set of all endomorphisms of BF-algebras X.  Definition 2.2 An interval-valued IFS )λ~,µ~(X,A AA=  is called interval-valued intuitionistic fuzzy dual ideal (shortly i-v IF dual ideal) of BF-algebra X if satisfies the following inequality (i-v IF1) (x)Aµ~(1)Aµ~ ≥  and (x)Aλ~(1)Aλ~≤  (i-v IF2) ( )( ){ }(y)Aµ~ ,NyNxNAµ~ min(x)Aµ~ ∗≥   (i-v IF2) ( )( ){ }(y)Aλ~ ,NyNxNAλ~ max(x)Aλ~ ∗≤ , for all Xz y, x, ∈ . Example 2.3 Consider a BF-algebra { }3 2, 1, 0,X =  with following table.          Let A be an interval valued fuzzy set in X defined by  [ ]0.7 0.6,(1)µ~(0)µ~ AA == ,  [ ]0.3 0.2,(3)µ~(2)µ~ AA == , [ ]0.2 0.1,(1)λ~(0)λ~ AA ==  and [ ]0.7 0.5,(3)λ~(2)λ~ AA == .  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 )λ~,µ~(A AA= of X is called a fully invariant if (x)µ~(f(x))µ~(x)µ~ AAfA ≤=  and (x)λ~(f(x))λ~(x)λ~ AAfA ≤=  for all Xx ∈  and ∗ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 56 End(X)f ∈ . Theorem 2.5 If I}i\{A i ∈  is a family of i-v intuitionistic fuzzy fully invariant dual ideals of X , then I Ii )iAλ~Ii ,iAµ~Ii(iA∈ ∈∨∈∧= is an interval-valued intuitionistic fully invariant dual ideal of X , where   X} xI,i\(x)iAµ~inf{(x)iAµ~Ii∈∈=∈∧  and X}.xI,i\(x)iAλ~sup{(x)iAλ~Ii∈∈=∈∨  Theorem 2.6 Let S be nonempty subsets of BF-algebra X and )λ~ ,µ~(A AA=  an i-v intuitionistic fuzzy dual ideals defined by    ∈=otherwise,] t,[sS xif] t,[s(x)µ~1122A  and   ∈=otherwise,]β ,[αS xif]β ,[α(x)λ~1122A  Where 1], [1,]β ,[α]β ,[α0] [0, 1], [1,] t,[s] t,[s0] [0, 11222211 ≤<≤≤<≤ 1] [1,]β ,[α] t,[s0] [0, iiii ≤+≤ for 2 1,i = .  If S  is an interval-valued intuitionistic fully invariant dual ideal of X , then )λ~ ,µ~(A AA=  is an interval-valued intuitionistic fully invariant dual ideal of X . Proof: We can easily see that )λ~ ,µ~(A AA=  is an i-v intuitionistic fuzzy dual ideal of X . Let Xx ∈  and End(X)f ∈ . If S,x ∈ then Sf(S)f(x) ⊆∈ .  Thus we have   ] t,[s(x)µ~(f(x))µ~(x)µ~ 22AAfA =≤= and ]β ,[α(x)λ~(f(x))λ~(x)λ~ 22AAfA =≤= .  For if otherwise, we have ] t,[s(x)µ~(f(x))µ~(x)µ~ 11AAfA =≤=  and ]β ,[α(x)λ~(f(x))λ~(x)λ~ 11AAfA =≤= . Thus, we have verified that )λ~ ,µ~(A AA=  is an interval-valued intuitionistic fully invariant dual ideal of X . Definition 2.7 An i-v intuitionistic fuzzy dual ideal )λ~ ,µ~(A AA=  of X has the same type as an i-v intuitionistic fuzzy dual ideal )λ~,µ~(B BB=  of X  if there exist End(X)f ∈  such that Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 57 fBA o= i. e (f(x))λ~(x)λ~ (f(x)),µ~(x)µ~ BABA ≥≥ for all Xx ∈ .  Theorem 2.8 Interval-valued intuitionistic fuzzy dual ideals of X have same type if and only if they are isomorphic. Proof: We only need to prove the necessity part because the sufficiency part is obvious. If an i-v intuitionistic fuzzy dual ideal )λ~ ,µ~(A AA= of X has the same type as )λ~ ,µ~(B BB= , then there exist End(X)∈φ  such that (x))(λ~(x)λ~ (x)),(µ~(x)µ~ BABA φφ ≥≥  for all Xx ∈ .  Let B(X)A(X):f → be a mapping defined by (x))B(f(A(x) φ=  for all Xx ∈ , i.e, (x))(λ~(x))λ~f(  (x)),(µ~(x))µ~f( BABA φφ == for Xx ∈ .   Then, it is clear that f is a surjective homomorphism. Also, f  is injective because (y))µ~f((x))µ~f( AA = for all  Xy x, ∈ implies (y))(µ~(x))(µ~ BB φφ = .  Hence (y)µ~(x)µ~ BA = .  Likewise, from (y))λ~f((x))λ~f( AA =  we conclude (y)λ~(x)λ~ BA =  for all Xx ∈ .  Hence )λ~ ,µ~(A AA= is isomorphic to )λ~,µ~(B BB= . This completes the proof.  Definition 2.9 An ideal C of X  is said to be characteristic if Cf(C) =  for all Aut(X)f ∈  where Aut(x)  is the set of all automorphisms of X . An i-v intuitionistic fuzzy dual ideal )λ~ ,µ~(A AA= of X  is called characteristic if (x)µ~(f(x)µ~ AA =  and (x)λ~(f(x))λ~ AA =  for all Xx ∈  and Aut(X)f ∈ .  Lemma 2.10 Let )λ~ ,µ~(A AA=  be an i-v intuitionistic fuzzy dual ideal of X and let Xx ∈ . Then t~(x)µ~A = , s~(x)λ~A =  if and only if )s~ ;µ~U(  x),t~ ;µ~U(x AA ∉∈  and )s~ ;λ~L(x A∈ , )t~ ;λ~L(x A∉  for all t~s~ > . Proof: Straight forward  Theorem 2.11 An i-v intuitionistic fuzzy dual ideal is characteristic if and only if each its level set is a characteristic dual ideal. Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 58 Proof: Let an i-v intuitionistic fuzzy dual ideal )λ~ ,µ~(A AA=  be characteristic, )t~;µ~U( xAut(X),f ),µ~Im(t~ AA ∈∈∈ .  Then t~(x)µ~(f(x))µ~ AA ≥= , which means that )t~ ;µ~U(f(x) A∈ .  Thus )t~ ;µ~U())t~ ;µ~f(U( AA ⊆ . Since for each  )t~;µ~U(x A∈ there exist Xy ∈  such that xf(y) =  we have t~(x)µ~(f(y))µ~(y)µ~ AAA ≥== , hence we conclude ).t~;µ~U(y A∈  Consequently, )t~ ;µ~f(U(f(y)x A∈= .  Hence )t~ ;µ~U()t~ ;µ~f(U( AA = . Similarly, )s~;λ~L())s~;λ~f(L( AA = .  This proves that )t~;µ~U( A  and )s~;λ~L( A are characteristic. Conversely, if all levels of )λ~,µ~(A AA= are characteristic dual ideals of X, then for Xx ∈ , Aut(X)f ∈  and (x)λ~s~t~(x)µ~ AA =<= , by lemma 2.10, we have )s~;µ~U( x),t~;µ~U(x AA ∉∈  and )t~;λ~L( x),s~;λ~L(x AA ∉∈ .  Thus )t~;µ~U())t~;µ~f(U(f(x) AA =∈  and )s~;λ~L()s~;λ~f(L(f(x) AA =∈ , i.e. t~(f(x))µ~A ≥  and s~(f(x))λ~A ≤ .  For t~t~(f(x))µ~ 1A >= , s~s~(f(x))λ~ 1A <= .  We have )t~;µ~f(U()t~;µ~U(f(x) 1A1A =∈ , ))s~;λ~f(L()s~;λ~L(f(x) 1A1A =∈ .  Hence )t~;µ~U(x 1A∈  , )s~;λ~L(x 1A∈  this is a contradiction.  Thus (x)µ~(f(x))µ~ AA =  and (x)λ~(f(x))λ~ AA = .So, )λ~ ,µ~(A AA=  is characteristic. References Atanassov, K.T. and G. Gargov, 1989. Interval valued intuitionistic fuzzy sets, Fuzzy  sets and systems, 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 in BF-algebras. Int. J. Math. Sci. Engg. Apple., 4: 263-274. Satyanarayana, B., D. Ramesh and R. Durga Prasad, 2011. On interval-valued intuitionistic fuzzy dual ideals 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 eigenvectors and its application. Inf. Technol. J., 10: 1850-1866. Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.2, No.3, 2012 59 Zadeh, 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.  This academic article was published by The International Institute for Science, Technology and Education (IISTE).  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Fully Invariant and Characteristic Interval-Valued Intuitionistic Fuzzy Dual Ideals of BF-algebras

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Fully Invariant and Characteristic Interval-Valued Intuitionistic Fuzzy Dual Ideals of BF-algebras

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