Intuitionistic Fuzzy Soft N-ideals

Transcription

1 Intenational Jounal of Fuzzy Mathematics and Systems. ISSN Volume 3, Numbe 4 (2013), pp Reseach India Publications Intuitionistic Fuzzy Soft N-ideals A. Solaiaju 1, P. Saangapani 2, R. Nagaajan 3 and P. Muuganantham 4 1 AssociatePofesso, PG & Reseach Depatment of Mathematics, Jamal Mohamed college, Tiuchiappalli-20, Tamilnadu, India 2 Assistant Pofesso, Depatment of Compute Science, Kuinji College of Ats & Science, Tiuchiappalli-02, Tamilnadu, India 3 Associate Pofesso, Depatment of Mathematics, J J College of Engineeing and Technology, Tiuchiappalli-09, Tamilnadu, India 4 Assistant Pofesso, Depatment of Mathematics, Kuinji College of Ats & Science, Tiuchiappalli-02, Tamilnadu, India Abstact The main pupose of this pape is to intoduce a basic vesion of Intuitionistic Fuzzy Soft N-ideal theoy, which extends the notion of ideals by intoducing some algebaic stuctues in soft set, Finally we investigate some basic popeties of maximal Intuitionistic Fuzzy Soft N-ideals. Keywods: Soft set Fuzzy soft set Soft N-ideal Intuitionistic Fuzzy set Nomal fuzzy set maximal Intuitionistic Fuzzy Soft N-ideals. Section 1: Intoduction The notion of a fuzzy set was intoduced L.A. Zadeh [24] and since then this has been applied to vaious algebaic stuctues. The idea of an intuitionistic fuzzy set was intoduced by K.T. Atanassov [3][4] as a genealization of the notion of fuzzy set. The concept - nea ing, a genealization of both the concepts nea ing and - ing was intoduced by Satyanaayana [19]. Late seveal autho such as Booth [6] and Satyanaayana [19] studied the eal theoy of - nea ings. Late Jun.et.al [8][9][10][11] consideed the fuzzification of left (espectively ight) ideals of - nea ings. In 1999 Molodtsav[17] poposed an appoach fo modeling vagueness and uncetainty called soft set theoy. Since its inception woks on Soft set theoy has been applied to many diffeent fields, such as function smoothness, Riemann integation, Peason integation, Measuement theoy, Game theoy and decision making. Maji et.al [15] defined some opeations on soft sets. Aktas and Naimcagman [1] genealized soft sets by defining the concept of soft goups. Afte them, Sun.et.al

2 260 A. Solaiaju et al [22] gave soft modules. Atagun and Sezgin [2] defined the concepts of soft sub ings of a ing, soft sub ideals of a field and soft submodules of a module and studied thei elative popeties with espect to soft set opeations. Atagun and Sezgin [20] defined soft N-subgoups and soft N-ideals of an N-goup. Naim cagman et.al [20] intoduced the concept of union substuctues of an nea ings and N-subgoups. A.Solaiaju et.al [21] studied the idea of union fuzzy soft n-goup. In this pape, we investigate basic vesion of popeties of maximal Intuitionistic Fuzzy Soft N-ideals and popeties of maximal Intuitionistic Fuzzy Soft N-ideals. Section 2: Peliminaies In this section we include some elementay aspects that ae necessay fo this pape, fom now on we denote a - nea ing and N is ideal unless othewise specified. Definition 2.1: A nonempty set R with two binay opeations + (addition) and. (multiplication) is called a Nea-ing if it satisfies the following axioms. (R, +) is a goup. (R,.) is a semigoup. (x+y). z = x. z + y.z fo all x, y, z R. Definition 2.2: A - nea ing is a tiple (M, +, ) whee (M, +) is a goup. is a nonempty set of binay opeations on M such that fo, (M, +, ) is a nea ing. x (yβz) = (x y)βz fo all x, y, z M and. Definition 2.3: A subset A of a - nea ing M is called a left (espectively ight) ideal of M if (A, +) is a nomal diviso of (M, +). u (x+v) - u v A ( espectively x u A) fo all x A, and u, v M Definition 2.4: A fuzzy set A in a - nea ing m is called a fuzzy left (espectively ight) ideal of M if A(x-y) min{ A(x), A(y) } A(y+x-y) A(x) fo all x, y M A( u (x+v) - u v) A(x) fo all x, u, v M and. Definition 2.5: Let X be an initial univesal Set and E be a set of paametes. A pai (F, E) is called a soft set ove X if and only if F is a mapping fom E into the set of all subsets of the set that is F:E P(x) whee P(x) is the powe set of X. Definition 2.6: The elation complement of the soft set F A ove U is denoted by F A whee F A : A P(U) is a mapping given as F A ( ) = U/ F A ( ) fo all A.

3 Intuitionistic Fuzzy Soft N-ideals 261 Definition 2.7: Let X be a non-empty fixed A : X [0, 1] and A : X [0, 1] denote the degee of membeship of each element x X to the set S espectively and 0 A (x)+ A (x) 1. Notation: Fo the sake of simplicity, we shall use the symbol A = < A, A > fo the IFS A = {< x, A (x), A (x)> / x X} Definition 2.8: Let A be an IFS in a - nea ing M, fo each pai <, > [0, 1] with + 1, the set A <, > = { x X / A (x) and A (x) } is called a <, > level subset of A. Definition 2.9: Let F A be a soft set ove. If fo all x, y A, n N F(x-y) min{f(x), F(y)} F(y+x-y) F(x) F(n(y+x)-ny) F(x) Then F A is called a soft N-ideal of N. Definition 2.10: Let U be an initial Univesal Set and E be a set of paametes. A pai FS(U) denotes the fuzzy powe set of U and A E. A pai (F, A) is called a fuzzy soft set ove U, whee F is a mapping given by F:A P(U). A fuzzy soft set is a paameteized family of fuzzy subsets of U. Definition 2.11: An intutionistic fuzzy set A on Univese X can be defined as follows A = {< x, A (x), A (x)> / x X} whee A (x): X [0, 1] and A (x): X [0, 1] with the popety 0 A (x)+ A (x) 1, the value of A (x) and A (x) denote the degee of membeship and non-membeship of x to A, espectively. A (x)= 1 - A (x) - A (x) is called the intutionistic fuzzy index. Definition 2.12: Let U be an initial Univesal Set and E be a set of paametes. A pai IFS(U) denotes the intutionistic fuzzy soft set ove U and A E. A pai (F, A) is called an intutionistic fuzzy soft set ove U, whee F is a mapping given by F:A IFS(U). Definition 2.13: An intutionistic fuzzy ideal A of M is called an intutionistic fuzzy soft N-ideal (IFSNI) of M if A (x-y) min{ A (x), A (y)} A (n(y+x-y) A (x) A (x - y) max{ A (x), A (y)} A (n(y+x)-ny) A (x), fo all x, y A, n N Example 1: Let R be the set of all integes. Then R is a ing. Take M Γ = R. Let a, b M,, Suppose a b is the poduct of a, b, R, then M is a Γ nea ing.

4 262 A. Solaiaju et al Define an IFS A = < A, A > in R as follows A (0) = 1 and A (±1) = A (±2) = A (±3) =.. = t and A (0) = 1 and A (±1) = A (±2) = A (±3) =.. = s whee t (0, 1) and s (0, 1) and t+s 1. By outine calculations, clealy A is an intutionistic fuzzy soft N-ideal of M. Example 2: Conside the additive goup (Z 6, +) is a multiplication given in the following table. (Z 6, +) is a nea ing But conside the N ideal * = {0, 3}, we define soft set (g, *) by G(0)={1, 3, 5} and G(3)={1, 5} Since G(4.(2+3) - 4.2) = G( ) = G(2-2) = G(0) = {1, 3, 5} G(3)={1, 5} Definition 2.14: Let M be a sub nea ing of R and F M be a soft set ove N. If fo all x, y M F(x-y) min{f(x), F(y)} F(nxy) min{f(x), F(y)} then the soft set F M is called a soft N-nea ing of R. Definition 2.15: An IFSNI A of a Γ nea ing M is said to be a nomal if A (0)=1, A (0)=0. Section 3: Popeties of Intutionistic fuzzy soft N-ideals Poposition 3.1: Given an IFSNI A of a Γ nea ing M. Let A* be the IFS in M defined by A* (x) = A (x)+1 - A (0), A* (x) = A (x)+1 - A (0) fo all x M then A* is a nomal IFSNI of M. Poof: Fo all x M use A* (x) = A (x)+1 - A (0) = 1 and A* (x) = A (x) - A (0) = 0 We have (IFSNI 1 ) A* (x-y) = A (x-y)+1 - A (0) min { A (x), A (y)}+1 - A (0) = min { A (x)+1- A (0), A (y) A (0)} = min { A* (x), A* (y)}

5 Intuitionistic Fuzzy Soft N-ideals 263 (IFSNI 2 ) A* (x-y) = A (x-y) - A (0) max { A (x), A (y)} - A (0) = max { A (x)- A (0), A (y) - A (0)} = max { A* (x), A* (y)} (IFSNI 3 ) A* (n(y+x)-ny) = A (n(y+x)-ny)+1 - A (0) A (x)+1 - A (0) = A* (x) (IFSNI 4 ) A* (n(y+x)-ny) = A (n(y+x)-ny) - A (0) A (x) - A (0) = A* (x) Coollay: Let A and A* be as in Popety 3.1, if thee exists x M such that A*(x)=0 then A(x)=0. Poposition 3.2: Let A be a IFSNI A of a Γ nea ing M and let f:[0, (0)] [0, 1], g:[0, (0)] [0, 1] ae inceasing functions. Then the IFS A f : M [0, 1] defined by Af =f( A (x)), Af =f( A (x)) is IFSNI of M. Poof: Let x, y M we have Af (x-y) = f( A (x-y)) f min { A (x), A (y)} = min {f ( A (x)), f( A (y))} = min { Af (x), Af (y)} Af (x-y) = f( A (x-y)) f max { A (x), A (y)} = max {f ( A (x)), f( A (y))} = max { Af (x), Af (y)} Af (n(y+x)-ny) = f( A (n(y+x)-ny) f( A (x)) = Af (x) Af (n(y+x)-ny) = f( A (n(y+x)-ny) f( A (x)) = Af (x) Theefoe A f is an IFSNI of M. If f[ A (0)] = 1, then Af (0)=1 and f[ A (x)] = 0 and Af (x) = 0 then A f is nomal. Assume that f(t) = f[ A (x)] A (x) and

6 264 A. Solaiaju et al f(t) = f[ A (x)] A (x) fo any x M which gives A A f. Poposition 3.3: Let A N(M) is a non constant maximal element of (N(M), ). Then A takes only the two values (0, 1) and (1, 0). Poof: Since A is nomal, we have A (0)=1 and A (0) = 0, and A (x) 1 and A (x) 0 fo some x M, we conside that A (0)=1 and A (0) = 0, If not then x 0 M such that 0 A (x 0 ) 1 and 0 A (x 0 ) 1. Define an IFS on M, by setting 0 (x) = [ A (x-y)+ A (x 0 )] / 2 and 0 (x) = [ A (x-y)+ A (x 0 )] / 2 fo all x M Then clealy is well defined and fo all x, y M We have 0 (x-y) = [ A (x-y)+ A (x 0 )] / 2 ½ { min { A (x), A (y)} + A (x 0 )} { min { [ A (x)+ A (x 0 )] / 2, [ A (y)+ A (x 0 )] / 2 } min { A (x 0 ), A (y 0 )} 0 (x-y) = [ A (x-y)+ A (x 0 )] / 2 ½ { min { A (x), A (y)} + A (x 0 )} { min { [ A (x)+ A (x 0 )] / 2, [ A (y)+ A (x 0 )] / 2 } min { A (x 0 ), A (y 0 )} 0 (n(y+x)-ny) = [ A (n(y+x)-ny)+ A (x 0 )] / 2 ½ { A (x) + A (x 0 )} = A (x)+ A (x 0 ) / 2 = 0 (x) 0 (n(y+x)-ny) = [ 0 (n(y+x)-ny)+ A (x 0 )] / 2 ½ { 0 (x) + 0 (x 0 )} = 0 (x) Theefoe is an IFSNI of M. Poposition 3.4: If {A i / i I} is a family of IFSNI on M, then ( i I A i ) IFSNI of M. Poof: Let {A i / i I} is a family of IFSNI on M. Fo all x, y M, we have ( i I A i ) (x-y) = inf { A i (x-y) / i I} inf { min { A i (x), A i (y)} / i I} = min { ( i I A i ) (x), ( i I A i ) (y)}

7 Intuitionistic Fuzzy Soft N-ideals 265 ( i I A i ) (x-y) = sup { A i (x-y) / i I} sup { max { A i (x), A i (y)} / i I} = max { ( i I A i ) (x), ( i I A i ) (y)} ( i I A i ) (n(y+x)-ny) = inf { A i (n(y+x)-ny) / i I} inf { A i (x) / i I} = ( i I A i ) (x) ( i I A i ) (n(y+x)-ny) = sup { A i (n(y+x)-ny) / i I} sup { A i (x) / i I} = ( i I A i ) (x) Hence ( i I A i ) IFSNI of M. Definition 3.1: An IFSNI A of M is said to be complete if it is nomal and if thee exists x M such that A(x)=0. Poposition 3.5: Let A be an IFSNI of M and let be a fixed element of M such that a fuzzy soft set A* in M by A*(x) = [A(x) A( )] / [A(1) A( )] fo all x M then A* is an complete IFSNI of M. Poof: Fo any x, y M, we have A*(x-y) = [A(x-y) A( )] / [A(1) A( )] [min{a(x), A(y)} A( )] / [A(1) A( )] = min{[a(x) A( )] / [A(1) A( )], [A(y) A( )] / [A(1) A( )]} = min{a*(x), A*(y)} Also A*(x-y) = [A(x-y) A( )] / [A(1) A( )] [max{a(x), A(y)} A( )] / [A(1) A( )] = max{[a(x) A( )] / [A(1) A( )], [A(y) A( )] / [A(1) A( )]} = max{a*(x), A*(y)} A*(n(y+x)-ny) = [A(n(y+x)-ny) A( )] / [A(1) A( )] [A(x) A( )] / [A(1) A( )] = A*(x) A*(n(y+x)-ny) = [A(n(y+x)-ny) A( )] / [A(1) A( )] [A(x) A( )] / [A(1) A( )] = A*(x) Theefoe A* is a complete IFSNI of M. Poposition 3.6: Let F A be a soft set ove M. Then A is a soft N-sub nea ing of M if

8 266 A. Solaiaju et al A is a union soft N-nea ing of M. Poof: Let A is a soft N-sub nea ing of M. Then fo all x, y M F A (x-y) = N/ F A (x-y) N/ min{f A (x), F A (y)} = max {N/ F A (x), N/ F A (y)} = max { F A (x), F A (y)} = F A (x) F A (y) And F A (xy) = N/ F A (xy) N/ min{f A (x), F A (y)} = max {N/ F A (x), N/ F A (y)} = max { F A (x), F A (y)} = F A (x) F A (y) Thus F A is a soft N-sub nea ing of M. Conclusion This pape summaized the basic concepts of Intuitionistic fuzzy soft-ideals and soft N-sub nea ings. By using these concepts, we studied the algebaic popeties of IFSNI. Futue wok To extend this wok one could study the popety of IF soft sets in othe algebaic stuctues such as goups and fields. Refeences: [1] Aktas H, N.Cagman, Soft sets and soft goups, Infom. Sci. 177 (2007), [2] Atagun AO, Sezgin A, Aygun E (2011) A Note on soft nea ings and idealistic soft nea ings. Filomat 25(1):53-68 [3] Atanassov K.T, Remaks on the intuitionistic fuzzy sets-iii. Fuzzy Sets Syst 75(1995), [4] Atanassov K.T, Some opeations on intuitionistic fuzzy sets.fuzzy SetsSyst 114(2000), [5] Aygunoglu A, H. Aygun, Intoduction to fuzzy soft goups, Comput. Math. Appl. 58(2009), [6] Booth G.L, A note on -nea-ings Stud. Sci. Math. Hung. 23(1988),

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