SOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS

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1 IFSCOM016 1 Proceeding Book No. 1 pp (016) ISBN: SOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU, AND ARIF BAL Abstract. K.T. Atanassov generalized fuzzy sets in to Intuitionistic Fuzzy Sets in 1983[1]. Intuitionistic Fuzzy Modal Operator was firstly defined by same author the other operators were defined by several researchers[, 3, 4]. Intuitionistic fuzzy algebraic stuctures their properties were studied in[5, 6, 7]. In this paper, we studied some intuitionistic fuzzy operators on intuitionistic fuzzy ideals groups. Received: 14 July 016 Accepted: 9 August Introduction The original concept of fuzzy sets in Zadeh [9] was introduced as an extension of crisp sets by enlarging the truth value set to the real unit interval [0, 1]. In fuzzy set theory, if the membership degree of an element x is µ(x) then the nonmembership degree is 1 µ(x) thus it is fixed. Intuitionistic fuzzy sets have been introduced by Atanassov in 1983 [1] form an extension of fuzzy sets by enlarging the truth value set to the lattice [0, 1] [0, 1]. Definition 1.1. [1] An intuitionistic fuzzy set (shortly IFS) on a set X is an object of the form A = {< x, µ A (x), ν A (x) >: x X} where µ A (x), (µ A : X [0, 1]) is called the degree of membership of x in A, ν A (x), (ν A : X [0, 1])is called the degree of non- membership of x in A, where µ A ν A satisfy the following condition: µ A (x) + ν A (x) 1, for all x X. The hesitation degree of x is defined by π A (x) = 1 µ A (x) ν A (x) Definition 1.. [1]An IFS A is said to be contained in an IFS B (notation A B) if only if, for all x X : µ A (x) µ B (x) ν A (x) ν B (x). It is clear that A = B if only if A B B A. 1 3 rd International Intuitionistic Fuzzy Sets Contemporary Mathemathics Conference 010 Mathematics Subject Classification. 03E7,47S40. Key words phrases. Intuitionistic Fuzzy Modal Operator, Intuitionistic Fuzzy Algebraic Structures. 84

2 SOME IF MODAL OPERATORS OVER IF IDEALS AND GROUPS 85 Definition 1.3. [1]Let A IF S let A = {< x, µ A (x), ν A (x) >: x X} then the above set is callede the complement of A A c = {< x, ν A (x), µ A (x) >: x X} Definition 1.4. [] Let X be a set A = {< x, µ A (x), ν A (x) >: x X} IF S(X). (1) A = {< x, µ A (x), 1 µ A (x) >: x X} () A = {< x, 1 ν A (x), ν A (x) >: x X} Definition 1.5. [3] Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), for α, β I { } (1) (A) = x. µ A(x), ν A(x)+1 : x X { } () (A) = x, µ A(x)+1, ν A(x) : x X (3) α (A) = { x, αµ A (x), αν A (x) + 1 α : x X} (4) α (A) = { x, αµ A (x) + 1 α, αν A (x) : x X} (5) for max{α, β} + γ I,,γ (A) = {< x, αµ A (x), βν A (x) + γ >: x X} (6) for max{α, β} + γ I,,γ (A) = {< x, αµ A (x) + γ, βν A (x) >: x X} Definition 1.6. [8] Let X be a set A = {< x, µ A (x), ν A (x) >: x X} IF S(X), α, β, α + β I (1) (A) = { x, αµ A (x), αν A (x) + β : x X} () (A) = { x, αµ A (x) + β, αν A (x) : x X} The operators,γ,,γ are an extensions of, (resp.). In 007, the author[4] defined a new operator studied some of its properties. This operator is named E defined as follows: Definition 1.7. [4]Let X be a set A = {< x, µ A (x), ν A (x) >: x X} IF S(X), α, β [0, 1]. We define the following operator: E (A) = {< x, β(αµ A (x) + 1 α), α(βν A (x) + 1 β) >: x X} If we choose α = 1 write α insteed of β we get the operator. Similarly, if β = 1 is chosen writen insteed of β, we get the operator α. In007, Atanassov introduced the operator,γ,δ which is a natural extension of all these operators in [3]. Definition 1.8. [3]Let Xbe a set, A IF S(X), α, β, γ, δ [0, 1] such that then the operator,γ,δ defined by max(α, β) + γ + δ 1,γ,δ (A) = {< x, αµ A (x) + γ, βν A (x) + δ >: x X} In 010, the author [4] defined a new operator which is a generalization of E.

3 86 SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU, AND ARIF BAL Definition 1.9. [4]Let X be a set A IF S(X), α, β, ω [0, 1]. We define the following operator: Z ω (A) = {< x, β(αµ A (x) + ω ω.α), α(βν A (x) + ω ω.β) >: x X} We have defined a new OTMO on IFS, that is generalization of the some OTMOs. defined as follows: Z Definition [4]Let X be a set A IF S(X), α, β, ω, θ [0, 1]. We define the following operator: Z (A) = {< x, β(αµ A(x) + ω ω.α), α(βν A (x) + θ θ.β) >: x X} The operator Z is a generalization of Zω, also, E,,. Definition [5]Let G be a groupoid, A IF S(G).If for all x, y G, A(xy) min(a(x), A(y)) then A called an intuitionistic fuzzy subgrupoid over G. Definition 1.1. [6]Let G be a grupoid, A IF S(G). If for all x, y G, A(xy) max(a(x), A(y)) then A called an intuitionistic fuzzy ideal over G, shortly IF I(G). Definition [6]Let G be a grup A IF S(G) a grupoid. If for all x G, A(x 1 ) A(x) then A called an intuitionistic fuzzy subgroup over G,shortly IF G(G).. Main Results Theorem.1. Let G be a groupoid A IF S(G). (1) If A IF I(G) then A IF I(G) () If A IF I(G) then A IF I(G) Proof. (1)For x, y G, µ A (xy) = µ A (xy) µ A (x) µ A (y) ν A (xy) = 1 µ A (xy) (1 µ A (x)) (1 µ A (y)) = ν A (x) ν A (y) A(xy) A(x) A(y) Theorem.. Let G be a groupoid A IF S(G). (1) If A IF I(G) then (A) IF I(G) () If A IF I(G) then (A) IF I(G)

4 SOME IF MODAL OPERATORS OVER IF IDEALS AND GROUPS 87 Proof. (1)For x, y G, µ (A) (xy) = µ A(xy) µ A(x) µ A(y) = µ (A) (x) µ (A) (y) ν (A) (xy) = ν A(xy) + 1 ν A(x) + 1 = ν (A) (x) ν (A) (y) (A)(xy) (A)(x) (A)(y) Theorem.3. Let G be a groupoid A IF S(G). (1) If A IF I(G) then α (A) IF I(G) () If A IF I(G) then α (A) IF I(G) Proof. (1)For x, y G, ν A(y) + 1 µ α(a)(xy) = αµ A (xy) + 1 α (αµ A (x) + 1 α) (αµ A (y) + 1 α) = µ α(a)(x) µ α(a)(y) ν α(a)(xy) = αν A (xy) (αν A (x)) (αν A (y)) = ν α(a)(x) ν α(a)(y) α (A)(xy) α (A)(x) α (A)(y) Theorem.4. Let G be a groupoid A IF S(G). (1) If A IF I(G) then (A) IF I(G) () If A IF I(G) then (A) IF SI(G) (3) If A IF I(G) then,γ (A) IF I(G) (4) If A IF I(G) then,γ (A) IF I(G) µ,γ (A)(xy) = αµ A (xy) αµ A (x) αµ A (y) = µ,γ (A)(x) µ,γ (A)(y) ν,γ (A)(xy) = βν A (xy) + γ (βν A (x) + γ) (βν A (y) + γ) = ν,γ (A)(x) ν,γ (A)(y),γ (A)(xy),γ (A)(x),γ (A)(y) The other properties can proof with same way. Theorem.5. Let G be a groupoid A IF S(G) an ideal then E (A) IF S(G) is an ideal.

5 88 SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU, AND ARIF BAL µ E (A)(xy) = β(αµ A (xy) + 1 α) β(αµ A (x) + 1 α) β(αµ A (y) + 1 α) = µ E (A)(x) µ E (A)(y) ν E (A)(xy) = α(βν A (xy) + 1 β) α(βν A (x) + 1 β) α(βν A (y) + 1 β) = ν E (A)(x) ν E (A)(y) E (A)(xy) E (A)(x) E (A)(y) Theorem.6. Let G be a groupoid A IF S(G) an ideal then,γ,δ (A) IF S(G) is an ideal. µ,γ,δ (A)(xy) = αµ A (xy) + γ (αµ A (x) + γ) (αµ A (y) + γ) = µ,γ,δ (A)(x) µ,γ,δ (A)(y) ν,γ,δ (A)(xy) = βν A (xy) + δ (βν A (x) + δ) (βν A (y) + δ) = ν,γ,δ (A)(x) ν,γ,δ (A)(y),γ,δ (A)(xy),γ,δ (A)(x),γ,δ (A)(y) Theorem.7. Let G be a groupoid A IF S(G) an ideal then Z (A) IF S(G) is an ideal. µ Z = β(αµ A(xy) + ω ω.α) β(αµ A (x) + ω ω.α) β(αµ A (y) + ω ω.α) = µ Z Z ν Z = α(βν A(xy) + θ θ.β) α(βν A (x) + θ θ.β) α(βν A (y) + θ θ.β) = ν Z Z Therefore, we obtain Z (A)(xy) Z (A)(x) Z (A)(y). Theorem.8. Let G be a group A IF S(G). (1) If A IF G(G) then A IF G(G). () If A IF G(G) then A IF G(G). Proof. It is clear that, if A IF G(G) then it means A IF I(G) for all x G, A(x 1 ) A(x). it will be enough to prove the correctness of the second condition. ()For x G µ A (x 1 ) = 1 ν A (x 1 ) 1 ν A (x) = µ A (x) ν A (x 1 ) = ν A (x 1 ) ν A (x) = ν A (x)

6 SOME IF MODAL OPERATORS OVER IF IDEALS AND GROUPS 89 The other property can be proved same way. Theorem.9. Let G be a group A IF S(G). (1) If A IF G(G) then (A) IF G(G) () If A IF G(G) then (A) IF G(G) Proof. ()For x, y G,If A IF G(G) then (A) IF I(G). (A)(xy) (A)(x) (A)(y). Now, Therefore, µ (A) (x 1 ) = µ A(x 1 ) + 1 ν (A) (x 1 ) = ν A(x 1 ) µ A(x) + 1 ν A(x) (A)(x 1 ) (A)(x) Theorem.10. Let G be a group A IF S(G). (1) If A IF G(G) then α (A) IF G(G) () If A IF G(G) then α (A) IF G(G) = µ (A) (x) = ν (A) (x) Proof. (1)For x, y G, it is clear that α (A)(xy) α (A)(x) α (A)(y). On the other h, µ α(a)(x 1 ) = αµ A (x 1 ) αµ A (x) = µ α(a)(x) ν α(a)(x 1 ) = αν A (x 1 ) + 1 α αν A (x) + 1 α = ν α(a)(x) α (A)(x 1 ) α (A)(x) Theorem.11. Let G be a group A IF S(G). (1) If A IF G(G) then (A) IF G(G) () If A IF G(G) then (A) IF G(G) (3) If A IF G(G) then,γ (A) IF G(G) (4) If A IF G(G) then,γ (A) IF G(G) µ,γ (A)(x 1 ) = αµ A (x 1 ) + γ αµ A (x) + γ = µ,γ (A)(x) ν,γ (A)(x 1 ) = βν A (x 1 ) βν A (x) = ν,γ (A)(x),γ (A)(x 1 ),γ (A)(x) The other properties can proof with same way. Theorem.1. Let G be a group A IF S(G). If A is an intuitionistic fuzzy subgroup on G then E (A) IF G(G).

7 90 SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU, AND ARIF BAL Proof. It is clear that for x, y G, E (A)(xy) E (A)(x) E (A)(y). µ E (A)(x 1 ) = β(αµ A (x 1 ) + 1 α) β(αµ A (x) + 1 α) = µ E (A)(x) ν E (A)(x 1 ) = α(βν A (x 1 ) + 1 β) α(βν A (x) + 1 β) E (A) IF G(G). = ν E (A)(x) Theorem.13. Let G be a group A IF S(G) an intuitionistic fuzzy group then,γ,δ (A) IF S(G) is an intuitionistic fuzzy subgroup. Proof. For x G, µ,γ,δ (A)(x 1 ) = αµ A (x 1 ) + γ αµ A (x) + γ = µ,γ,δ (A)(x) ν,γ,δ (A)(x 1 ) = βν A (x 1 ) + δ βν A (x) + δ Therefore,γ,δ (A) IF G(G). = ν,γ,δ (A)(x) Theorem.14. Let G be a group A IF S(G). If A is an intuitionistic fuzzy subgroup on G then Z (A) IF G(G). Proof. It can shown easily. References [1] K.T. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR.s Session, Sofia, June [] K. T. Atanassov, Intuitionistic Fuzzy Sets,,Spinger, Heidelberg, [3] K.T. Atanassov, Studies in Fuzziness Soft Computing-On Intuitionistic Fuzzy Sets Theory, ISBN , Springer Heidelberg, New York, 01. [4] G. Çuvalcıoğlu, New Tools Based on Intuitionistic Fuzzy Sets Generalized Nets, ISBN , Springer International Publishing Switzerl, (016) [5] K.Hur,Y. Jang H. W. Kang, Intuitionistic fuzzy Subgroupoid, Int. Jour. of Fuzzy Logic Int. Sys., 3(1) (003) [6] R. Biswas, Intuitionistic fuzzy subgroups, Mathematical Forum, Vol. X,(1989) [7] S. Yılmaz, Intuitionistic fuzzy Modal operatörlerin, Intuitionistic fuzzy Yapılarka İlişkileri, Mersin Üniversitesi Fen Bilimleri Enstitüsü, Yüksek lisans Tezi, 01, 75 s. [8] Dencheva K., Extension of intuitionistic fuzzy modal operators,proc.of the Second Int. IEEE Symp. Intelligent systems, Varna, June -4, (004), Vol. 3, 1-. [9] L.A. Zadeh, Fuzzy Sets, Information Control, 8(1965) Mersin University Faculty of Arts Sciences Department of Mathematics address: sinemyilmaz@gmail.com address: gcuvalcioglu@gmail.com address: arif.bal.math@gmail.com