On (, qk)-intuitionistic (fuzzy ideals, fuzzy soft ideals) of subtraction algebras Journal: Songklanakarin Journal of Science Technology Manuscript ID: SJST-0-00.R Manuscript Type: Original Article Date Submitted by the Author: 0-Apr-0 Complete List of Authors: Khan, Madad; COMSATS Institute of Information Technology, Mathematics Davvaz, Bijan; Yazd University, Mathematics Yaqoob, Naveed; Majmaah University, Mathematics Gulistan, Muhammad; Hazara University, Mathematics Keyword: Physical Sciences Mathematics Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. KDYG.tex
Page of 0 0 0 0 0 0 On (; _q k )-intuitionistic (fuzzy ideals, fuzzy soft ideals) of subtraction algebras a Madad Khan, b Bijan Davvaz, c Naveed Yaqoob d Muhammad Gulistan a Department of Mathematics, COMSATS Institute of I.T, Abbottabad, Pakistan b Department of Mathematics, Yazd University, Yazd, Iran c Department of Mathematics, College of Science in Al-Zul, Majmaah University, Al-Zul, Saudi Arabia d Department of Mathematics, Hazara University, Mansehra, Pakistan E-mail: a madadmath@yahoo.com, b bdavvaz@yahoo.com, c nayaqoob@ymail.com, d gulistanmath@hu.edu.pk Abstract: The intent of this article is to study the concept of an (; _q k )-intuitionistic fuzzy ideal (; _q k )-intuitionistic fuzzy soft ideal of subtraction algebras to introduce some related properties. 00 AMS Classi cation: 0F, 0G, 0A. Keywords: Subtraction algebras, (; _q k )-intuitionistic fuzzy (soft) subalgebras, (; _q k )-intuitionistic fuzzy (soft) ideals.
Page of 0 0 0 0 0 0 Introduction Schein () cogitate the system of the form (X; ; n), where X is the set of functions closed under the composition " " of functions ( hence (X; ) is a function semigroup) the set theoretical subtraction "n" ( hence (X; n) is a subtraction algebra in the sense of (Abbot, )). He proved that every subtraction semigroup is isomorphic to a di erence semigroup of invertible functions. He suggested problem concerning the structure of multiplication in a subtraction semigroup. It was explained by Zelinka (), he had solved the problem for subtraction algebras of a special type known as the "atomic subtraction algebras". The notion of ideals in subtraction algebras was introduced by Jun et al. (00). For detail study of subtraction algebras see (Ceven Ozturk, 00 Jun et al., 00). The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh (). Since then the notion of fuzzy sets is actively applicable in di erent algebraic structures. The fuzzi cation of ideals in subtraction algebras were discussed in (Lee Park, 00). Atanassov () introduced the idea of intuitionistic fuzzy set which in more general one as compared to a fuzzy set. Bhakat Das () introduced a new type of fuzzy subgroups, that is, the (; _q)-fuzzy subgroups. Jun et al. (0) introduced the notion of (; _q k )-fuzzy subgroup. In fact, the (; _q k )-fuzzy subgroup is an important generalization of Rosenfeld s fuzzy subgroup. Shabir et al. (00) characterized semigroups by (; _q k )-fuzzy ideals, also see Shabir Mahmood (0, 0). Molodtsov () introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that are free from the di culties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. (00) described the application of soft set theory to a decision-making problem. Maji et al. (00) also studied several operations on the theory of soft sets. Maji et al. (00b, 00) studied intuitionistic fuzzy soft sets. The notion of fuzzy soft sets, as a generalization of the stard soft sets, was introduced in (Maji, 00a), an application of fuzzy soft sets in a decision-making problem was presented. Ahmad et al. (00)
Page of 0 0 0 0 0 0 have introduced arbitrary fuzzy soft union fuzzy soft intersection. Aygunoglu et al. (00) introduced the notion of fuzzy soft group studied its properties. Yaqoob et al. (0) studied the properties of intuitionistic fuzzy soft groups in terms of intuitionistic double t-norm. Jun et al. (00) introduced the notion of fuzzy soft BCK/BCI-algebras (closed) fuzzy soft ideals, then derived their basic properties. Williams Saeid (0) studied fuzzy soft ideals in subtraction algebras. Recently, Yang (0) have studied fuzzy soft semigroups fuzzy soft (left, right) ideals, have discussed fuzzy soft image fuzzy soft inverse image of fuzzy soft semigroups (ideals) in detail. Liu Xin (0) studied the idea of generalized fuzzy soft groups fuzzy normal soft groups. In this article, we study the concept of (; _q k )-intuitionistic fuzzy (soft) ideals of subtraction algebras. Here we consider some basic properties of (; _q k )-intuitionistic fuzzy (soft) ideals of subtraction algebras. Preliminaries In this section we recall some of the basic concepts of subtraction algebra which will be very helpful in further study of the paper. Throughout the paper X denotes the subtraction algebra unless otherwise speci ed. De nition. (Jun et al., 00) A non-empty set X together with a binary operation " " is said to be a subtraction algebra if it satis es the following: (S ) x (y x) = x; (S ) x (x y) = y (y x) ; (S ) (x y) z = (x z) y; for all x; y; z X: The last identity permits us to omit parentheses in expression of the form (x y) z: The subtraction determines an order relation on X : a b, a b = 0; where 0 = a a is an element that does not depends upon the choice of a X: The ordered set (X; ) is a semi-boolean algebra in the sense of (Abbot, ); that is, it is a meet semi lattice with zero, in which every interval [0; a] is a boolean algebra with respect to
Page of 0 0 0 0 0 0 the induced order. Here a ^ b = a (a b); the complement of an element b [0; a] is a b is denoted by b = ; if b; c [0; a] ; then b _ c = b = ^ c = = = ((a b) ^ (a c)) = a (a b) ((a b) (a c)) : In a subtraction algebra, the following are true see (Jun et al., 00): (a) (x y) y = x y; (a) x 0 = x 0 x = 0; (a) (x y) x = 0; (a) x (x y) y; (a) (x y) (y x) = x y; (a) x (x (x y)) = x y; (a) (x y) (z y) x z; (a) x y if only if x = y w for some w X; (a) x y implies x z y z z y z x; for all z X; (a0) x; y z implies x y = x ^ (z y) ; (a) (x ^ y) (x ^ z) x ^ (y z) ; (a) (x y) z = (x z) (y z) : De nition. (Jun et al., 00) A non-empty subset A of a subtraction algebra X is called an ideal of X; denoted by A C X : if it satis es: (b) a x A for all a A x X; (b) for all a; b A; whenever a _ b exists in X then a _ b A: Proposition. (Jun et al., 00) A non-empty subset A of a subtraction algebra X is called an ideal of X; if only if it satis es:
Page of 0 0 0 0 0 0 (b) 0 A; (b) for all x X for all y A; x y A ) x A: Proposition. (Jun et al., 00) Let X be a subtraction algebra x; y X: If w X is an upper bound for x y; then the element x _ y = w ((w y) x) is the least upper bound for x y: De nition. (Jun et al., 00) Let Y be a non-empty subset of X then Y is called a subalgebra of X if x y Y; whenever x; y Y: De nition. (Lee Park, 00) Let f be a fuzzy subset of X. Then f is called a fuzzy subalgebra of X if it satis es (FS) f(x y) minff(x); f(y)g, whenever x; y X: De nition. (Lee Park, 00) A fuzzy subset f is said to a fuzzy ideal of X if satis es: (FI) f(x y) f(x); (FI) If there exists x _ y, then f(x _ y) minff(x); f(y)g; for all x; y X: (; _q k )-intuitionistic fuzzy ideals In this section we will discuss some properties related to (; _q k )-intuitionistic fuzzy ideals of subtraction algebras. De nition. An intuitionistic fuzzy set A in X is an object of the form A = f(x; A (x); A (x)) : x Xg; where the function A : X! [0; ] A : X! [0; ] denote the degree of membership degree of non-membership of each element x X; 0 A (x) + A (x) for all x X: For simplicity, we will use the symbol A = ( A ; A ) for the intuitionistic fuzzy set A = f(x; A (x); A (x)) : x Xg: We de ne 0(x) = 0 (x) = for all x X:
Page of 0 0 0 0 0 0 De nition. Let X be a subtraction algebra. An intuitionistic fuzzy set A = f(x; A (x); A (x)) : x Xg; of the form >< (; ) if y = x x (;) y = >: (0; ) if y = x is said to be an intuitionistic fuzzy point with support x value (; ) is denoted by x (;). A fuzzy point x (;) is said to intuitionistic belongs to (resp., intuitionistic quasi-coincident) with intuitionistic fuzzy set A = f(x; A (x); A (x)) : x Xg written x (;) A resp: x (;) qa if A (x) A (x) (resp., A (x) + > A (x) + < ) : By the symbol x (;) q k A we mean A (x) + + k > A (x) + + k < ; where k (0; ) : We use the symbol x t A implies A (x) t t x [] A implies A (x) t; in the whole paper. De nition. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy subalgebra of X if x (t;t ) A; y (t;t ) A ) (x y) (t^t ;t _t ) _q ka; for all x; y X; t ; t ; t ; t ; k (0; ): OR An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy subalgebra of X if it satisfy the following conditions, (i) x t A ; y t A ) (x y) t^t _q k A ; for all x; y X; t ; t ; k (0; ]; (ii) t x [] A; t y [] A ) t_t x y [] ^ [q k] A ; for all x; y X; t ; t ; k [0; ): Example. Let X = f0; a; bg be a subtraction algebra with the following Cayley table 0 a b 0 0 0 0 a a 0 a b b b 0 ;
Page of 0 0 0 0 0 0 Let us de ne the intuitionistic fuzzy set A = ( A ; A ) of X as X 0 a b A (x) 0: 0: 0: A (x) 0: 0: 0: t 0: t 0: t 0: t 0: k 0: then A = ( A ; A ) is an (; _q 0: )-intuitionistic fuzzy subalgebra of X: De nition. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy ideal of X if it satisfy the following conditions, (i) x (t;t) A; y X ) (x y) t _q k A; (ii) If there exist x _ y; then x (t;t ) A; y (t;t ) A ) (x _ y) (t^t ;t _t ) _q ka; for all x; y X; t; t ; t ; t ; t ; k (0; ): OR An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy ideal of X if it satisfy the following conditions, (i) x t A ; y X ) (x y) t _q k A ; (ii) If there exist x _ y in X then x t A ; y t A ) (x _ y) t^t _q k A ; for all x; y X; t; t ; t ; k (0; ]; (iii) t x [] A; y X ) t x y [] ^ [q k] A ; (iv) If there exist x _ y in X then t x [] A; t y [] A ) t_t x_y [] ^ [q k] A ; for all x; y X; t; t ; t ; k [0; ): ;
Page of 0 0 0 0 0 0 Example. Let X = f0; a; bg be a subtraction algebra with the Cayley table de ne in Example., let us de ne an intuitionistic fuzzy set A = ( A ; A ) of X as X 0 a b A (x) 0: 0: 0: A (x) 0: 0: 0: t 0: t 0: t 0: t 0: t 0: k 0: then A = ( A ; A ) is an (; _q 0: )-intuitionistic fuzzy ideal of X: Theorem. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy subalgebra of X if only if A (x y) minf A (x) ; A (y); k g A(x y) maxf A (x) ; A (y); k g: Proof. Let A = ( A ; A ) be an (; _q k )-intuitionistic fuzzy subalgebra of X; assume on contrary that there exist some t (0; ] r [0; ) such that This implies that A (x y) < t < minf A (x) ; A (y); k g A (x y) > r > maxf A (x) ; A (y); k g: A (x) t; A (y) t; A (x y) < t ) A (x y) + t + k < k ) (x y) t _q k A ; ; + k + k =
Page 0 of 0 0 0 0 0 0 which is a contradiction. Hence Also from we get which is a contradiction. Hence A (x y) minf A (x) ; A (y); k g: A (x y) > r > maxf A (x) ; A (y); k g A (x) r; A (y) r; A (x y) > r ) A (x y) + r + k > k + k + k = r ) x y [] ^ [q k] A (x); A (x y) maxf A (x) ; A (y); k g: Conversely, let A (x y) minf A (x) ; A (y); k g A(x y) maxf A (x) ; A (y); k g: Let x t A ; y t A for all x; y X; t ; t ; k (0; ] t x [] A; t y [] A for all x; y X; t ; t ; k [0; ): This implies that A (x) t ; A (y) t A (x) t ; A (y) t : Consider A (x y) min A (x); A (y); k min t ; t ; k : If t ^ t > k ; then A(x y) k : So A(x y) + (t ^ t ) + k > k + k + k = ; which implies that (x y) t^t q k A : If t ^ t k ; then A(x y) t ^ t : So (x y) t^t A : Thus (x y) t^t _q k A : Also consider A (x y) maxf A (x) ; A (y); k g maxft ; t ; k g: If t _ t < k ; then A(x y) k : So A(x y) + (t _ t ) + k < k + k + k = ; which implies that t_t x y [q k] A : If t _ t > k ; then A(x y) t _ t : So t_t x y [] A: Thus t_t x y [] _ [q k] A : Hence A = ( A ; A ) is an (; _q k )-intuitionistic fuzzy subalgebra of X: Theorem. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy ideal of X if only if (i) A (x y) minf A (x) ; k g; (ii) A (x y) maxf A (x) ; k g;
Page of 0 0 0 0 0 0 (ii) A (x _ y) minf A (x) ; A (y); k g; (iv) A (x _ y) maxf A (x) ; A (y); k g: Proof. The proof is similar to the proof of the Theorem.: Proposition. Every (; _q k )-intuitionistic fuzzy ideal of X is an (; _q k )-intuitionistic fuzzy subalgebra of X; but converse is not true as shown in the following example. Example.0 Let X = f0; a; bg be a subtraction algebra with the Cayley table de ne in Example., let X 0 a b A (x) 0: 0: 0: A (x) 0: 0: 0: then by Theorem., A = ( A ; A ) is an (; _q k )-intuitionistic fuzzy subalgebra of X: But by Theorem., we observe that A = ( A ; A ) is not a (; _q 0: ) intuitionistic fuzzy ideal of X: As A (a _ b) = A (0 ((0 b) a))) = A (0) = 0: minf A (a) ; A (b); k g = minf0:; 0:; 0:g = 0:: De nition. Let A = ( A ; A ) is an intuitionistic fuzzy set of X: De ne the intuitionistic level set as A (;) = fx Xj A (x) ; A (x) ; where (0; k ]; [ k ; )g: Theorem. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy subalgebra of X if only if A (;) = ; is a subalgebra of X: Proof. Let A = ( A ; A ) be an (; _q k )-intuitionistic fuzzy subalgebra of X: Suppose that x; y A (;) then A (x) ; A (y) A (x) ; A (y) : Since A = ( A ; A ) is an (; _q k )-intuitionistic fuzzy subalgebra of X; so A (x y) minf A (x) ; A (y); k g minf; ; k g = minf; k g = 0
Page of 0 0 0 0 0 0 A (x y) maxf A (x) ; A (y); k g maxf; ; k g = maxf; k g = : Thus (x y) A (;) : Hence A (;) = ; is a subalgebra of X: Conversely, assume that A (;) = ; is a subalgebra of X: Assume on contrary that there exist some x; y X such that A (x y) < minf A (x) ; A (y); k g A(x y) > maxf A (x) ; A (y); k g: Choose (0; k ]; [ k ; ) such that A(x y) < < minf A (x) ; A (y); k g A(x y) > > maxf A (x) ; A (y); k g: This implies that (x y) = A (;); which is a contradiction to the hypothesis. Hence A (x y) minf A (x) ; A (y); k g A(x y) maxf A (x) ; A (y); k g: Thus A = ( A; A ) is an (; _q k )-intuitionistic fuzzy subalgebra of X: Theorem. An intuitionistic fuzzy set A = ( A ; A ) of X is said to be an (; _q k )-intuitionistic fuzzy ideal of X if only if A (;) = ; is an ideal of X: Proof. The proof is similar to the proof of the Theorem.: De nition. Let X be a subtraction algebra A X, an (; _q k )-intuitionistic characteristic function A = fhx; A (x); A (x)ijx Sg; where A A are fuzzy sets respectively, de ned as follows: >< A : X! [0; ]jx! A (x) := >: >< A : X! [0; ]jx! A (x) := >: k if x A 0 if x = A if x A k if x = A Lemma. For a non-empty subset A of a subtraction algebra X; we have :
Page of 0 0 0 0 0 0 E (i) A is a subalgebra of X if only if the characteristic intuitionistic set A = D A ; A of A in X is an (; _q k )-intuitionistic fuzzy subalgebra of X: E (ii) A is an ideal of X if only if the characteristic intuitionistic set A = D A ; A of A in X is an (; _q k )-intuitionistic fuzzy ideal of X: Proof. (i) Let A is a subalgebra of X. We consider the following cases. Case () Let x; y A then x y A so A (x) = k = A (y) = A (x y) A (x) = = A (y) = A (x y):hence A (x y) = k minf A (x) ; A (y); k g minf k ; k ; k g = k A (x y) = maxf A (x) ; A (y); k g maxf; ; k g = : E Which implies that A = D A ; A of A is an (; _q k )-intuitionistic fuzzy subalgebra of X: Case () Let x; y = A then x y = A so A (x) = 0 = A (y) = A (x y) A (x) = k = A (y) = A (x y): Hence A (x y) = 0 minf A (x) ; A (y); k g minf0; 0; k g = 0 A (x y) = k maxf A (x) ; A (y); k g maxf k ; k ; k g = k : E Which again implies that A = D A ; A of A in X is an (; _q k )-intuitionistic fuzzy subalgebra of X: E Hence characteristic intuitionistic set A = D A ; A of A is an (; _q k )-intuitionistic fuzzy subalgebra of X: E Conversely, let the characteristic intuitionistic set A = D A ; A of A in X is an (; _q k )-intuitionistic fuzzy subalgebra of X: Let x; y A then A (x) = k = A (y) A (x) = = A (y):
Page of 0 0 0 0 0 0 Hence A (x y) minf A (x) ; A (y); k g minf k ; k ; k g = A (x y) maxf A (x) ; A (y); k g maxf; ; k g = : Which implies that (x y) A: Hence A is a subalgebra of X: (ii) The proof of this part is similar to the proof of part (i). (; _q k )-intuitionistic fuzzy soft ideals Molodtsov de ned the notion of a soft set as follows. De nition. (Molodtsov, ) A pair (F; A) is called a soft set over U, where F is a mapping given by F : A! P (U): In other words a soft set over U is a parametrized family of subsets of U: The class of all intuitionistic fuzzy sets on X will be denoted by IF (X): De nition. (Maji et al., 00b) Let U be an initial universe E be the set of parameters. Let A E: A pair ( e F ; A) is called an intuitionistic fuzzy soft set over U, where e F is a mapping given by e F : A k! IF (U): In general, for every A: F e E [] = D is an intuitionistic fuzzy set in U it is called intuitionistic fuzzy value set of parameter : De nition. Let U be an initial universe E be a set of parameters. Suppose that A; B E; ( e F ; A) ( e G; B) are two intuitionistic fuzzy soft sets, we say that ( e F ; A) is an intuitionistic fuzzy soft subset of ( e G; B) if only
Page of 0 0 0 0 0 0 () A B; () for all A, e F [] is an intuitionistic fuzzy subset of e G[]; that is, for all x U A (x) eg[] (x); (x) eg[] (x): This relationship is denoted by ( e F ; A) b ( e G; B): De nition. Let ( e F ; A) ( e G; B) be two intuitionistic fuzzy soft sets over a common universe U. Then ( e F ; A)e^( e G; B) is de ned by ( e F ; A)e^( e G; B) = ( e ; A B); where e [; "] = e F [] \ e G["] for all (; ") A B; that is, for all (; ") A B; x U: e[; "] = D E (x) ^ eg["] (x) (x) _ eg["] (x) ; De nition. Let ( e F ; A) ( e G; B) be two intuitionistic fuzzy soft sets over a common universe U. Then ( e F ; A)e_( e G; B) is de ned by ( e F ; A)e_( e G; B) = ( e ; A B); where e [; "] = e F [] [ e G["] for all (; ") A B; that is, for all (; ") A B; x U: D E e[; "] = (x) _ eg["] (x) (x) ^ eg["] (x) ; De nition. Let ( e F ; A) ( e G; B) be two intuitionistic fuzzy soft sets over a common universe U. Then the intersection ( e ; C); where C = A [ B for all C x U; We denote it by ( e F ; A) e ( e G; B) = ( e ; C): (x) if A B >< e[] (x) = eg[] (x) if B A >: >: (x) ^ eg[] (x) if A \ B; (x) if A B >< e[] (x) = eg[] (x) if B A (x) _ eg[] (x) if A \ B:
Page of 0 0 0 0 0 0 De nition. Let ( F e ; A) ( G; e B) be two intuitionistic fuzzy soft sets over a common universe U. Then the union ( ; e C); where C = A [ B for all C x U; (x) if A B >< e[] (x) = eg[] (x) if B A >: (x) _ eg[] (x) if A \ B; (x) if A B >< e[] (x) = eg[] (x) if B A >: (x) ^ eg[] (x) if A \ B: We denote it by ( F e ; A) d ( G; e B) = ( ; e C): In contrast with the above de nitions of IF-soft set union intersection, we may sometimes adopt di erent de nitions of union intersection given as follows. De nition. Let ( F e ; A) ( G; e B) be two IF-soft sets over a common universe U A \ B = ;: Then the bi-intersection of ( F e ; A) ( G; e B) is de ned to be the fuzzy soft set ( ; e C); where C = A \ B e[] = F e [] \ G[] e for all C: This is denoted by ( ; e C) = ( F e ; A)eu( G; e B): De nition. Let ( e F ; A) ( e G; B) be two IF-soft sets over a common universe U A \ B = ;: Then the bi-union of ( e F ; A) ( e G; B) is de ned to be the fuzzy soft set ( e ; C); where C = A \ B e[] = F e [] [ G[] e for all C: This is denoted by ( ; e C) = ( F e ; A)et( G; e B): De nition.0 Let ( F e ; A) be an IF-soft set over X. Then ( F e ; A) is called an (; _q k )-intuitionistic fuzzy soft subtraction algebra if F e E [] = D is an (; _q k )-intuitionistic fuzzy subalgebra of X; for all A:
Page of 0 0 0 0 0 0 Example. Let X = f0; a; bg be a subtraction algebra with the following Cayley table 0 a b 0 0 0 0 a a 0 a b b b 0 let U is the set of cellular brs of mobile companies in the market. Let = fattractive, expensive, cheapg is a parameter space A = fattractive, cheapg: De ne ef ([attractive]) = fh0; 0:; 0:i ; ha; 0:; 0:i ; hb; 0:; 0:ig; ef ([cheap]) = fh0; 0:; 0:i ; ha; 0:; 0:i ; hb; 0:; 0:ig: Let k = 0: then h e F ; Ai is an (; _q 0: )-intuitionistic fuzzy soft subalgebra of X: De nition. An intuitionistic fuzzy soft set hf e ; Ai of X is called an (; _q k )-intuitionistic fuzzy soft D E subalgebra of X; if for all A; F e [] = is an (; _q k )-intuitionistic fuzzy subalgebra of X; if (i) (x y) minf (x) (y); k g; (ii) (x y) maxf (x) (y); k g; for all x; y X: De nition. An intuitionistic fuzzy soft set hf e ; Ai of X is called an (; _q k )-intuitionistic fuzzy soft ideal of X; if for all A; F e E [] = D is an (; _q k )-intuitionistic fuzzy soft ideal of X; if (i) (x y) minf (x) ; k g; (ii) (x y) maxf (x) ; k g; (iii) (x _ y) minf (x) (y); k g; (iv) (x _ y) maxf (x) (y); k g; for all x; y X:
Page of 0 0 0 0 0 0 Proposition. An (; _q k )-intuitionistic fuzzy soft ideal of X is an (; _q k )-intuitionistic fuzzy soft subalgebra of X; but converse is not true as shown in the following example. Example. From Example., h e F ; Ai is not an (; _q k )-intuitionistic fuzzy soft ideal of X: As ef ([attractive]) (a _ b) = ef ([attractive]) (0 ((0 b) a))) = ef ([attractive]) (0) = 0: minf ef ([attractive]) (a); ef ([attractive]) (b); k g = 0: = minf0:; 0:; 0:g; for k = 0:: Theorem. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft subalgebras (resp., ideals) of X. Then h e F ; Aie^h e G; Bi is also an (; _q k )-intuitionistic fuzzy soft subalgebra (resp., ideal) of X: Proof. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft ideals of X: We know that h e F ; Aie^h e G; Bi = h e ; A Bi; where e [; "] = e F [] \ e G["] for all (; ") A B; that is for all x X: Let x; y X; we have D E e[; "](x) = (x) ^ eg[] (x) (x) _ eg[] (x) ^ eg[] (x y) = (x y) ^ eg[] (x y) minf (x) ; k g ^ minf eg[] (x) ; k g = minf (x) ^ eg[] (x) ; k g = minf ^ eg[] (x) ; k g; _ eg[] (x y) = (x y) _ eg[] (x y) maxf (x) ; k g ^ = maxf (x) _ eg[] (x) ; = maxf _ eg[] (x) ; maxf eg[] (x) ; k k g: g k g
Page of 0 0 0 0 0 0 Also we have ^ eg[] (x _ y) = (x _ y) ^ eg[] (x _ y) minf (x) (y) ; k = g ^ minf (x) ^ eg[] (x) (y) ^ eg[] (y) ; = minf ^ eg[] (x) ; ^ eg[] (y) ; _ eg[] (x _ y) = (x _ y) _ eg[] (x _ y) maxf (x) (y) ; k g ^ = maxf (x) _ eg[] (x) (y) _ eg[] (y) ; k = maxf _ eg[] (x) ; ^ eg[] (y) ; k g: minf eg[] (x) ; eg[] (y) ; k k g k g; maxf eg[] (x) ; eg[] (y) ; Hence h e F ; Aie^h e G; Bi is also an (; _q k )-intuitionistic fuzzy soft ideal of X: The other case can be seen in a similar way. Theorem. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft subalgebras (resp., ideals) of X. Then h e F ; Aie_h e G; Bi is also an (; _q k )-intuitionistic fuzzy soft subalgebra (resp., ideal) of X: Proof. The proof is similar to the proof of the Theorem.: Theorem. Let h e F ; Ai be an intuitionistic fuzzy soft set of X: Then h e F ; Ai is an (; _q k )-intuitionistic fuzzy soft subalgebra (resp., ideal) of X if only if h e F ; Ai (;) = h e F (;) ; Ai = fx Xj (x) ; (x) ; where (0; k ]; [ k ; )g is a soft subalgebra (resp., ideal) of X: Proof. Let h e F ; Ai be an (; _q k )-intuitionistic fuzzy soft subalgebra of X: Let x; y h e F ; Ai (;) then (x) (y) (x) (y) ; where (0; k ]; [ k ; ): Since h F e ; Ai is an (; _q k )-intuitionistic fuzzy soft subalgebra of X; so (x y) minf (x) (y) ; k g = minf; ; k g = ; g k g g
Page 0 of 0 0 0 0 0 0 (x y) maxf (x) (y) ; k g = maxf; ; k g = : Thus (x y) h e F ; Ai (;) : Hence h e F ; Ai (;) is a soft subalgebra of X: Conversely assume that h e F ; Ai (;) is a soft subalgebra of X: Let there exist some r (0; ] s [0; ) such that This implies that (x (x y) < r < minf (x) (y) ; k g (x y) > s > minf (x) (y) ; k g: y) = h e F ; Ai (;) ; which is contradiction. Thus (x y) minf (x) (y) ; k g (x y) maxf (x) (y) ; k g: Hence h e F ; Ai is an (; _q k )-intuitionistic fuzzy soft subalgebra of X: Proposition. Every (; _q k )-intuitionistic fuzzy soft ideal h e F ; Ai of X satis es the following, (i) (0) minf (x); k g; (ii) (0) maxf (x); k g, for all x X: Proof: By letting x = y in the De nition., we get the required proof. Lemma.0 If an (; _q k )-intuitionistic fuzzy soft set h e F ; Ai of X satis es the followings, (i) (0) minf (x); k g; (ii) (0) maxf (x); k g;
Page of 0 0 0 0 0 0 (iii) (x z) minf ((x y) z) (y) ; k g; (iv) (x z) maxf ((x y) z) (y) ; k g. Then we have x a ) (x) minf (a); k g F e [] (x) maxf (a); k g; for all a; x; y; z X: Proof. Let a; x X x a: Consider Also consider This complete the proof. (x) = (x 0) by (a); minf ((x a) 0) (a) ; k g by (iii); = minf (0) (a) ; k g by x a i x a = 0; = minf (a); k g by (i): (x) = (x 0) by (a); maxf ((x a) 0) (a) ; k g by (iv), = maxf (0) (a) ; k g by x a i x a = 0; = maxf (a); k g by (ii): Theorem. An (; _q k )-intuitionistic fuzzy soft set h e F ; Ai of X satis es the conditions (i)-(iv) of the Lemma.0, if only if h e F ; Ai is an (; _q k )-intuitionistic fuzzy soft ideal of X: Proof. Suppose that h e F ; Ai; i.e, e F [] = D E satis es the conditions (i)-(iv) of the Lemma.0. Let x; y X; then by using (a) we have x y x: Now by the use of lemma.0, (x y) minf (x); k g F e [] (x y) maxf (x); k g for all x; y X: Also (x _ y) minf (x); k g F e [] (x _ y) maxf (x); k g whenever x _ y exists in X by using the lemma.0, we have (x _ y) minf (x) (y); k g F e [] (x _ y) 0
Page of 0 0 0 0 0 0 maxf (x) (x); k g for all x; y X: Hence h F e ; Ai is an (; _q k )-intuitionistic fuzzy soft ideal of X: Conversely, assume that h e F ; Ai is an (; _q k )-intuitionistic fuzzy soft ideal of X: Conditions (i) (ii) directly follows from the Proposition.. let x; y; z X: Put x = y y = x z in (a) (a); We get from (a); (x z) y x z from (a); we have y (y ((x z)) x z: Which indicates that x z is an upper bound for (x z) y y (y ((x z)): By using the Proposition., we have Thus ((x z) y) _ (y (y ((x z))) = (x z) (((x z) ((x z) y)) (y (y (x z)))) = x z: (x z) = (((x z) y) _ (y (y ((x z)))) Which is (iii). And minf ((x z) y) (y (y (x z)) ; k g = minf ((x y) z) (y) ; k g by (S ) (a) x = y: (x z) = (((x z) y) _ (y (y ((x z)))) Which is (iv). Hence it completes the proof. maxf ((x z) y) (y (y (x z)) ; k g = maxf ((x y) z) (y) ; k g by (S ) (a) x = y: Theorem. An (; _q k )-intuitionistic fuzzy soft set h e F ; Ai of X is an (; _q k )-intuitionistic fuzzy soft ideal of X if only if it satis es (i) (x ((x a) b)) minf (a) (b); k g; (ii) (x ((x a) b)) maxf (a) (b); k g for all x; a; b X:
Page of 0 0 0 0 0 0 Proof. Let (; _q k )-intuitionistic fuzzy set A = of X satis es (i) (ii). Consider (x y) = ((x y) (((x y) x) x))) by (i) minf (x) (x); k g = minf (x); k g (x y) = ((x y) (((x y) x) x))) by (ii) max cf (x) (x); k g = maxf (x); k g: Also suppose that x _ y exists in X by using the Proposition., we have x _ y = w ((w x) y): Now using the given conditions (x _ y) = (w ((w x) y)) minf (x) (y); k g by (i); (x _ y) = (w ((w x) y)) maxf (x) (y); k g by (ii): Hence A = is an (; _q k )-intuitionistic fuzzy soft ideal of X: Conversely, let A = be an (; _q k )-intuitionistic fuzzy soft ideal of X: By Theorem., (x z) minf ((x y) z) (y); k g (x z) maxf ((x y) z) (y); k g:
Page of 0 0 0 0 0 0 Let z = (x a) b y = b; then (x ((x a) b)) minf ((x y) ((x a) b)) (b); k g = minf ((x y) ((x b) a)) (b); k g by (S ) = minf (a) (b); k g by putting x = x b y = a in (a) (x ((x a) b)) maxf ((x y) ((x a) b)) (b); k g = maxf ((x y) ((x b) a)) (b); k g by (S ) = maxf (a) (b); k g by putting x = x b y = a in (a): Hence A = satisfy condition (i) (ii). Lemma. An intuitionistic fuzzy set A = of X is an (; _q k )-intuitionistic fuzzy soft ideal of X if only if fuzzy sets are (; _q k )- fuzzy soft ideals of X: Proof. The proof is straightforward. Theorem. An intuitionistic fuzzy set A = of X is an (; _q k )-intuitionistic fuzzy soft ideal of X if only if B A = C A = are (; _q k )- intuitionistic fuzzy soft ideal of X: Proof. The proof is straightforward. Theorem. If A = is an (; _q k )-intuitionistic fuzzy soft ideal of X; then are soft ideals of X: X = fx X : (x) = minf (x); k g = ef [] (0)g X = fx X : (x) = maxf (x); k g = ef [] (0)g
Page of 0 0 0 0 0 0 Proof. Let a X x X: Then by de nition minf (x); k g = F e [] (0): Since A = is an (; _q k )-intuitionistic fuzzy soft ideal of X so we have (i) (a x) minf (a) ; k g = e F [] (0) ; which implies that (a x) X : (ii) (a x) maxf (a) ; k g = e F [] (0) ; which implies that (a x) X : Let x _ y exists in X; we get (ii) (x _ y) minf (x) (y); k g = F e [] (0) ;which implies that (x _ y) X ; (iv) (x _ y) maxf (x) (y); k g = F e [] (0) ;which implies that (x _ y) X : Hence are soft ideals of X: X = fx X : (x) = minf (x); k g = ef [] (0)g X = fx X : (x) = maxf (x); k g = ef [] (0)g Theorem. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft subalgebras (resp., ideals) of X. Then so is h e F ; Ai \ h e G; Bi: Proof. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft subalgebras of X. We know that h e F ; Ai \ h e G; Bi = h e ; Ci; where C = A [ B: Now for any C x; y X; we consider the following cases Case : For any A B; we have e[] (x y) = (x y) minf (x) (y); k g = minf e[] (x) ; e[] (y); k g;
Page of 0 0 0 0 0 0 Case : For any B A; we have Case : e[] (x y) = (x y) maxf (x) (y); k g = maxf e[] (x) ; e[] (y); k g: e[] (x y) = eg[] (x y) minf eg[] (x) ; eg[] (y); k g = minf e[] (x) ; e[] (y); k g; e[] (x y) = eg[] (x y) maxf eg[] (x) ; eg[] (y); k g = maxf e[] (x) ; e[] (y); k g: For any A \ B; we have \ eg["] [ eg["] : Analogous to the proof of Theorem.. Hence h e F ; Ai \ h e G; Bi is an (; _q k )-intuitionistic fuzzy soft subalgebra of X. Theorem. Let h e F ; Ai h e G; Bi be two (; _q k )-intuitionistic fuzzy soft subalgebra (resp., ideals) of X. Then so is h e F ; Ai [ h e G; Bi: Proof. The proof is similar to the proof of the Theorem.: Acknowledgement: We highly appreciate the detailed valuable comments of the referees which greatly improve the quality of this paper. References [] Abbot, J.C.. Sets, lattices Boolean algebra, Allyn Bacon, Boston. [] Ahmad, B. Athar, K. 00. Mappings on fuzzy soft classes. Advances in Fuzzy Systems. -. [] Atanassov, K.T.. Intuitionistic fuzzy sets. Fuzzy sets System. 0, -.
Page of 0 0 0 0 0 0 [] Aygunoglu, A. Aygun, H. 00. Introduction to fuzzy soft groups. Computers Mathematics with Applications., -. [] Bhakat, S.K. Das, P.. (; _q) fuzzy subgroups. Fuzzy Sets Systems. 0, -. [] Ceven, Y. Ozturk, M.A. 00. Some results on subtraction algebras. Hacettepe Journal of Mathematica Statistics., -0. [] Jun, Y.B., Kim, H.S. Roh, E.H. 00. Ideal theory of subtraction algebras. Scientiae Mathematicae Japonicae., -. [] Jun, Y.B. Kim, H.S. 00. On ideals in subtraction algebras. Scientiae Mathematicae Japonicae., -. [] Jun, Y.B., Kang, M.S. Park, C.H. 0. Fuzzy subgroups based on fuzzy points. Communications of the Korean Mathematical Society., -. [0] Jun, Y.B., Lee, K.J. Park, C.H. 00. Fuzzy soft set theory applied to BCK/BCI-algebras. Computers Mathematics with Applications., 0-. [] Lee, K.J. Park, C.H. 00. Some questions on fuzzi cations of ideals in subtraction algebras. Communications of the Korean Mathematical Society., -. [] Liu, Y. Xin, X. 0. General fuzzy soft groups fuzzy normal soft groups, Annals of Fuzzy Mathematics Informatics., -00. [] Maji, P.K., Roy, A.R. Biswas, R. 00. An application of soft sets in a decision making problem, Computers Mathematics with Applications., 0-0. [] Maji, P.K., Biswas, R. Roy, A.R. 00. Soft set theory. Computers Mathematics with Applications., -. [] Maji, P.K., Biswas, R. Roy, A.R. 00a. Fuzzy soft sets. Journal of Fuzzy Mathematics., -0. [] Maji, P.K., Biswas, R. Roy, A.R. 00b. Intuitionistic fuzzy soft sets. Journal of Fuzzy Mathematics., -.
Page of 0 0 0 0 0 0 [] Maji, P.K., Roy, A.R. Biswas, R. 00. On intuitionistic fuzzy soft sets. Journal of Fuzzy Mathematics., -. [] Molodtsov, D.. Soft set theory- rst results. Computers Mathematics with Applications., -. [] Schein, B.M.. Di erence semigroups, Communications in Algebra. 0, -. [0] Shabir, M., Jun, Y.B. Nawaz, Y. 00. Semigroups characterized by (; _q k )-fuzzy ideals. Computers Mathematics with Applications. 0, -. [] Shabir, M. Mahmood, T., 0. Characterizations of hemirings by (; _q k )-fuzzy ideals. Computers & Mathematics with Applications. (), 0 0. [] Shabir, M. Mahmood, T., 0. Semihypergroups characterized by (; _q k )-fuzzy hyperideals. Information Sciences Letters. (), 0-. [] Williams, D.R.P. Saeid, A.B. 0. Fuzzy soft ideals in subtraction algebras. Neural Computing Applications., -. [] Yang, C. 0. Fuzzy soft semigroups fuzzy soft ideals. Computers Mathematics with Applications., -. [] Yaqoob, N., Akram, M. Aslam, M. 0. Intuitionistic fuzzy soft groups induced by (t; s)-norm. Indian Journal of Science Technology,, -. [] Zadeh, L.A.. Fuzzy sets, Information Control., -. [] Zelinka, B.. Subtraction semigroup. Mathematica Bohemica. 0, -.