for review onlyrdo.psu.ac.th/sjstweb/ar-press/58-june/6.pdf(2011) have studied fuzzy soft semigroups...

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On (∈∈∈∈,∈∨∈∨∈∨∈∨qk)-intuitionistic (fuzzy ideals, fuzzy soft ideals) of

subtraction algebras

Journal: Songklanakarin Journal of Science and Technology

Manuscript ID: SJST-2015-0028.R1

Manuscript Type: Original Article

Date Submitted by the Author: 07-Apr-2015

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 and 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

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Songklanakarin Journal of Science and Technology SJST-2015-0028.R1 Yaqoob

For Review O

nlyOn (2;2 _qk)-intuitionistic (fuzzy ideals, fuzzy soft

ideals) of subtraction algebras

aMadad Khan, bBijan Davvaz, cNaveed Yaqoob and dMuhammad Gulistan

aDepartment of Mathematics, COMSATS Institute of I.T, Abbottabad, Pakistan

bDepartment of Mathematics, Yazd University, Yazd, Iran

cDepartment of Mathematics, College of Science in Al-Zul�,

Majmaah University, Al-Zul�, Saudi Arabia

dDepartment of Mathematics, Hazara University, Mansehra, Pakistan

E-mail: [email protected], [email protected], [email protected],

[email protected]

Abstract: The intent of this article is to study the concept of an (2;2 _qk)-intuitionistic fuzzy ideal and

(2;2 _qk)-intuitionistic fuzzy soft ideal of subtraction algebras and to introduce some related properties.

2010 AMS Classi�cation: 06F35, 03G25, 08A72.

Keywords: Subtraction algebras, (2;2 _qk)-intuitionistic fuzzy (soft) subalgebras, (2;2 _qk)-intuitionistic

fuzzy (soft) ideals.

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1 Introduction

Schein (1992) cogitate the system of the form (X; �; n), where X is the set of functions closed under the

composition " � " of functions (and hence (X; �) is a function semigroup) and the set theoretical subtrac-

tion "n" (and hence (X; n) is a subtraction algebra in the sense of (Abbot, 1969)). 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 (1995),

and 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. (2005). For detail study

of subtraction algebras see (Ceven and Ozturk, 2009 and Jun et al., 2007).

The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh

(1965). 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 and Park, 2007). Atanassov (1986)

introduced the idea of intuitionistic fuzzy set which in more general one as compared to a fuzzy set.

Bhakat and Das (1996) introduced a new type of fuzzy subgroups, that is, the (2;2 _q)-fuzzy subgroups.

Jun et al. (2011) introduced the notion of (2;2 _qk)-fuzzy subgroup. In fact, the (2;2 _qk)-fuzzy subgroup

is an important generalization of Rosenfeld�s fuzzy subgroup. Shabir et al. (2010) characterized semigroups

by (2;2 _qk)-fuzzy ideals, also see Shabir and Mahmood (2011, 2013).

Molodtsov (1999) 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. (2002) described the application of soft set theory to a decision-making problem. Maji

et al. (2003) also studied several operations on the theory of soft sets. Maji et al. (2001b, 2004) studied

intuitionistic fuzzy soft sets.

The notion of fuzzy soft sets, as a generalization of the standard soft sets, was introduced in (Maji, 2001a),

and an application of fuzzy soft sets in a decision-making problem was presented. Ahmad et al. (2009)

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have introduced arbitrary fuzzy soft union and fuzzy soft intersection. Aygunoglu et al. (2009) introduced

the notion of fuzzy soft group and studied its properties. Yaqoob et al. (2013) studied the properties

of intuitionistic fuzzy soft groups in terms of intuitionistic double t-norm. Jun et al. (2010) introduced

the notion of fuzzy soft BCK/BCI-algebras and (closed) fuzzy soft ideals, and then derived their basic

properties. Williams and Saeid (2012) studied fuzzy soft ideals in subtraction algebras. Recently, Yang

(2011) have studied fuzzy soft semigroups and fuzzy soft (left, right) ideals, and have discussed fuzzy soft

image and fuzzy soft inverse image of fuzzy soft semigroups (ideals) in detail. Liu and Xin (2013) studied

the idea of generalized fuzzy soft groups and fuzzy normal soft groups.

In this article, we study the concept of (2;2 _qk)-intuitionistic fuzzy (soft) ideals of subtraction algebras.

Here we consider some basic properties of (2;2 _qk)-intuitionistic fuzzy (soft) ideals of subtraction algebras.

2 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 2.1 (Jun et al., 2005) A non-empty set X together with a binary operation "�" is said to be a

subtraction algebra if it satis�es the following:

(S1) x� (y � x) = x;

(S2) x� (x� y) = y � (y � x) ;

(S3) (x� y)� z = (x� z)� y; for all x; y; z 2 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 2 X: The ordered set (X;�) is a semi-Boolean algebra in the sense of (Abbot, 1969);

that is, it is a meet semi lattice with zero, in which every interval [0; a] is a boolean algebra with respect to

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the induced order. Here a ^ b = a� (a� b); the complement of an element b 2 [0; a] is a� b and is denoted

by b=; and if b; c 2 [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., 2005):

(a1) (x� y)� y = x� y;

(a2) x� 0 = x and 0� x = 0;

(a3) (x� y)� x = 0;

(a4) x� (x� y) � y;

(a5) (x� y)� (y � x) = x� y;

(a6) x� (x� (x� y)) = x� y;

(a7) (x� y)� (z � y) � x� z;

(a8) x � y if and only if x = y � w for some w 2 X;

(a9) x � y implies x� z � y � z and z � y � z � x; for all z 2 X;

(a10) x; y � z implies x� y = x ^ (z � y) ;

(a11) (x ^ y)� (x ^ z) � x ^ (y � z) ;

(a12) (x� y)� z = (x� z)� (y � z) :

De�nition 2.2 (Jun et al., 2005) A non-empty subset A of a subtraction algebra X is called an ideal of X;

denoted by ACX : if it satis�es:

(b1) a� x 2 A for all a 2 A and x 2 X;

(b2) for all a; b 2 A; whenever a _ b exists in X then a _ b 2 A:

Proposition 2.3 (Jun et al., 2005) A non-empty subset A of a subtraction algebra X is called an ideal of

X; if and only if it satis�es:

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(b3) 0 2 A;

(b4) for all x 2 X and for all y 2 A; x� y 2 A) x 2 A:

Proposition 2.4 (Jun et al., 2005) Let X be a subtraction algebra and x; y 2 X: If w 2 X is an upper

bound for x and y; then the element x _ y = w � ((w � y)� x) is the least upper bound for x and y:

De�nition 2.5 (Jun et al., 2005) Let Y be a non-empty subset of X then Y is called a subalgebra of X if

x� y 2 Y; whenever x; y 2 Y:

De�nition 2.6 (Lee and Park, 2007) 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 2 X:

De�nition 2.7 (Lee and Park, 2007) A fuzzy subset f is said to a fuzzy ideal of X if satis�es:

(FI1) f(x� y) � f(x);

(FI2) If there exists x _ y, then f(x _ y) � minff(x); f(y)g; for all x; y 2 X:

3 (2;2 _qk)-intuitionistic fuzzy ideals

In this section we will discuss some properties related to (2;2 _qk)-intuitionistic fuzzy ideals of subtraction

algebras.

De�nition 3.1 An intuitionistic fuzzy set A in X is an object of the form A = f(x; �A(x); A(x)) : x 2 Xg;

where the function �A : X ! [0; 1] and A : X ! [0; 1] denote the degree of membership and degree of

non-membership of each element x 2 X; and 0 � �A(x) + A(x) � 1 for all x 2 X: For simplicity, we will

use the symbol A = (�A; A) for the intuitionistic fuzzy set A = f(x; �A(x); A(x)) : x 2 Xg: We de�ne

0(x) = 0 and 1(x) = 1 for all x 2 X:

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De�nition 3.2 Let X be a subtraction algebra. An intuitionistic fuzzy set A = f(x; �A(x); A(x)) : x 2 Xg;

of the form

x(�;�)y =

8>><>>:(�; �) if y = x

(0; 1) if y 6= x;

is said to be an intuitionistic fuzzy point with support x and value (�; �) and 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 2 Xg written x(�;�) 2 A�resp: x(�;�)qA

�if �A(x) � � and A(x) �

� (resp., �A(x) + � > 1 and A(x) + � < 1) : By the symbol x(�;�)qkA we mean �A(x) + � + k > 1 and

A(x) + � + k < 1; where k 2 (0; 1) :

We use the symbol xt 2 �A implies �A(x) � t and tx [2] A implies A(x) � t; in the whole paper.

De�nition 3.3 An intuitionistic fuzzy set A = (�A; A) of X is said to be an (2;2 _qk)-intuitionistic fuzzy

subalgebra of X if

x(t1;t3) 2 A; y(t2;t4) 2 A) (x� y)(t1^t2;t3_t4) 2 _qkA; for all x; y 2 X; t1; t2; t3; t4; k 2 (0; 1):

OR

An intuitionistic fuzzy set A = (�A; A) of X is said to be an (2;2 _qk)-intuitionistic fuzzy subalgebra of X

if it satisfy the following conditions,

(i) xt1 2 �A; yt2 2 �A ) (x� y)t1^t2 2 _qk�A; for all x; y 2 X; t1; t2; k 2 (0; 1];

(ii) t3x [2] A;

t4y [2] A )

t3_t4x�y [2] ^ [qk] A; for all x; y 2 X; t3; t4; k 2 [0; 1):

Example 3.4 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

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Let us de�ne the intuitionistic fuzzy set A = (�A; A) of X as

X 0 a b

�A(x) 0:5 0:6 0:7

A(x) 0:1 0:2 0:21

;

and

t1 0:4

t2 0:5

t3 0:22

t4 0:23

k 0:5

then A = (�A; A) is an (2;2 _q0:5)-intuitionistic fuzzy subalgebra of X:

De�nition 3.5 An intuitionistic fuzzy set A = (�A; A) of X is said to be an (2;2 _qk)-intuitionistic fuzzy

ideal of X if it satisfy the following conditions,

(i) x(t;t) 2 A; y 2 X ) (x� y)t 2 _qkA;

(ii) If there exist x _ y; then x(t1;t3) 2 A; y(t2;t4) 2 A ) (x _ y)(t1^t2;t3_t4) 2 _qkA; for all x; y 2 X;

t; t1; t2; t3; t4; k 2 (0; 1):

OR

An intuitionistic fuzzy set A = (�A; A) of X is said to be an (2;2 _qk)-intuitionistic fuzzy ideal of X if it

satisfy the following conditions,

(i) xt 2 �A; y 2 X ) (x� y)t 2 _qk�A;

(ii) If there exist x _ y in X then xt1 2 �A; yt2 2 �A ) (x _ y)t1^t2 2 _qk�A; for all x; y 2 X; t; t1; t2; k 2

(0; 1];

(iii) tx [2] A; y 2 X ) t

x�y [2] ^ [qk] A ;

(iv) If there exist x_y in X then t3x [2] A;

t4y [2] A )

t3_t4x_y [2]^ [qk] A; for all x; y 2 X; t; t3; t4; k 2 [0; 1):

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Example 3.6 Let X = f0; a; bg be a subtraction algebra with the Cayley table de�ne in Example 3.4, and

let us de�ne an intuitionistic fuzzy set A = (�A; A) of X as

X 0 a b

�A(x) 0:5 0:6 0:7

A(x) 0:1 0:2 0:3

;

and

t 0:4

t1 0:4

t2 0:45

t3 0:42

t4 0:47

k 0:2

then A = (�A; A) is an (2;2 _q0:2)-intuitionistic fuzzy ideal of X:

Theorem 3.7 An intuitionistic fuzzy set A = (�A; A) of X is said to be an (2;2 _qk)-intuitionistic fuzzy

subalgebra of X if and only if �A(x�y) � minf�A (x) ; �A(y); 1�k2 g and A(x�y) � maxf A (x) ; A(y);1�k2 g:

Proof. Let A = (�A; A) be an (2;2 _qk)-intuitionistic fuzzy subalgebra of X; and assume on contrary

that there exist some t 2 (0; 1] and r 2 [0; 1) such that

�A(x� y) < t < minf�A (x) ; �A(y);1� k2

g

and

A(x� y) > r > maxf A (x) ; A(y);1� k2

g:

This implies that

�A (x) � t; �A (y) � t; �A(x� y) < t

) �A(x� y) + t+ k <1� k2

+1� k2

+ k = 1

) (x� y)t2 _qk�A;

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which is a contradiction. Hence

�A(x� y) � minf�A (x) ; �A(y);1� k2

g:

Also from

A(x� y) > r > maxf A (x) ; A(y);1� k2

g

we get

A (x) � r; A (y) � r; A(x� y) > r

) A(x� y) + r + k >1� k2

+1� k2

+ k = 1

) r

x� y [2] ^ [qk] A(x);

which is a contradiction. Hence

A(x� y) � maxf A (x) ; A(y);1� k2

g:

Conversely, let �A(x�y) � minf�A (x) ; �A(y); 1�k2 g and A(x�y) � maxf A (x) ; A(y);1�k2 g: Let xt1 2 �A;

yt2 2 �A for all x; y 2 X; t1; t2; k 2 (0; 1] and t3x [2] A;

t4y [2] A for all x; y 2 X; t3; t4; k 2 [0; 1): This implies

that �A (x) � t1; �A (y) � t2 and A (x) � t3; A (y) � t4: Consider �A(x�y) � min��A(x); �A(y);

1�k2

�

min�t1; t2;

1�k2

: If t1 ^ t2 > 1�k

2 ; then �A(x� y) �1�k2 : So �A(x� y) + (t1 ^ t2) + k >

1�k2 + 1�k

2 + k = 1;

which implies that (x � y)t1^t2qk�A: If t1 ^ t2 � 1�k2 ; then �A(x � y) � t1 ^ t2: So (x � y)t1^t2 2 �A:

Thus (x � y)t1^t2 2 _qk�A: Also consider A(x � y) � maxf A (x) ; A(y); 1�k2 g � maxft3; t4; 1�k2 g: If

t3 _ t4 < 1�k2 ; then A(x � y) �

1�k2 : So A(x � y) + (t3 _ t4) + k <

1�k2 + 1�k

2 + k = 1; which implies

that t3_t4x�y [qk] A: If t3 _ t4 >

1�k2 ; then A(x � y) � t3 _ t4: So t3_t4

x�y [2] A: Thust3_t4x�y [2] _ [qk] A: Hence

A = (�A; A) is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X: �


ideal of X if and only if

(i) �A(x� y) � minf�A (x) ; 1�k2 g;

(ii) A(x� y) � maxf A (x) ; 1�k2 g;

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(ii) �A(x _ y) � minf�A (x) ; �A(y); 1�k2 g;

(iv) A(x _ y) � maxf A (x) ; A(y); 1�k2 g:

Proof. The proof is similar to the proof of the Theorem 3.7: �

Proposition 3.9 Every (2;2 _qk)-intuitionistic fuzzy ideal of X is an (2;2 _qk)-intuitionistic fuzzy sub-

algebra of X; but converse is not true as shown in the following example.

Example 3.10 Let X = f0; a; bg be a subtraction algebra with the Cayley table de�ne in Example 3.4, and

let

X 0 a b

�A(x) 0:9 0:3 0:6

A(x) 0:2 0:6 0:4

then by Theorem 3.7, A = (�A; A) is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X: But by Theorem

3.8, we observe that A = (�A; A) is not a (2;2 _q0:4)�intuitionistic fuzzy ideal of X: As �A(a _ b) =

�A(0� ((0� b)� a))) = �A(0) = 0:9 � minf�A (a) ; �A(b); 1�k2 g = minf0:3; 0:6; 0:3g = 0:3:

De�nition 3.11 Let A = (�A; A) is an intuitionistic fuzzy set of X: De�ne the intuitionistic level set as

A(�;�) = fx 2 Xj�A(x) � �; A(x) � �; where � 2 (0; 1�k2 ]; � 2 [1�k2 ; 1)g:


subalgebra of X if and only if A(�;�) 6= ; is a subalgebra of X:

Proof. Let A = (�A; A) be an (2;2 _qk)-intuitionistic fuzzy subalgebra of X: Suppose that x; y 2 A(�;�)

then �A(x) � �; �A(y) � � and A(x) � �; A(y) � �: Since A = (�A; A) is an (2;2 _qk)-intuitionistic

fuzzy subalgebra of X; so

�A(x� y) � minf�A (x) ; �A(y);1� k2

g

� minf�; �; 1� k2

g = minf�; 1� k2

g = �

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and

A(x� y) � maxf A (x) ; A(y);1� k2

g

� maxf�; �; 1� k2

g = maxf�; 1� k2

g = �:

Thus (x� y) 2 A(�;�): Hence A(�;�) 6= ; is a subalgebra of X:

Conversely, assume that A(�;�) 6= ; is a subalgebra of X: Assume on contrary that there exist some x; y 2

X such that �A(x � y) < minf�A (x) ; �A(y); 1�k2 g and A(x � y) > maxf A (x) ; A(y); 1�k2 g: Choose

� 2 (0; 1�k2 ]; � 2 [ 1�k2 ; 1) such that �A(x � y) < � < minf�A (x) ; �A(y); 1�k2 g and A(x � y) > � >

maxf A (x) ; A(y); 1�k2 g: This implies that (x � y) =2 A(�;�); which is a contradiction to the hypothesis.

Hence �A(x� y) � minf�A (x) ; �A(y); 1�k2 g and A(x� y) � maxf A (x) ; A(y);1�k2 g: Thus A = (�A; A)

is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X: �


ideal of X if and only if A(�;�) 6= ; is an ideal of X:

Proof. The proof is similar to the proof of the Theorem 3.12: �

De�nition 3.14 Let X be a subtraction algebra and A � X, an (2;2 _qk)-intuitionistic characteristic

function

�A = fhx; ��A(x); �A(x)ijx 2 Sg;

where ��A and �A are fuzzy sets respectively, de�ned as follows:

��A : X �! [0; 1]jx �! ��A(x) :=

8>><>>:1�k2 if x 2 A

0 if x =2 A

and

�A : X �! [0; 1]jx �! �A(x) :=

8>><>>:1 if x 2 A

1�k2 if x =2 A

:

Lemma 3.15 For a non-empty subset A of a subtraction algebra X; we have

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(i) A is a subalgebra of X if and only if the characteristic intuitionistic set �A =D��A ; �A

Eof A in X is

an (2;2 _qk)-intuitionistic fuzzy subalgebra of X:

(ii) A is an ideal of X if and only if the characteristic intuitionistic set �A =D��A ; �A

Eof A in X is an

(2;2 _qk)-intuitionistic fuzzy ideal of X:

Proof. (i) Let A is a subalgebra of X. We consider the following cases.

Case (1) Let x; y 2 A then x� y 2 A so ��A(x) =1�k2 = ��A(y) = ��A(x� y) and �A(x) = 1 = �A(y) =

�A(x� y):Hence

��A(x� y) =1� k2

� minf�A (x) ; �A(y);1� k2

g

� minf1� k2

;1� k2

;1� k2

g = 1� k2

and

�A(x� y) = 1 � maxf A (x) ; A(y);1� k2

g � maxf1; 1; 1� k2

g = 1:

Which implies that �A =D��A ; �A

Eof A is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X:

Case (2) Let x; y =2 A then x� y =2 A so ��A(x) = 0 = ��A(y) = ��A(x� y) and �A(x) =1�k2 = �A(y) =

�A(x� y): Hence

��A(x� y) = 0 � minf�A (x) ; �A(y);1� k2

g � minf0; 0; 1� k2

g = 0

and

�A(x� y) =1� k2

� maxf A (x) ; A(y);1� k2

g

� maxf1� k2

;1� k2

;1� k2

g = 1� k2

:

Which again implies that �A =D��A ; �A

Eof A in X is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X:

Hence characteristic intuitionistic set �A =D��A ; �A

Eof A is an (2;2 _qk)-intuitionistic fuzzy subalgebra

of X:

Conversely, let the characteristic intuitionistic set �A =D��A ; �A

Eof A in X is an (2;2 _qk)-intuitionistic

fuzzy subalgebra of X: Let x; y 2 A then ��A(x) =1�k2 = ��A(y) and �A(x) = 1 = �A(y):

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Hence

��A(x� y) � minf�A (x) ; �A(y);1� k2

g

� minf1� k2

;1� k2

;1� k2

g = 1� k2

and

�A(x� y) � maxf A (x) ; A(y);1� k2

g

� maxf1; 1; 1� k2

g = 1:

Which implies that (x� y) 2 A: Hence A is a subalgebra of X:

(ii) The proof of this part is similar to the proof of part (i). �

4 (2;2 _qk)-intuitionistic fuzzy soft ideals

Molodtsov de�ned the notion of a soft set as follows.

De�nition 4.1 (Molodtsov, 1999) 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 4.2 (Maji et al., 2001b) Let U be an initial universe and E be the set of parameters. Let A � E:

A pair ( eF ;A) is called an intuitionistic fuzzy soft set over U , where eF is a mapping given by eF : A �! IF (U):

In general, for every � 2 A: eF [�] = D� eF [�]; eF [�]E is an intuitionistic fuzzy set in U and it is called intuitionisticfuzzy value set of parameter �:

De�nition 4.3 Let U be an initial universe and E be a set of parameters. Suppose that A;B � E; ( eF ;A)and ( eG;B) are two intuitionistic fuzzy soft sets, we say that ( eF ;A) is an intuitionistic fuzzy soft subset of( eG;B) if and only

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(1) A � B;

(2) for all � 2 A, eF [�] is an intuitionistic fuzzy subset of eG[�]; that is, for all x 2 U and � 2 A; � eF [�](x) �� eG[�](x); and eF [�](x) � eG[�](x): This relationship is denoted by ( eF ;A) b ( eG;B):De�nition 4.4 Let ( eF ;A) and ( eG;B) be two intuitionistic fuzzy soft sets over a common universe U . Then( eF ;A)e( eG;B) is de�ned by ( eF ;A)e( eG;B) = (e�; A � B); where e�[�; "] = eF [�] \ eG["] for all (�; ") 2 A � B;that is,

e�[�; "] = D� eF [�](x) ^ � eG["](x); eF [�](x) _ eG["](x)E ;for all (�; ") 2 A�B; x 2 U:

De�nition 4.5 Let ( eF ;A) and ( eG;B) be two intuitionistic fuzzy soft sets over a common universe U . Then( eF ;A)e_( eG;B) is de�ned by ( eF ;A)e_( eG;B) = (e; A � B); where e[�; "] = eF [�] [ eG["] for all (�; ") 2 A � B;that is,

e[�; "] = D� eF [�](x) _ � eG["](x); eF [�](x) ^ eG["](x)E ;for all (�; ") 2 A�B; x 2 U:

De�nition 4.6 Let ( eF ;A) and ( eG;B) be two intuitionistic fuzzy soft sets over a common universe U . Thenthe intersection (e�; C); where C = A [B and for all � 2 C and x 2 U;

�e�[�](x) =

8>>>>>><>>>>>>:� eF [�](x) if � 2 A�B

� eG[�](x) if � 2 B �A

� eF [�](x) ^ � eG[�](x) if � 2 A \B;

e�[�](x) =

8>>>>>><>>>>>>: eF [�](x) if � 2 A�B

eG[�](x) if � 2 B �A

eF [�](x) _ eG[�](x) if � 2 A \B:

We denote it by ( eF ;A) e ( eG;B) = (e�; C):14

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De�nition 4.7 Let ( eF ;A) and ( eG;B) be two intuitionistic fuzzy soft sets over a common universe U . Thenthe union (e�; C); where C = A [B and for all � 2 C and x 2 U;

�e�[�](x) =

8>>>>>><>>>>>>:� eF [�](x) if � 2 A�B

� eG[�](x) if � 2 B �A

� eF [�](x) _ � eG[�](x) if � 2 A \B;

e�[�](x) =

8>>>>>><>>>>>>: eF [�](x) if � 2 A�B

eG[�](x) if � 2 B �A

eF [�](x) ^ eG[�](x) if � 2 A \B:

We denote it by ( eF ;A) d ( eG;B) = (e�; C):

In contrast with the above de�nitions of IF-soft set union and intersection, we may sometimes adopt di¤erent

de�nitions of union and intersection given as follows.

De�nition 4.8 Let ( eF ;A) and ( eG;B) be two IF-soft sets over a common universe U and A\B 6= ;: Then

the bi-intersection of ( eF ;A) and ( eG;B) is de�ned to be the fuzzy soft set (e�; C); where C = A \ B and

e�[�] = eF [�] \ eG[�] for all � 2 C: This is denoted by (e�; C) = ( eF ;A)eu( eG;B):

De�nition 4.9 Let ( eF ;A) and ( eG;B) be two IF-soft sets over a common universe U and A \ B 6= ;:

Then the bi-union of ( eF ;A) and ( eG;B) is de�ned to be the fuzzy soft set (e�; C); where C = A \ B and

e�[�] = eF [�] [ eG[�] for all � 2 C: This is denoted by (e�; C) = ( eF ;A)et( eG;B):

De�nition 4.10 Let ( eF ;A) be an IF-soft set over X. Then ( eF ;A) is called an (2;2 _qk)-intuitionisticfuzzy soft subtraction algebra if eF [�] = D

� eF [�]; eF [�]Eis an (2;2 _qk)-intuitionistic fuzzy subalgebra of X;

for all � 2 A:

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Example 4.11 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

and let U is the set of cellular brands of mobile companies in the market. Let � = fattractive, expensive,

cheapg is a parameter space and A = fattractive, cheapg: De�ne

eF ([attractive]) = fh0; 0:9; 0:1i ; ha; 0:8; 0:2i ; hb; 0:7; 0:3ig;

eF ([cheap]) = fh0; 0:5; 0:12i ; ha; 0:6; 0:15i ; hb; 0:7; 0:2ig:

Let k = 0:4 then h eF ;Ai is an (2;2 _q0:4)-intuitionistic fuzzy soft subalgebra of X:De�nition 4.12 An intuitionistic fuzzy soft set h eF ;Ai of X is called an (2;2 _qk)-intuitionistic fuzzy soft

subalgebra of X; if for all � 2 A; eF [�] = D� eF [�]; eF [�]E is an (2;2 _qk)-intuitionistic fuzzy subalgebra of X;if

(i) � eF [�](x� y) � minf� eF [�] (x) ; � eF [�](y); 1�k2 g;(ii) eF [�](x� y) � maxf eF [�] (x) ; eF [�](y); 1�k2 g;for all x; y 2 X:

De�nition 4.13 An intuitionistic fuzzy soft set h eF ;Ai of X is called an (2;2 _qk)-intuitionistic fuzzy soft

ideal of X; if for all � 2 A; eF [�] = D� eF [�]; eF [�]E is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X; if(i) � eF [�](x� y) � minf� eF [�] (x) ; 1�k2 g;(ii) eF [�](x� y) � maxf eF [�] (x) ; 1�k2 g;(iii) � eF [�](x _ y) � minf� eF [�] (x) ; � eF [�](y); 1�k2 g;(iv) eF [�](x _ y) � maxf eF [�] (x) ; eF [�](y); 1�k2 g;for all x; y 2 X:

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Proposition 4.14 An (2;2 _qk)-intuitionistic fuzzy soft ideal of X is an (2;2 _qk)-intuitionistic fuzzy soft

subalgebra of X; but converse is not true as shown in the following example.

Example 4.15 From Example 4.11, h eF ;Ai is not an (2;2 _qk)-intuitionistic fuzzy soft ideal of X: As� eF ([attractive])(a _ b) = � eF ([attractive])(0� ((0� b)� a))) = � eF ([attractive])(0) = 0:9

� minf� eF ([attractive])(a); � eF ([attractive])(b); 1� k2 g

= 0:3 = minf0:8; 0:7; 0:3g; for k = 0:4:

Theorem 4.16 Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft subalgebras (resp., ideals)of X. Then h eF ;Aieh eG;Bi is also an (2;2 _qk)-intuitionistic fuzzy soft subalgebra (resp., ideal) of X:Proof. Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft ideals of X: We know that

h eF ;Aieh eG;Bi = he�; A�Bi; where e�[�; "] = eF [�] \ eG["] for all (�; ") 2 A�B; that ise�[�; "](x) = D� eF [�] (x) ^ � eG[�] (x) ; eF [�] (x) _ eG[�] (x)E

for all x 2 X: Let x; y 2 X; we have

�� eF [�] ^ � eG[�]

�(x� y) = � eF [�] (x� y) ^ � eG[�] (x� y)

��minf� eF [�] (x) ; 1� k2 g

�^�minf� eG[�] (x) ; 1� k2 g

�=

�minf� eF [�] (x) ^ � eG[�] (x) ; 1� k2 g

�= minf

�� eF [�] ^ � eG[�]

�(x) ;

1� k2

g;

and

� eF [�] _ eG[�]

�(x� y) = eF [�] (x� y) _ eG[�] (x� y)

��maxf eF [�] (x) ; 1� k2 g

�^�maxf eG[�] (x) ; 1� k2 g

�=

�maxf eF [�] (x) _ eG[�] (x) ; 1� k2 g

�= maxf

� eF [�] _ eG[�]

�(x) ;

1� k2

g:

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Also we have�� eF [�] ^ � eG[�]

�(x _ y) = � eF [�] (x _ y) ^ � eG[�] (x _ y)

��minf� eF [�] (x) ; � eF [�] (y) ; 1� k2 g

�^�minf� eG[�] (x) ; � eG[�] (y) ; 1� k2 g

�=

�minf� eF [�] (x) ^ � eG[�] (x) ; � eF [�] (y) ^ � eG[�] (y) ; 1� k2 g

�= minf

�� eF [�] ^ � eG[�]

�(x) ;

�� eF [�] ^ � eG[�]

�(y) ;

1� k2

g;

and� eF [�] _ eG[�]

�(x _ y) = eF [�] (x _ y) _ eG[�] (x _ y)

��maxf eF [�] (x) ; eF [�] (y) ; 1� k2 g

�^�maxf eG[�] (x) ; eG[�] (y) ; 1� k2 g

�=

�maxf eF [�] (x) _ eG[�] (x) ; eF [�] (y) _ eG[�] (y) ; 1� k2 g

�= maxf

� eF [�] _ eG[�]

�(x) ;

� eF [�] ^ eG[�]

�(y) ;

1� k2

g:

Hence h eF ;Aieh eG;Bi is also an (2;2 _qk)-intuitionistic fuzzy soft ideal of X: The other case can be seen ina similar way. �

Theorem 4.17 Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft subalgebras (resp., ideals)of X. Then h eF ;Aie_h eG;Bi is also an (2;2 _qk)-intuitionistic fuzzy soft subalgebra (resp., ideal) of X:Proof. The proof is similar to the proof of the Theorem 4.16: �

Theorem 4.18 Let h eF ;Ai be an intuitionistic fuzzy soft set of X: Then h eF ;Ai is an (2;2 _qk)-intuitionisticfuzzy soft subalgebra (resp., ideal) of X if and only if h eF ;Ai(�;�) = h eF (�;�); Ai = fx 2 Xj� eF [�] (x) � �;

eF [�] (x) � �; where � 2 (0; 1�k2 ]; � 2 [ 1�k2 ; 1)g is a soft subalgebra (resp., ideal) of X:

Proof. Let h eF ;Ai be an (2;2 _qk)-intuitionistic fuzzy soft subalgebra of X: Let x; y 2 h eF ;Ai(�;�) then� eF [�] (x) � �; � eF [�] (y) � � and eF [�] (x) � �; eF [�] (y) � �; where � 2 (0; 1�k2 ]; � 2 [ 1�k2 ; 1): Since h eF ;Ai isan (2;2 _qk)-intuitionistic fuzzy soft subalgebra of X; so

� eF [�] (x� y) � minf� eF [�] (x) ; � eF [�] (y) ; 1� k2 g

= minf�; �; 1� k2

g = �;

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and

eF [�] (x� y) � maxf eF [�] (x) ; eF [�] (y) ; 1� k2 g

= maxf�; �; 1� k2

g = �:

Thus (x� y) 2 h eF ;Ai(�;�): Hence h eF ;Ai(�;�) is a soft subalgebra of X:Conversely assume that h eF ;Ai(�;�) is a soft subalgebra of X: Let there exist some r 2 (0; 1] and s 2 [0; 1)such that

� eF [�] (x� y) < r < minf� eF [�] (x) ; � eF [�] (y) ; 1� k2 g

and

eF [�] (x� y) > s > minf eF [�] (x) ; eF [�] (y) ; 1� k2 g:

This implies that (x� y) =2 h eF ;Ai(�;�); which is contradiction. Thus� eF [�] (x� y) � minf� eF [�] (x) ; � eF [�] (y) ; 1� k2 g

and

eF [�] (x� y) � maxf eF [�] (x) ; eF [�] (y) ; 1� k2 g:

Hence h eF ;Ai is an (2;2 _qk)-intuitionistic fuzzy soft subalgebra of X: �

Proposition 4.19 Every (2;2 _qk)-intuitionistic fuzzy soft ideal h eF ;Ai of X satis�es the following,

(i) � eF [�](0) � minf� eF [�](x); 1�k2 g;(ii) eF [�](0) � maxf eF [�](x); 1�k2 g, for all x 2 X:

Proof: By letting x = y in the De�nition 4.13, we get the required proof. �

Lemma 4.20 If an (2;2 _qk)-intuitionistic fuzzy soft set h eF ;Ai of X satis�es the followings,

(i) � eF [�](0) � minf� eF [�](x); 1�k2 g;(ii) eF [�](0) � maxf eF [�](x); 1�k2 g;

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(iii) � eF [�](x� z) � minf� eF [�]((x� y)� z); � eF [�] (y) ; 1�k2 g;(iv) eF [�](x� z) � maxf eF [�]((x� y)� z); eF [�] (y) ; 1�k2 g.Then we have x � a) � eF [�](x) � minf� eF [�](a); 1�k2 g and eF [�](x) � maxf eF [�](a); 1�k2 g; for all a; x; y; z 2X:

Proof. Let a; x 2 X and x � a: Consider

� eF [�](x) = � eF [�](x� 0) by (a2);� minf� eF [�]((x� a)� 0); � eF [�] (a) ; 1� k2 g by (iii);

= minf� eF [�](0); � eF [�] (a) ; 1� k2 g by x � a i¤ x� a = 0;

= minf� eF [�](a); 1� k2 g by (i):

Also consider

eF [�](x) = eF [�](x� 0) by (a2);� maxf eF [�]((x� a)� 0); eF [�] (a) ; 1� k2 g by (iv),

= maxf eF [�](0); eF [�] (a) ; 1� k2 g by x � a i¤ x� a = 0;

= maxf eF [�](a); 1� k2 g by (ii):

This complete the proof. �

Theorem 4.21 An (2;2 _qk)-intuitionistic fuzzy soft set h eF ;Ai of X satis�es the conditions (i)-(iv) of the

Lemma 4.20, if and only if h eF ;Ai is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X:Proof. Suppose that h eF ;Ai; i.e, eF [�] = D

� eF [�]; eF [�]Esatis�es the conditions (i)-(iv) of the Lemma 4.20.

Let x; y 2 X; then by using (a3) we have x � y � x: Now by the use of lemma 4.20, � eF [�](x � y) �

minf� eF [�](x); 1�k2 g and eF [�](x� y) � maxf eF [�](x); 1�k2 g for all x; y 2 X:

Also � eF [�](x _ y) � minf� eF [�](x); 1�k2 g and eF [�](x _ y) � maxf eF [�](x); 1�k2 g whenever x _ y exists in

X and by using the lemma 4.20, we have � eF [�](x _ y) � minf� eF [�](x); � eF [�](y); 1�k2 g and eF [�](x _ y) �

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maxf eF [�](x); eF [�](x); 1�k2 g for all x; y 2 X: Hence h eF ;Ai is an (2;2 _qk)-intuitionistic fuzzy soft ideal ofX:

Conversely, assume that h eF ;Ai is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X: Conditions (i) and (ii)directly follows from the Proposition 4.19. let x; y; z 2 X: Put x = y and y = x � z in (a3) and (a4); We

get from (a3); (x� z)� y � x� z and from (a4); we have y � (y � ((x� z)) � x� z: Which indicates that

x� z is an upper bound for (x� z)� y and y � (y � ((x� z)): By using the Proposition 2.4, we have

((x� z)� y) _ (y � (y � ((x� z)))

= (x� z)� (((x� z)� ((x� z)� y))� (y � (y � (x� z))))

= x� z:

Thus

� eF [�](x� z) = � eF [�](((x� z)� y) _ (y � (y � ((x� z))))� minf� eF [�] ((x� z)� y) ; � eF [�] (y � (y � (x� z)) ; 1� k2 g

= minf� eF [�] ((x� y)� z) ; � eF [�] (y) ; 1� k2 g by (S3) and (a2) and x = y:

Which is (iii). And

eF [�](x� z) = eF [�](((x� z)� y) _ (y � (y � ((x� z))))� maxf eF [�] ((x� z)� y) ; eF [�] (y � (y � (x� z)) ; 1� k2 g

= maxf eF [�] ((x� y)� z) ; eF [�] (y) ; 1� k2 g by (S3) and (a2) and x = y:

Which is (iv). Hence it completes the proof. �

Theorem 4.22 An (2;2 _qk)-intuitionistic fuzzy soft set h eF ;Ai of X is an (2;2 _qk)-intuitionistic fuzzy

soft ideal of X if and only if it satis�es

(i) � eF [�](x� ((x� a)� b)) � minf� eF [�](a); � eF [�](b); 1�k2 g;(ii) eF [�](x� ((x� a)� b)) � maxf eF [�](a); eF [�](b); 1�k2 g for all x; a; b 2 X:

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Proof. Let (2;2 _qk)-intuitionistic fuzzy set A =�� eF [�]; eF [�]

�of X satis�es (i) and (ii). Consider

� eF [�](x� y) = � eF [�] ((x� y)� (((x� y)� x)� x))) by (i)� minf� eF [�](x); � eF [�](x); 1� k2 g

= minf� eF [�](x); 1� k2 g

and

eF [�](x� y) = eF [�] ((x� y)� (((x� y)� x)� x))) by (ii)� max cf eF [�](x); eF [�](x); 1� k2 g

= maxf eF [�](x); 1� k2 g:

Also suppose that x _ y exists in X and by using the Proposition 2.4, we have x _ y = w � ((w � x) � y):

Now using the given conditions

� eF [�](x _ y) = � eF [�] (w � ((w � x)� y))� minf� eF [�](x); � eF [�](y); 1� k2 g by (i);

and

eF [�](x _ y) = eF [�] (w � ((w � x)� y))� maxf eF [�](x); eF [�](y); 1� k2 g by (ii):

Hence A =�� eF [�]; eF [�]

�is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X:

Conversely, let A =�� eF [�]; eF [�]

�be an (2;2 _qk)-intuitionistic fuzzy soft ideal of X: By Theorem 4.21,

� eF [�](x� z) � minf� eF [�]((x� y)� z); � eF [�](y); 1� k2 g

and

eF [�](x� z) � maxf eF [�]((x� y)� z); eF [�](y); 1� k2 g:

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Let z = (x� a)� b and y = b; then

� eF [�](x� ((x� a)� b)) � minf� eF [�]((x� y)� ((x� a)� b)); � eF [�](b); 1� k2 g

= minf� eF [�]((x� y)� ((x� b)� a)); � eF [�](b); 1� k2 g by (S3)

= minf� eF [�](a); � eF [�](b); 1� k2 g by putting x = x� b and y = a in (a4)

and

eF [�](x� ((x� a)� b)) � maxf eF [�]((x� y)� ((x� a)� b)); eF [�](b); 1� k2 g

= maxf eF [�]((x� y)� ((x� b)� a)); eF [�](b); 1� k2 g by (S3)

= maxf eF [�](a); eF [�](b); 1� k2 g by putting x = x� b and y = a in (a4):

Hence A =�� eF [�]; eF [�]

�satisfy condition (i) and (ii). �

Lemma 4.23 An intuitionistic fuzzy set A =�� eF [�]; eF [�]

�of X is an (2;2 _qk)-intuitionistic fuzzy soft

ideal of X if and only if fuzzy sets � eF [�] and eF [�] are (2;2 _qk)- fuzzy soft ideals of X:

Proof. The proof is straightforward. �

Theorem 4.24 An intuitionistic fuzzy set A =�� eF [�]; eF [�]

�of X is an (2;2 _qk)-intuitionistic fuzzy soft

ideal of X if and only if B A =�� eF [�]; � eF [�]

�and C A =

� eF [�]; eF [�]

�are (2;2 _qk)- intuitionistic fuzzy

soft ideal of X:

Proof. The proof is straightforward. �

Theorem 4.25 If A =�� eF [�]; eF [�]

�is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X; then

X� eF [�] = fx 2 X : � eF [�](x) = minf� eF [�](x); 1� k2 g = � eF [�](0)g

and

X eF [�] = fx 2 X : eF [�](x) = maxf eF [�](x); 1� k2 g = eF [�](0)gare soft ideals of X:

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Proof. Let a 2 X� eF [�] and x 2 X: Then by de�nitionminf� eF [�](x); 1�k2 g = � eF [�](0): Since A =�� eF [�]; eF [�]

�is an (2;2 _qk)-intuitionistic fuzzy soft ideal of X so we have

(i) � eF [�](a� x) � minf� eF [�] (a) ; 1�k2 g = � eF [�] (0) ; which implies that (a� x) 2 X� eF [�] :

(ii) eF [�](a� x) � maxf eF [�] (a) ; 1�k2 g = eF [�] (0) ; which implies that (a� x) 2 X eF [�] :

Let x _ y exists in X; we get

(ii) � eF [�](x _ y) � minf� eF [�] (x) ; � eF [�](y); 1�k2 g = � eF [�] (0) ;which implies that (x _ y) 2 X� eF [�] ;

(iv) eF [�](x _ y) � maxf eF [�] (x) ; eF [�](y); 1�k2 g = eF [�] (0) ;which implies that (x _ y) 2 X eF [�] :

Hence

X� eF [�] = fx 2 X : � eF [�](x) = minf� eF [�](x); 1� k2 g = � eF [�](0)g

and

X eF [�] = fx 2 X : eF [�](x) = maxf eF [�](x); 1� k2 g = eF [�](0)g

are soft ideals of X: �

Theorem 4.26 Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft subalgebras (resp., ideals)of X. Then so is h eF ;Ai\h eG;Bi:

Proof. Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft subalgebras of X. We know thath eF ;Ai\h eG;Bi = he�; Ci; where C = A [ B: Now for any � 2 C and x; y 2 X; we consider the following

cases

Case 1: For any � 2 A�B; we have

�e�[�](x� y) = � eF [�](x� y) � minf� eF [�] (x) ; � eF [�](y); 1� k2 g

= minf�e�[�] (x) ; �e�[�](y); 1� k2 g;

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and

e�[�](x� y) = eF [�](x� y) � maxf eF [�] (x) ; eF [�](y); 1� k2 g

= maxf e�[�] (x) ; e�[�](y); 1� k2 g:

Case 2: For any � 2 B �A; we have

�e�[�](x� y) = � eG[�](x� y) � minf� eG[�] (x) ; � eG[�](y); 1� k2 g

= minf�e�[�] (x) ; �e�[�](y); 1� k2 g;

and

e�[�](x� y) = eG[�](x� y) � maxf eG[�] (x) ; eG[�](y); 1� k2 g

= maxf e�[�] (x) ; e�[�](y); 1� k2 g:

Case 3: For any � 2 A \ B; we have � eF [�] \ � eG["] and eF [�] [ eG["]: Analogous to the proof of Theorem4.16. Hence h eF ;Ai\h eG;Bi is an (2;2 _qk)-intuitionistic fuzzy soft subalgebra of X. �

Theorem 4.27 Let h eF ;Ai and h eG;Bi be two (2;2 _qk)-intuitionistic fuzzy soft subalgebra (resp., ideals)of X. Then so is h eF ;Ai[h eG;Bi:Proof. The proof is similar to the proof of the Theorem 4.26: �

Acknowledgement: We highly appreciate the detailed valuable comments of the referees which greatly

improve the quality of this paper.

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for review onlyrdo.psu.ac.th/sjstweb/ar-press/58-june/6.pdf(2011) have studied fuzzy soft semigroups...

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