Direct product of general intuitionistic fuzzy sets of

Direct product of general intuitionistic fuzzy sets of subtraction algebras Muhammad Gulistan, Shah Nawaz and Syed Zaheer Abbas Accepted Manuscript Version

This is the unedited version of the article as it appeared upon acceptance by the journal. A final edited version of the article in the journal format will be made available soon.

As a service to authors and researchers we publish this version of the accepted manuscript (AM) as soon as possible after acceptance. Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). Please note that during production and pre-press, errors may be discovered which could affect the content.

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Publisher: Cogent OA Journal: Cogent Mathematics DOI: http://dx.doi.org/10.1080/23311835.2016.1176619

Direct product of general intuitionistic fuzzy sets of subtraction algebras 1,∗

Muhammad Gulistan, 2 Shah Nawaz, 3 Syed Zaheer Abbas

1,2,3

Department of Mathematics, Hazara University, Mansehra, Pakistan.

Downloaded by [39.54.63.164] at 09:13 18 April 2016

Abstract: We define direct product of (∈, ∈ ∨qk )-intuitionistic fuzzy sets and direct product of (∈, ∈ ∨qk )intuitionistic fuzzy soft sets of subtraction algebras and investigate some related properties. 2010 AMS Classification: 06F35, 03G25, 08A72. Keywords: Subtraction algebras, Direct product, an (∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebras, an (∈, ∈ ∨qk )-intuitionistic fuzzy soft ideals.

1

Introduction

The system (X; ◦, \) by Schein [26], is a set of functions closed under the composition ” ◦ ” under the composition of function(and hence (X, ◦) is a function semigroup) and the set theoretical subtraction ”\” (and hence (X, \) is a subtraction algebra in the sense of [1]). He proved that every subtraction semigroup is isomorphic to a difference semigroup of invertible functions. Zelinka [37] discussed a problem proposed by B. M. Schein concerning the structure of multiplication in a subtraction semigroup. He solved the problem for subtraction algebras of a special type called the atomic subtraction algebras. Jun et al. [12] introduced the notion of ideals in subtraction algebras and discussed characterization of ideals. To study more about subtraction algebras see [8, 13]. The fuzzifications of ideals in subtraction algebras were discussed in [18]. Bhakat and Das [7], introduced a new type of fuzzy subgroups, that is, the (∈, ∈ ∨q) fuzzy subgroups. In fact, the (∈, ∈ ∨qk ) fuzzy subgroup is an important generalization of Rosenfeld’s fuzzy subgroup. Shabir et ∗ Corresponding

author.

E-mail address: [email protected]

1

al. characterized semigroups by (∈, ∈ ∨qk )-fuzzy ideals in [27]. Gulistan et al. [9] studied (∈, ∈ ∨qk )-fuzzy KU-ideals of KU-algebras. Molodtsov [25] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that are free from the difficulties 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. [20] described the application of soft set theory to a decision-making problem. Maji et al. [21] also studied several operations on the theory of soft sets. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [36]. The notion of fuzzy soft sets, as a generalization of the standard soft sets, was introduced in [22], and an application of fuzzy soft sets in a decision-making problem was presented. In [2], Ahmad et al. have introduced arbitrary fuzzy soft union and fuzzy soft intersection. Aygunoglu et al.[6] introduced the notion of fuzzy soft group and studied its properties. In [11], Jun et al. have introduced the notion of fuzzy soft BCK/BCI-algebras and (closed) fuzzy soft ideals, and then derived their basic properties. Recently, Yang [28] 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. Recently, Khan et al. gave the idea of (∈, ∈ ∨qk )-intuitionistic (fuzzy ideals, fuzzy soft ideals) of subtraction algebras [15]. Gulistan et al. [10] defined direct product of generalized cubic sets in Hv-LA-semigroups. Yaqoob [29], studied interval-valued intuitionistic fuzzy ideals of regular LA-semigroups. In [3, 30], the authors applied the concept of intuitionistic fuzzy soft sets to ordered ternary semigroups and groups. Also see ([4], [16], [17], [31], [32], [33], [34], [35]). The aim of this article is to study the concept of Direct product of (∈, ∈ ∨qk )-intuitionistic fuzzy sets and Direct product of (∈, ∈ ∨qk )-intuitionistic fuzzy soft sets of subtraction algebras and investigate some related properties.

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. Through out the paper X denotes the subtraction algebra unless otherwise specified. Definition 2.1 [12] A nonempty set X together with a binary operation ”-” is said to be a subtraction algebra if it satisfies 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 ∈ 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 [1], that is, it is 2

a meet semi lattice with zero, in which every interval [0, a] is a boolean algebra with respect to the induced order. Here a ∧ b = a − (a − b); the complement of an element b ∈ [0, a] is a − b and is denoted by b/ ; and 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 [12]) : (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 ∈ X, (a9) x ≤ y implies x − z ≤ y − z and z − y ≤ z − x, for all z ∈ 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) . Definition 2.2 [12] A nonempty subset A of a subtraction algebra X is called an ideal of X, denoted by A ✁ X : if it satisfies: (b1) a − x ∈ A for all a ∈ A and x ∈ X, (b2) for all a, b ∈ A, whenever a ∨ b exists in X then a ∨ b ∈ A. Proposition 2.3 [12] A nonempty subset A of a subtraction algebra X is called an ideal of X, if and only if it satisfies: (b3) 0 ∈ A, (b4) for all x ∈ X and for all y ∈ A, x − y ∈ A ⇒ x ∈ A. Proposition 2.4 [12] Let X be a subtraction algebra and x, y ∈ X. If w ∈ 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. Definition 2.5 [12] Let Y is a nonempty subset of X then Y is called a subalgebra of X if x − y ∈ Y, whenever x, y ∈ Y. Definition 2.6 [18] Let f be a fuzzy set of X. Then f is called a fuzzy subalgebra of X if it satisfies 3

(FS) f (x − y) ≥ min{f (x), f (y)}, whenever x, y ∈ X. Definition 2.7 [18] A fuzzy set f is said to a fuzzy ideal of X if satisfies: (FI1) f (x − y) ≥ f (x), (FI2) If there exists x ∨ y, then f (x ∨ y) ≥ min{f (x), f (y)}, for all x, y ∈ X. Here we mentioned some of the related definitions and results which are directly used in our work. For detail we refer the reader [15]. Definition 2.8 [5] An intuitionistic fuzzy set A in X is an object of the form A = {(x, µA (x), γA (x)) : x ∈ X}, where the function µA : X → [0, 1] and γA : X → [0, 1] denote the degree of membership and degree of non-membership of each element x ∈ X, and 0 ≤ µA (x) + γA (x) ≤ 1 for all x ∈ X. For simplicity, we will use the symbol A = (µA , γA ) for the intuitionistic fuzzy set A = {(x, µA (x), γA (x)) : x ∈ X}. We define


0(x) = 0 and 1(x) = 1 for all x ∈ X. Definition 2.9 [15] Let X be a subtraction algebra. An intuitionistic fuzzy set A = {(x, µA (x), γA (x)) : x ∈ X}, of the form x(α,β) y =

(

(α, β)

if y = x

(0, 1)

if y 6= x

,

is said to be an intuitionistic fuzzy point with support x and value (α, β) and is denoted by x(α,β) . A fuzzy point x(α,β) is said to intuitionistic belongs to (resp., intuitionistic quasi-coincident) with intuitionistic fuzzy set A = {(x, µA (x), γA (x)) : x ∈ X} written x(α,β) ∈ A resp: x(α,β) qA if µA (x) ≥ α and γA (x) ≤ β (resp., µA (x) + α > 1 and γA (x) + β < 1) . By the symbol x(α,β) qk A we mean µA (x) + α + k > 1 and

γA (x) + β + k < 1, where k ∈ (0, 1) . We use the symbol xt ∈ µA implies µA (x) ≥ t and

t x [∈]γA

implies γA (x) ≤ t, in the whole paper.

Definition 2.10 [15] An intuitionistic fuzzy set A = (µA , γA ) of X is said to be an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X if x(t1, t3 ) ∈ A, y(t2, t4 ) ∈ A ⇒ (x − y)(t1 ∧t2 ,t3 ∨t4 ) ∈ ∨qk A, for all x, y ∈ X, t1 , t2 , t3 , t4 , k ∈ (0, 1). Definition 2.11 [15] An intuitionistic fuzzy set A = (µA , γA ) of X is said to be an (∈, ∈ ∨qk )-intuitionistic fuzzy ideal of X if it satisfy the following conditions, (i) x(t,t) ∈ A, y ∈ X ⇒ (x − y)t ∈ ∨qk A, (ii) If there exist x ∨ y, then x(t1, t3 ) ∈ A, y(t2, t4 ) ∈ A ⇒ (x ∨ y)(t1 ∧t2 ,t3 ∨t4 ) ∈ ∨qk A, for all x, y ∈ X, t, t1 , t2 , t3 , t4 , k ∈ (0, 1). Molodtsov defined the notion of a soft set as follows. 4

Definition 2.12 [25] 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).

Definition 2.13 [23, 24] Let U be an initial universe and E be the set of parameters. Let A ⊆ E. A pair (Fe, A) is called an intuitionistic fuzzy soft set over U , where Fe is a mapping given by Fe : A −→ IF (U ). E D In general, for every ǫ ∈ A. Fe [ǫ] = µFe[ǫ] , γFe[ǫ] is an intuitionistic fuzzy set in U and it is called intuitionistic

fuzzy value set of parameter ǫ.

Definition 2.14 [15] An intuitionistic fuzzyDsoft set hFe ,EAi of X is called an (∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebra of X, if for all ǫ ∈ A, Fe [ǫ] = µFe[ǫ] , γFe[ǫ] is an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X, if


(i) µFe[ǫ] (x − y) ≥ min{µFe[ǫ] (x) , µFe[ǫ] (y), 1−k 2 }, (ii) γFe[ǫ] (x − y) ≤ max{γFe[ǫ] (x) , γFe[ǫ] (y), 1−k 2 }, for all x, y ∈ X. e Definition 2.15 [15] An intuitionistic Dfuzzy soft set E hF , Ai of X is called an (∈, ∈ ∨qk )-intuitionistic fuzzy soft ideal of X, if for all ǫ ∈ A, Fe [ǫ] = µFe[ǫ] , γFe[ǫ] is an (∈, ∈ ∨qk )-intuitionistic fuzzy soft ideal of X, if

(i) µFe[ǫ] (x − y) ≥ min{µFe[ǫ] (x) , 1−k 2 },

(ii) γFe[ǫ] (x − y) ≤ max{γFe[ǫ] (x) , 1−k 2 }, (iii) µFe[ǫ] (x ∨ y) ≥ min{µFe[ǫ] (x) , µFe[ǫ] (y), 1−k 2 }, (iv) γFe[ǫ] (x ∨ y) ≤ max{γFe[ǫ] (x) , γFe[ǫ] (y), 1−k 2 }, for all x, y ∈ X.

3

Direct Product of an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra/Ideals

In this section, we define Direct product of an (∈, ∈ ∨qk )-intuitionistic fuzzy sets and investigate some related properties.

Definition 3.1 Let A = (µA (x), γA (x)) and B = (µB (x), γB (x)) be two (∈, ∈ ∨qk )-intuitionistic fuzzy sets of X1 and X2 , respectively. Then the Direct product of A = (µA (x), γA (x)) and B = (µB (x), γB (x)) is defined as A × B = hµA×B , γA×B i, where µA×B (x, y) = µA (x) ∧ µB (y) and γA×B (x, y) = γA (x) ∨ γB (y) for all (x, y) ∈ X1 × X2 . 5

Definition 3.2 An intuitionistic fuzzy set A × B of X1 × X2 is called an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 × X2 if it satisfies, (i) µA×B ((x1 , y1 ) − (x2 , y2 )) ≥ min{µA×B (x1 , y1 ), µA×B (x2 , y2 ), 1−k 2 }, (ii) γA×B ((x1 , y1 ) − (x2 , y2 )) ≤ max{γA×B (x1 , y1 ), γA×B (x2 , y2 ), 1−k 2 }. Example 4.1: Let X1 = {0, a, b} and X2 = {0, a, b, c} are two subtraction algebras with the following Cayley tables −

0

a

b

0

0

0

0

a

a

0

a

b

b

b

0

and

−

0

a

b

c

0

0

0

0

0

a

a

0

a

0

b

b

b

0

0

c

c

b

a

0


Let us define the intuitionistic fuzzy sets A = (µA (x), γA (x)) of X1 and B = (µB (x), γB (x)) of X2 as follows X

0

a

b

µA (x)

0.5

0.6

0.7

γA (x)

0.1

0.2

0.21

and

X

0

a

b

c

µB (x)

0.6

0.5

0.4

0.3

γB (x)

0.2

0.21

0.23

0.24

.

Then X1 × X2 = {(0, 0), (0, a), (0, b), (0, c), (a, 0), (a, a), (a, b), (a, c), (b, 0), (b, a), (b, b), (b, c)} is a subtraction algebra. Now define the direct product A×B on X1 ×X2 as A×B = hµA×B , γA×B i where µA×B : X1 ×X2 → [0, 1] and γA×B : X1 × X2 → [0, 1], X1 × X2

µA×B

γA×B

(0, 0)

0.5

0.2

(0, a)

0.5

0.21

(0, b)

0.4

0.23

(0, c)

0.3

0.24

(a, 0)

0.6

0.2

(a, a)

0.5

0.21

(a, b)

0.4

0.23

(a, c)

0.3

0.24

(b, 0)

0.6

0.21

(b, a)

0.5

0.21

(b, b)

0.4

0.23

(b, c)

0.3

0.24

then A × B is an (∈, ∈ ∨q0.4 ) −intuitionistic fuzzy subalgebra of X1 × X2 . Definition 3.3 An intuitionistic fuzzy set A × B of X1 × X2 is called an (∈, ∈ ∨qk )-intuitionistic fuzzy ideal of X1 × X2 if it satisfies (i) µA×B ((x1 , y1 ) − (x2 , y2 )) ≥ min{µA×B (x1 , y1 ), 1−k 2 }, 6

(ii) γA×B ((x1 , y1 ) − (x2 , y2 )) ≤ max{γA×B (x1 , y1 ), 1−k 2 }, (iii) µA×B ((x1 , y1 ) ∨ (x2 , y2 )) ≥ min{µA×B (x1 , y1 ), µA×B (x2 , y2 ), 1−k 2 }, (iv) γA×B ((x1 , y1 ) ∨ (x2 , y2 )) ≤ max{γA×B (x1 , y1 ), γA×B (x2 , y2 ), 1−k 2 }. Theorem 3.4 Let A and B be two (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 and X2 , respectively. Then the Direct product A × B is an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 × X2 . Proof: Let A and B be two (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebras of X1 and X2 , respectively. For any (x1 , y1 ), (x2 , y2 ) ∈ X1 × X2 . We have µA×B ((x1 , y1 ) − (x2 , y2 ))

=

µA×B (x1 − x2 , y1 − y2 )

=

µA (x1 − x2 ) ∧ µB (y1 − y2 ) 1−k 1−k } ∧ min{µB (y1 ), µB (y2 ), } min{µA (x1 ), µA (x2 ), 2 2 1−k 1−k } ∧ min{µA (x2 ), µB (y2 ), } min{µA (x1 ), µB (y1 ), 2 2 1−k }. {µA×B (x1 , y1 ), µA×B (x2 , y2 ), 2

≥


= = Also γA×B ((x1 , y1 ) − (x2 , y2 ))

= γA×B (x1 − x2 , y1 − y2 ) = γA (x1 − x2 ) ∨ γB (y1 − y2 ) 1−k 1−k } ∨ {γB (y1 ), γB (y2 ), } ≤ max{γA (x1 ), γA (x2 ), 2 2 1−k 1−k } ∨ max{γA (x2 ), γB (y2 ), } = max{γA (x1 ), γB (y1 ), 2 2 1−k = {γA×B (x1 , y1 ), γA×B (x2 , y2 ), }. 2

Hence this shows that A × B is an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 × X2 . ✷ Theorem 3.5 Let A and B be two (∈, ∈ ∨qk )-intuitionistic fuzzy ideals of X1 and X2 , respectively. Then the Direct product A × B is an (∈, ∈ ∨qk )-intuitionistic fuzzy ideal of X1 × X2 . Proof: Straightforward.

✷

Proposition 3.6 Every an (∈, ∈ ∨qk )-intuitionistic fuzzy ideal A × B = (µA×B (x), γA×B (x)) of X1 × X2 satisfies the following, (i) µA×B (0, 0) ≥ min{µA×B (x1 , y1 ), 1−k 2 }, (ii) γA×B (0, 0) ≤ max{γA×B (x1 , y1 ),

1−k 2 }.

Proof: By letting (x1 , y1 ) = (x2 , y2 ) in (i) and (ii) condition in the Definition 3.3, we get the required proof. 7

Lemma 3.7 If an (∈, ∈ ∨qk )-intuitionistic fuzzy set A × B = (µA×B (x), γA×B (x)) of X1 × X2 satisfies the followings, (i) µA×B (0, 0) ≥ min{µA×B (x1 , y1 ), 1−k 2 }, (ii) γA×B (0, 0) ≤ max{γA×B (x1 , y1 ),

1−k 2 },

(iii) µA×B ((x1 , y1 ) − (x3 , y3 )) ≥ min{µA (((x1 , y1 ) − (x2 , y2 )) − (x3 , y3 )), µA (x2 , y2 ), 1−k 2 }, (iv) γA×B ((x1 , y1 ) − (x3 , y3 )) ≤ max{γA×B (((x1 , y1 ) − (x2 , y2 )) − (x3 , y3 )), γA×B (x2 , y2 ), 1−k 2 }, 1−k then we have (x1 , y1 ) ≤ (a, b) ⇒ µA×B (x1 , y1 ) ≥ min{µA×B (a, b) , 1−k 2 } and γA×B (x) ≤ max{γA×B (a, b) , 2 }

for all (a, b) , (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) ∈ X1 × X2 . Proof: Let (a, b) , (x1 , y1 ) ∈∈ X1 × X2 and (x1 , y1 ) ≤ (a, b) . Consider µA×B (x1 , y1 )

= µA×B ((x1 , y1 ) − (0, 0)),


≥ min{µA×B (((x1 , y1 ) − (a, b)) − (0, 0)), µA×B (a, b) , = min{µA×B (0, 0) , µA×B (a, b) , = min{µA×B (a, b) ,

1−k } by (iii), 2

1−k }, 2

1−k } by (i). 2

Also consider γA×B (x1 , y1 ) =

γA×B ((x1 , y1 ) − (0, 0)),

≤

max{γA×B (((x1 , y1 ) − (a, b)) − (0, 0)), γA×B (a, b) ,

=

max{γA×B (0, 0) , γA×B (a, b) ,

=

max{γA×B (a, b) ,

1−k } by (iv), 2

1−k }, 2

1−k } by (ii). 2

Definition 3.8 Let A = (µA (x), γA (x)) and B = (µB (x), γB (x)) are intuitionistic fuzzy sets of X1 and X2 , respectively. Define the intuitionistic level set for the A×B as (A × B)(α,β) = {(x, y) ∈ X1 ×X2 |µA×B (x, y) ≥ 1−k α, γA×B (x, y) ≤ β} where α ∈ (0, 1−k 2 ], β ∈ [ 2 , 1)}.

Theorem 3.9 Let A and B be two (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 and X2 , respectively. Then the Direct product A × B is an (∈, ∈ ∨qk )-intuitionistic fuzzy subalgebra of X1 × X2 if and only if (A × B)(α,β) 6= Φ is an subalgebra of X1 × X2 . Proof: Straightforward.

✷

Theorem 3.10 Let A and B be two (∈, ∈ ∨qk )-intuitionistic fuzzy ideals of X1 and X2 , respectively. Then the Direct product A× B is an (∈, ∈ ∨qk )-intuitionistic fuzzy ideal of X1 × X2 if and only if (A × B)(α,β) 6= Φ is an ideal of X1 × X2 . 8

Proof: Straightforward.

4

✷

Direct Product of (∈, ∈ ∨qk )-intuitionistic Fuzzy Soft subalgebras

In this section, we define Direct product of an (∈, ∈ ∨qk )-intuitionistic fuzzy soft sets and investigate some related properties. e Bi be two (∈, ∈ ∨qk )-intuitionistic fuzzy soft sets of X1 and X2 , reDefinition 4.1 Let hFe, Ai and hG, e Bi is defined spectively. Then the Direct product of (∈, ∈ ∨qk )-intuitionistic fuzzy soft sets hFe , Ai and hG,


as

e Bi = (Ub, A × B), hFe , Ai ⊗ hG, E D e = µe for all [ǫ, ε] ∈ A × B. where Ub[ǫ, ε] = Fe [ǫ] × G[ε] , γ e e[ǫ]×G[ε] e F [ǫ]×G[ε] F

Here µFe[ǫ]×G[ε] e (x, y) = µF e[ǫ] (x) ∧ µG[ε] e (y) and γF e[ǫ]×G[ε] e (x, y) = γF e[ǫ] (x) ∨ γG[ε] e (y) for all (x, y) ∈ X1 × X2 and [ǫ, ε] ∈ A × B. e Bi of X1 × X2 is called an an Definition 4.2 An (∈, ∈ ∨qk )-intuitionistic fuzzy soft set hFe , Ai ⊗ hG,

(∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebra of X1 × X2 if it satisfies

1−k (i) µFe[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 )) ≥ min{µF e[ǫ]×G[ε] e (x1 , y1 ), µF e[ǫ]×G[ε] e (x2 , y2 ), 2 }; 1−k (ii) γFe[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 )) ≤ max{γF e[ǫ]×G[ε] e (x1 , y1 ), γF e[ǫ]×G[ε] e (x2 , y2 ), 2 }.

e Bi of X1 × X2 is called an an Definition 4.3 An (∈, ∈ ∨qk )-intuitionistic fuzzy soft set hFe , Ai ⊗ hG,

(∈, ∈ ∨qk )-intuitionistic fuzzy soft ideal of X1 × X2 if it satisfies 1−k (i) µFe[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 )) ≥ min{µF e[ǫ]×G[ε] e (x1 , y1 ), 2 }, 1−k (ii) γFe[ǫ]×G[ε] e[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 )) ≤ max{γF e (x1 , y1 ), 2 },

1−k (iii) µFe[ǫ]×G[ε] e ((x1 , y1 ) ∨ (x2 , y2 )) ≥ min{µF e[ǫ]×G[ε] e (x1 , y1 ), µF e[ǫ]×G[ε] e (x2 , y2 ), 2 }, 1−k (iv) γFe[ǫ]×G[ε] e ((x1 , y1 ) ∨ (x2 , y2 )) ≤ max{γF e[ǫ]×G[ε] e (x1 , y1 ), γF e[ǫ]×G[ε] e (x2 , y2 ), 2 }.

e Bi be two (∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebras of X1 and X2 , Theorem 4.4 Let hFe, Ai and hG, e Bi is an (∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebra of respectively. Then the Direct product hFe, Ai ⊗ hG, X1 × X2 .

e Bi be two (∈, ∈ ∨qk )-intuitionistic fuzzy soft groups of X1 and X2 , respectively. Proof: Let hFe , Ai and hG, 9

For any (x1 , y1 ), (x2 , y2 ) ∈ X1 × X2 and [ǫ, ε] ∈ A × B. We have µFe[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 ))

=

µFe[ǫ]×G[ε] e (x1 − x2 , y1 − y2 )

=

µFe[ǫ] (x1 − x2 ) ∧ µG[ε] e (y1 − y2 )

≥ = =

1−k 1−k } ∧ min{µG[ε] } e (y1 ), µG[ε] e (y2 ), 2 2 1−k 1−k } ∧ min{µFe[ǫ] (x2 ), µG[ε] } min{µFe[ǫ] (x1 ), µG[ε] e (y2 ), e (y1 ), 2 2 1−k }. {µFe[ǫ]×G[ε] e (x1 , y1 ), µF e[ǫ]×G[ε] e (x2 , y2 ), 2 min{µFe[ǫ] (x1 ), µFe[ǫ] (x2 ),

Also γFe[ǫ]×G[ε] e ((x1 , y1 ) − (x2 , y2 )) = = ≤


= =

γFe[ǫ]×G[ε] e (x1 − x2 , y1 − y2 ) γFe[ǫ] (x1 − x2 ) ∨ γG[ε] e (y1 − y2 ) 1−k 1−k } ∨ {γG[ε] } e (y1 ), γG[ε] e (y2 ), 2 2 1−k 1−k } ∨ max{γFe[ǫ] (x2 ), γG[ε] } max{γFe[ǫ] (x1 ), γG[ε] e (y2 ), e (y1 ), 2 2 1−k }. {γFe[ǫ]×G[ε] e (x1 , y1 ), γF e [ǫ]×G[ε] e (x2 , y2 ), 2 max{γFe[ǫ] (x1 ), γFe[ǫ] (x2 ),

e Bi is an (∈, ∈ ∨qk )-intuitionistic fuzzy soft subalgebra of X1 × X2 . ✷ Hence this shows that hFe, Ai ⊗ hG,

e Bi be two (∈, ∈ ∨qk )-intuitionistic fuzzy soft ideals of X1 and X2 , respecTheorem 4.5 Let hFe , Ai and hG, e Bi is an (∈, ∈ ∨qk )-intuitionistic fuzzy soft ideal of X1 × X2 . tively. Then the Direct product hFe, Ai ⊗ hG, Proof: Straightforward.

✷

Conclusion. In this paper we established some new results related to the direct product of (∈, ∈ ∨qk )intuitionistic fuzzy sets and direct product of (∈, ∈ ∨qk )-intuitionistic fuzzy soft sets of subtraction algebras. We investagated several related properties.

References [1] J.C. Abbot, Sets, lattices and Boolean algebra. Allyn and Bacon, Boston (1969). [2] B. Ahmad and K. Athar, Mappings on fuzzy soft classes, Advances in Fuzzy Systems, (2009) 1-6. [3] M. Akram and N. Yaqoob, Intuitionistic fuzzy soft ordered ternary semigroups, International Journal of Pure and Applied Mathematics, 84(2) (2013), 93-107. [4] M. Aslam, S. Abdullah, B. Davvaz and N. Yaqoob, Rough M-hypersystems and fuzzy M-hypersystems in Γ-semihypergroups, Neural Computing and Applications, 21(1) (2012) 281-287. [5] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and System, 20(1986), 87-96. 10

[6] A. Aygunoglu and H. Aygun, Introduction to fuzzy soft groups, Computers and Mathematics with Applications, 58 (2009) 1279-1286. [7] S.K. Bhakat and P. Das, (∈, ∈ ∨q) fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996) 359-368. [8] Y. Ceven and M.A. Ozturk, Some results on subtraction algebras, Hacettepe Journal of Mathematics and Statistics, 38 (2009) 299-304. [9] M. Gulistan, M. Shahzad and N. Yaqoob, On (∈, ∈ ∨qk )-fuzzy KU-ideals of KU-algebras, Acta Universitatis Apulensis, 39 (2014), 75-83. [10] M. Gulistan, M. Khan, N. Yaqoob, M. Shahzad and U. Ashraf, Direct product of generalized cubic sets in Hv-LA-semigroups, Science International (Lahore), 28(2) (2016) 767-779. [11] Y.B. Jun, K.J. Lee and C.H. Park, Fuzzy soft set theory applied to BCK/BCI-algebras, Computers and Mathematics with Applications, 59 (2010) 3180-3192. [12] Y.B. Jun, H.S. Kim and E.H. Roh, Ideal theory of subtraction algebras, Scientiae Mathematicae Japon-


icae, 61 (2005) 459-464. [13] Y.B. Jun and H.S. Kim, On ideals in subtraction algebras, Scientiae Mathematicae Japonicae, 65 (2007) 129-134. [14] Y.B. Jun, M.S. Kang and C.H. Park, Fuzzy subgroups based on fuzzy points, Communications of the Korean Mathematical Society, 26(3) (2011) 349-371. [15] M. Khan, B. Davvaz, N. Yaqoob, M. Gulistan and M. M. Khalaf, On (∈, ∈ ∨qk )-intuitionistic (fuzzy ideals, fuzzy soft ideals) of subtraction algebras, Songklanakarin Journal of Science and Technology, 37 (4)(2015), 465-475. [16] M. Khan, Y.B. Jun, M. Gulistan and N. Yaqoob, The generalized version of Jun’s cubic sets in semigroups, Journal of Intelligent and Fuzzy Systems, 28 (2015), 947-960. [17] A. Khan, F. Yousafzai, W. Khan and N. Yaqoob, Ordered LA-semigroups in terms of interval valued fuzzy ideals, Journal of Advanced Research in Pure Mathematics, 5(1) (2013) 100-117. [18] K.J. Lee and C.H. Park, Some questions on fuzzifications of ideals in subtraction algebras, Communications of the Korean Mathematical Society, 22 (2007) 359-363. [19] Y. Liu and X. Xin, General fuzzy soft groups and fuzzy normal soft groups, Annals of Fuzzy Mathematics and Informatics, 6(2) (2013) 391-400. [20] P.K. Maji, A.R. Roy and R. Biswas, An application of soft sets in a decision making problem, Computers and Mathematics with Applications, 44 (2002) 1077-1083. [21] P.K. Maji, R. Biswas and A.R. Roy, Soft set theory, Computers and Mathematics with Applications, 45 (2003) 555-562. [22] P.K. Maji, R. Biswas and A.R. Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics, 9 (2001) 589-602. 11

[23] P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics, 9(3) (2001) 677-692. [24] P.K. Maji, A.R. Roy and R. Biswas, On intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics, 12(3) (2004) 669-683. [25] D. Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications, 37 (1999) 19-31. [26] B.M. Schein, Difference semigroups, Communications in Algebra, 20 (1992) 2153-2169. [27] M. Shabir, Y. B. Jun and Y. Nawaz, Semigroups characterized by (∈, ∈ ∨qk )-fuzzy ideals, Computers and Mathematics with Applications, 60 (2010) 1473-1493. [28] C. Yang, Fuzzy soft semigroups and fuzzy soft ideals, Computers and Mathematics with Applications, 61 (2011) 255-261. [29] N. Yaqoob, Interval-valued intuitionistic fuzzy ideals of regular LA-semigroups, Thai Journal of Math-


ematics, 11(3) (2013), 683-695. [30] N. Yaqoob, M. Akram and M. Aslam, Intuitionistic fuzzy soft groups induced by (t,s)-norm, Indian Journal of Science and Technology, 6(4) (2013), 4282-4289. [31] N. Yaqoob, M. Aslam, B. Davvaz and A. Ghareeb, Structures of bipolar fuzzy Γ-hyperideals in Γsemihypergroups, Journal of Intelligent and Fuzzy Systems, 27(6) (2014) 3015-3032. [32] N. Yaqoob, R. Chinram, A. Ghareeb and M. Aslam, Left almost semigroups characterized by their interval valued fuzzy ideals, Afrika Matematika, 24(2) (2013) 231-245. [33] N. Yaqoob, S.M. Mostafa and M.A. Ansari, On cubic KU-ideals of KU-algebras, ISRN Algebra, Article ID 935905 (2013) 10 pages. [34] F. Yousafzai, N. Yaqoob and A. Ghareeb, Left regular AG-groupoids in terms of fuzzy interior ideals, Afrika Matematika, 24(4) (2013) 577-587. [35] F. Yousafzai, N. Yaqoob and K. Hila, On fuzzy (2,2)-regular ordered Γ-AG**-groupoids, UPB Scientific Bulletin, Series A, 74(2) (2012) 87-104. [36] L.A. Zadeh, Fuzzy sets, Informations and Control, 8 (1965) 338-353. [37] B. Zelinka, Subtraction semigroup, Mathematica Bohemica, 120 (1995) 445-447.

12

Public Interest Statement (PIS)

ABOUT THE AUTHORS


1) Dr. Muhammad Gulistan, Works as an assistant professor in the Department of Mathematics, Hazara University Mansehra, Pakistan. He published more than 25 research papers in the field of fuzzy sets, cubic sets and hyper structures. 2) Mr. Shah Nawaz is the M.Phil’s student of Dr. Muhammad Gulistan. He is working in the same filed. 3) Mr. Syed Zaheer Abbas Works as an assistant professor in the Department of Mathematics, Hazara University Mansehra, Pakistan. His field of interest is fuzzy sets and abstract algebras.

PUBLIC INTEREST STATEMENT Real world is featured with complex phenomenon. As uncertainty is inevitably involved in problems arise in various fields of life and classical methods failed to handle these types of problems. Dealing with imprecise, uncertain or imperfect information was a big task for many years. Many models were presented in order to properly incorporate uncertainty into system description; L. A. Zadeh in 1965 introduced the idea of a fuzzy set. Zadeh replaced conventional characteristic function of classical crisp sets which takes on its values in {0,1} by membership function which takes on its values in closed interval [0, 1]. But it seems to be the limited case so this was generalized by K.T. Atanassov in 1986. Soft sets are also considered a very handy tool in order to handle imprecise information.