Soft intersection Γ-nearrings and its applications

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1 615 Soft intersection Γ-nearrings and its applications Saleem Abdullah Muhammad Aslam Muhammad Naeem Kostaq Hila ICM 2012, March, Al Ain Abstract Fuzzy set theory, rough set theoru and soft set theory are all mathematical tools to describe uncertainties, while the soft set theory more adequate than the fuzzy set theory and rough set theory. In this paper, we concentrate on the inclusion relation and intersection with set theory and Γ-nearring. We use the concept of Molodtsov soft set to Γ-nearring and we define soft int Γ-nearring. The structure of soft intersection Γ-nearring is based on the inclusion relation and intersection of sets and since this new concept brings the soft set theory, set theory and Γ-nearring theory together, it is very functional by means of improving the soft set theory with respect to Γ-nearring structure. From this view, it functions as a bridge among soft set theory, set theory and Γ-nearring theory. Based on the soft int Γ-nearring definition, we introduce the concepts of soft intersection Γ-nearring, soft intersection Γ-ideal and soft intersection prime (semi-prime) Γ-ideal. Moreover, investigate these notions with respect to soft image, soft pre-image and α-inclusion of soft sets. Finally, we give some applications of soft intersection Γ-nearring to Γ-nearring theory. Keywords: Γ-nearring, soft set, soft intersection Γ-nearring, soft int Γ-ideal 1 Introduction The notion of nearring was first intersectionroduced by Dickson and Leonard in 1905 [1]. He showed that there do exist fields with one distributive law (near fields). It was Zassenhaus who was able to determine all finite nearrings. Now a day, near fields are mighly tools in characterizing doubly transitive groups, incidence groups and Frobenius groups. We note that the ideals of nearrings play a central role in the structure theory, however, they do not in general coincide with the usual ring ideals of a ring. In 1984, Satyanarayana intersectionroduced Γ-nearring in his doctoral thesis and give some basic results [34]. For further see [35, 36]. To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertaintersectionies presented in these problems. While probability theory, fuzzy set theory [3], rough set theory [4, 5], and other mathematical tools are well known and often useful approaches to describing uncertaintersectiony, each of these theories has its inherent difficulties as pointersectioned out in [7, 6]. In 1999, Molodtsov [6] intersectionroduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertaintersectionies. This so-called soft set theory is free from the difficulties affecting existing methods. Presently, works on soft set theory are progressing rapidly. Maji et al. [8] defined several operations on soft sets and made a theoretical study on the theory of soft sets. Ali et al. [23] introduced several operations of soft sets. Sezgin and Atagün [24] studied on soft set operations. Babitha and Sunil introduced Supported by Higher Education Commission of Pakistan. Supported by Higher Education Commission of Pakistan.

2 616 Soft intersection Γ-nearrings and its applications soft set relations and functions [25]. Majumdar and Samanta, worked on soft mappings [26] were proposed and many related concepts were discussed too. Moreover, the theory of soft sets has gone through remarkably rapid strides with a wide-ranging applications especially in soft decision making as in the following studies: [27, 28] and some other fields such as [30, 31, 32, 33]. Since its inception, it has received much attention in the mean of algebraic structures. In [9], Aktaṣ and C. ağman applied the concept of soft set to groups theory and introduced soft group of a group. Feng et.al, studied soft semirings by using soft set [10]. Recently, Acar studied soft rings [11]. Jun et. al, applied the concept of soft set to BCK/BCI-algebras [12, 13, 14]. Sezgin and Atagün initiated th concept of normalistic soft groups [15]. Zhan et.al, worked on soft ideal of BL-algebras [16]. In [17], Kazancı et. al, used the concept of soft set to BCH-algebras. Sezgin et. al, studied soft nearrings [18]. Atagün and Sezgin [19] defined the concepts of soft subrings and ideals of a ring, soft subfields of a field and soft submodules of a module and studied their related properties with respect to soft set operations also union soft substructues of nearrings and nearring modules are studied in [20]. C. ağman et al. defined two new types of group actions on a soft set, called group SI-action and group SU-action [21], which are based on the inclusion relation and the intersectionersection of sets and union of sets, respectively.. C. ağman and Enginoğlu [28] redefined the operations of soft sets to develop the soft set theory. By using their definitions, in this paper, we define soft intersection Γ-nearring. The structure of soft intersection Γ-nearring is based on the inclusion relation and intersection of sets and since this new concept brings the soft set theory, set theory and Γ-nearring theory together, it is very functional by means of improving the soft set theory with respect to Γ-nearring structure. From this view, it functions as a bridge among soft set theory, set theory and Γ-nearring theory. Based on the soft intersection Γ-nearring definition, we introduce the concepts of soft intersection subγ-nearring, soft intersection Γ-ideal and soft intersection prime (semi-prime) Γ-ideal. We give some interesting results of these notions. Moreover, we investigate these notions with respect to soft image, soft pre-image and α-inclusion of soft sets. Finally, we give some applications of soft intersection Γ-nearring to Γ-nearring theory. 2 Preliminaries We first recall some basic concepts for the sake of completeness from [2]. A nearring N is an algebraic system (N, +, ) consisting of a non-empty set N together with two binary operations addition + and multiplication such that (1) (N, +) is a group and (2) (N, ) is a semigroup and (3) (x + y) z = x z + y z for all x, y, z N. We will use word nearring to mean right distributive nearring. A Γ-nearring [34] is a triple (N, +, Γ), where (i) (N, +) is a group, (ii) Γ is a non-empty set of binary operators on N such that for each α Γ, (N, +, α) is a nearring, (iii) xα(yαz) = (xαy)βz for all x, y, z N and α, β Γ. A subset A of a Γ-nearring N is called a left (resp. right) Γ-ideal of N if (i) (A, +) is a normal subgroup of (N, +), (ii) xα(y + z) xαz A (resp. xαy A) for all x A, α Γ and y, z N. We will use word Γ-nearring to mean right distributive Γ-nearring. A Γ-nearring N is a zerosymmetrc Γ-nearring if N = N 0, where is called the zerosymmetric part of N. N 0 = {n N : nα0 = 0, for all α Γ},

3 S. Abdullah, M. Aslam, M. Naeem and K. Hila 617 We now review some soft set concepts. From now on, X refers to an initial universe, E is a set of parameters, P (X) is the power set of X and A, B, C E. A soft set λ A over X is a set defined by λ A : E P (X) such that λ A (x) = if x / A. Here λ A is also called an approximate function. A soft set over X can be represented by the set of ordered pairs λ A = {(x, λ A (x)) : x E, λ A (x) P (X)} It is clear to see that a soft set is a parametrized family of subsets of the set X. Note that the set of all soft sets over X will be denoted by S(X). Definition 1 [7] Let λ A and λ B be two soft sets over a common universe X. Then, we say that λ A is a soft subset of λ B if 1. A B and 2. λ A (e) λ B (e) for all e A. We write λ A λ B. A soft set λ B is said to be a soft super set of λ A, if λ A is a soft subset of λ B. We denote it by λ B λ A. Definition 2 [7] Let λ A,λ B S(X). Then, union of λ A and λ B, denoted by λ A λ B, is defined as λ A λ B = λ A B, where λ A B (x) = λ A (x) λ B (x) for all x E. Definition 3 [7] Let λ A,λ B S(X). Then, intersectionersection of λ A and λ B, denoted by λ A λ B, is defined as λ A λ B = λ A B, where λ A B (x) = λ A (x) λ B (x) for all x E. Definition 4 [7] Let λ A,λ B S(X). Then, -product of λ A and λ B, denoted by λ A λ B, is defined as λ A λ B = λ A B, where λ A B (x, y) = λ A (x) λ B (y) for all (x, y) E E. Definition 5 [7] A soft set λ A over X is said to be a NULL soft set denoted by Φ if for all e A, λ A (e) = (empty set). Definition 6 [7] A soft set λ A over X is said to be an absolute soft set denoted by λ A if for all e A, λ A (e) = X. Definition 7 [21] Let λ A,λ B S(X). and f be a function from A to B. Then, soft image of λ A under f, denoted by f(λ A ), is a soft set over X by { {λa (y) : y N, f (y) = x}, f (f(λ A )) (x) = 1 (y) otherwise for all y B. And soft pre-image (or soft inverse image) of λ B under f, denoted by f 1 (λ B ), is a soft set over X by ( f 1 (λ B ) ) = λ B (f (a)) for all x A. 3 Soft intersection Γ-nearring In this section, we first define soft intersection Γ-nearring. We then define soft intersection subγ-nearring of soft intersection Γ-nearring, soft intersection-γ-ideal of a Γ-nearring with some examples and study their fundamental properties with respect to soft set operations. In what follows, N will denote right distributive Γ-nearring, unless otherwise specified.

4 618 Soft intersection Γ-nearrings and its applications Definition 8 Let N be a Γ-nearring and λ N be a soft set over X. Then, λ N is said to be soft intersection Γ-nearring over X if it satisfies the following conditions hold: (i) λ N (x + y) λ N (x) λ N (y), (ii) λ N ( x) = λ N (x), (iii) λ N (xαy) λ N (x) λ N (y), for all x, y M and α Γ. Example 1 Let N = {0, a, b, c} and Γ = {α,β} be non-empty sets. The binary operations defined as; + 0 a b c α 0 a b c β 0 a b c 0 0 a b c a a 0 c b a 0 a 0 b a 0 a 0 0 b b c 0 a b b 0 0 b 0 c c b a 0 c 0 b 0 b c b {[ ] x x Clearly (N, +, Γ) is a Γ-nearring. Assume that N is the set of parameters and X = y y 2 2 matrices with Z 4 terms, is the universal set. We construct a soft set λ N over X by {[ ] [ ] [ ] [ ] [ ]} λ N (0) =,,,,, {[ ] [ ] [ ] [ ]} λ N (a) =,,,, {[ ] [ ] [ ] } λ N (b) =,,,, {[ ] [ ] [ ]} λ N (c) =,, Then, one can easily show that the soft set λ N is a soft intersection Γ-nearring over X. Example 2 In Example 1, assume that N = {0, a, b, c} is again the set of parameters and X = S 3, symmetric group, is the universal set. we defined a soft set λ N by λ N (0) = {(12), (13)}, λ N (a) = {(12), (13), (123)}, λ N (b) = {(12), (23) (123)}, λ N (c) = {(12), (13) (123)}. x, y Z 4 }, λ N is a not a soft intersection Γ-nearring, because λ N (a.a) = λ N (0) = {(12), (13)} {(12), (13), (123)}. Example 3 Let N = {0, a, b, c} and Γ = {α,β} be non-empty sets. The binary operations defined as; + 0 a b c α 0 a b c β 0 a b c 0 0 a b c a a 0 c b a 0 a b c a a a a a b b c 0 a b b b c c b a 0 c 0 a b c c a a a c Thus, { (N, +, Γ) is a Γ-nearring. Assume } that N is the set of parameters and X = D 2 = x, y : x 2 = y 2 = (xy) 2 = e, xy = yx = {e, x, y, xy}, dihedral group, be the universal set. We define a soft set λ N over X by λ N (0) = D 2, λ N (a) = {e, x, xy}, λ N (b) = λ N (c) = {e, x}. By routine calculation λ N is a soft intersection Γ-nearring over X.

5 S. Abdullah, M. Aslam, M. Naeem and K. Hila 619 Proposition 1 Let S be a subγ-nearring of Γ-nearring N. Let λ N is a soft set of N over X and defne as: { α1 If x S, λ N (x) =: α 2 otherwse. Then, λ N is a soft intersection Γ-nearring over X, where α 2 α 1, for all α 1, α 2 X. Remark 1 If λ N is a soft intersection Γ-nearring over X, then λ N (0) λ N (x), for all x N. Theorem 1 Let N be a Γ-nearring and λ N a soft set over X. Then, λ N is a soft intersection Γ-nearring if and only if λ N (x y) λ N (x) λ N (y) and λ N (xαy) λ N (x) λ N (y), for all x, y N and α Γ. Proof Assume that λ N is a soft intersection Γ-nearring over X. Then, by Definition of a soft intersection Γ-nearring, we have λ N (x y) λ N (x) λ N ( y) = λ N (x) λ N (y) λ N (x y) λ N (x) λ N (y) and λ N (xαy) λ N (x) λ N (y), for all x, y M and α Γ. Conversely, assume that λ N (x y) λ N (x) λ N (y) and λ N (xαy) λ N (x) λ N (y), for all x, y M and α Γ..If we choose x = 0, then λ N (0 y) = λ N ( y) λ N (0) λ N (y) = λ N (y). Now, λ N (y) = λ N ( ( y)) λ N ( y). Thus, λ N (y) = λ N ( y) for all y M. Also, by assumption, we have λ N (x + y) = λ N (x ( y)) λ N (x) λ N ( y) = λ N (x) λ N (y) λ N (x + y) λ N (x + y). Thus, λ N is a soft intersection Γ-nearring over X. Remark 2 Let λ N be a soft intersection Γ-nearring over X. (i) If λ N (x y) = 0 for any x, y N, then λ N (x) = λ N (y). (ii) If λ N (x y) = λ N (0) for any x, y N, then λ N (x) = λ N (y). It is known that if (N, +, Γ) is a Γ-nearring, then (N, +) is a group but not necessarily abelian. That is, for any x, y N, x + y needs not be equal to y + x. However, we have the following: Theorem 2 Let λ N be a soft intersection Γ-nearring over X and x N. Then, for all y N. λ N (x) = λ N (0) λ N (x + y) = λ N (y + x) Proof Straightforward. Theorem 3 Let N be a Γ-near-field and λ N a soft set over X. If λ N (0) λ N (1 N ) = λ N (x) for all 0 x N, then λ N is a soft intersection Γ-nearring over X.

6 620 Soft intersection Γ-nearrings and its applications Proof Suppose that λ N (0) λ N (1 N ) = λ N (x) for all 0 x N, In oreder to prove that λ N is soft intersection Γ-nearring over X, it is enough to prove that λ N (x y) λ N (x) λ N (y) and λ N (xαy) λ N (x) λ N (y). Let x, y N and α Γ. Then, we have the following cases: Case 1: Suppose that x 0 and y = 0 or x = 0 and y 0. Since M is a Γ-near-field, so it follows that xαy = 0 and λ N (xαy) = λ N (0). Since λ N (0) λ N (x), for all x N, so λ N (xαy) = λ N (0) λ N (x) and λ N (xαy) = λ N (0) λ N (y). This imply λ N (xαy) λ N (x) λ N (y). Case 2: Suppose that x 0 and y 0. It follows that xαy 0. Then, λ N (xαy) = λ N (1 N ) = λ N (x) and λ N (xαy) = λ N (1 N ) = λ N (y), which implies that λ N (xαy) λ N (x) λ N (y). Case 3: Suppose that x = 0 and y = 0, then cleaerly λ N (xαy) λ N (x) λ N (y). Hence, λ N (xαy) λ N (x) λ N (y), for all x, y M and α Γ. Now, let x, y N. Then, x y = 0 or x y 0. If x y = 0, then either x = y = 0 or x 0, y 0 and x = y. But, since λ N (x y) = λ N (0) λ N (x), for all x N, it follows that λ N (x y) = λ N (0) λ N (x) λ N (y). If x y 0, then either x 0 y 0 and x y or x 0 and y = 0 or x = 0 and y 0. Assume that x 0 y 0 and x y. This follows that λ N (x y) = λ N (1 N ) = λ N (x) λ N (x) λ N (y). Now, let x 0 and y = 0. Then,λ N (x y) λ N (x) λ N (y). Finally, let x = 0 and y 0. Then, λ N (x y) λ N (x) λ N (y). Hence, λ N (x y) λ N (x) λ N (y), for all x, y N. Thus, λ N is a soft intersection Γ-nearring over X. Theorem 4 If λ N and λ M are soft intersection Γ-nearrings over X, then λ N λ M is also soft intersection Γ-nearring over X. Proof By Definition, let λ N λ N = λ N M, where λ N M (x, y) = λ N (x) λ M (y) for all (x, y) N M. Since N and M are Γ-nearrings, so is N M. Now, let (x 1, y 1 ), (x 2, y 2 ) N M. Then, λ N M ((x 1, y 2 ) (x 2, y 2 )) = λ N M (x 1 x 2, y 1 y 2 ) = λ N (x 1 x 2 ) λ M (y 1 y 2 ) (λ N (x 1 ) λ N (x 2 )) λ M (y 1 ) λ M (y 2 ) = (λ N (x 1 ) λ M (y 1 )) (λ N (x 2 ) λ M (y 2 )) = λ N M (x 1, y 1 ) λ N M (x 2, y 2 ) λ N M ((x 1, y 2 ) (x 2, y 2 )) λ N M (x 1, y 1 ) λ N M (x 2, y 2 ). Now, let (x 1, y 1 ), (x 2, y 2 ) N M and (α 1,α 2 ) Γ 1 Γ 2. Then, λ N M ((x 1, y 2 ) (α 1,α 2 ) (x 2, y 2 )) = λ N M (x 1 α 1 x 2, y 1 α 2 y 2 ) = λ N (x 1 α 1 x 2 ) λ M (y 1 α 2 y 2 ) (λ N (x 1 ) λ N (x 2 )) λ M (y 1 ) λ M (y 2 ) = (λ N (x 1 ) λ M (y 1 )) (λ N (x 2 ) λ M (y 2 )) = λ N M (x 1, y 1 ) λ N M (x 2, y 2 ) λ N M ((x 1, y 2 ) (x 2, y 2 )) λ N M (x 1, y 1 ) λ N M (x 2, y 2 ). Hence, λ N λ M is a soft intersection Γ-nearring over X.

7 S. Abdullah, M. Aslam, M. Naeem and K. Hila 621 Definition 9 Let λ N and µ M be soft intersection Γ-nearrings over X 1 and X 2, respectively. Then, the direct product of soft intersection Γ-nearrings λ N and µ M over X 1 and X 2, respectively, is denoted by λ N µ M = h N M and defined as: h N M (x, y) = λ N (x) µ M (y) for all (x, y) N M. Theorem 5 If λ N and µ M are soft intersection Γ-nearrings over X 1 and X 2, then h N M = λ N λ M is also soft intersection Γ-nearring over X 1 X 2. Proof Let (x 1, y 1 ), (x 2, y 2 ) N M. Then, h N M ((x 1, y 2 ) (x 2, y 2 )) = h N M (x 1 x 2, y 1 y 2 ) = λ N (x 1 x 2 ) µ M (y 1 y 2 ) (λ N (x 1 ) λ N (x 2 )) µ M (y 1 ) µ M (y 2 ) = (λ N (x 1 ) µ M (y 1 )) (λ N (x 2 ) µ M (y 2 )) = h N M (x 1, y 1 ) h N M (x 2, y 2 ) h N M ((x 1, y 2 ) (x 2, y 2 )) h N M (x 1, y 1 ) h N M (x 2, y 2 ). Now, let (x 1, y 1 ), (x 2, y 2 ) N M and (α 1,α 2 ) Γ 1 Γ 2. Then, h N M ((x 1, y 2 ) (α 1,α 2 ) (x 2, y 2 )) = h N M (x 1 α 1 x 2, y 1 α 2 y 2 ) = λ N (x 1 α 1 x 2 ) µ M (y 1 α 2 y 2 ) (λ N (x 1 ) λ N (x 2 )) (µ M (y 1 ) µ M (y 2 )) = (λ N (x 1 ) µ M (y 1 )) (λ N (x 2 ) µ M (y 2 )) = h N M (x 1, y 1 ) h N M (x 2, y 2 ) h N M ((x 1, y 2 ) (x 2, y 2 )) h N M (x 1, y 1 ) h N M (x 2, y 2 ). Hence, λ N λ M is a soft intersection Γ-nearring over X 1 X 2. Theorem 6 If λ M and µ N are soft intersection Γ-nearrings over X, then λ M µ M is also soft intersection Γ-nearring over X. Proof Now, let x, y N. Then, Now, let x, y N and α Γ. Then, λ N λ N (x y) = λ N (x y) µ N (x y) (λ N (x) λ N (y)) (µ N (x) µ N (y)) = (λ N (x) µ N (x)) (λ N (y) µ N (y)) = λ N λ N (x) λ N λ N (y) λ N λ M (x y) λ N λ N (x 1, y 1 ) λ N λ N (x 2, y 2 ). λ N λ M (xαy) = λ N (xαy) µ N (xαy) (λ N (x) λ N (y)) (µ N (x) µ N (y)) = (λ N (x) µ N (x)) (λ N (y) µ N (y)) = λ N λ N (x) λ N λ N (y) λ N λ N (xαy) λ N λ N (x 1, y 1 ) λ N λ N (x 2, y 2 ). Hence, λ N λ M is a soft intersection Γ-nearring over X.

8 622 Soft intersection Γ-nearrings and its applications Definition 10 Let S be a subγ-nearring of Γ-nearring N, λ N a soft intersection Γ-nearring over X and let λ S be a soft subset of λ N over X. Then, λ S is called soft intersection subγ-nearring of λ N over X, if λ S is itself a soft intersection Γ-nearring over X and denoted by λ S i λ N. Example 4 In Example 1, assume that N = {0, } a, b, c} is again the set of parameters and X = D 3 = { x, y : x 3 = y 2 = (xy) 2 = e, xy = yx 2 = { e, x, x 2, y, yx, yx 2}, dihedral group, the universal set. We define a soft set λ N over X by λ N (0) = D 3, λ N (a) = { e, x, x 2, y, yx }, λ N (b) = { e, x, x 2, y }, λ N (c) = { e, x, x 2, y }. Then, λ N is a soft intersection Γ-nearring over X. Now, let S = {0, a} be a subγ-nearring of N, the set of parameters and we defined a soft subset λ S of λ N over X by λ N (0) = { e, x, x 2, y }, λ N (a) = { e, x, x 2}. It is clear that λ N is a soft intersection subγ-nearring of soft intersection Γ-nearring λ N over X. Theorem 7 If λ N is a soft intersection Γ-nearring over X, λ M i λ N and λ K i λ N over X, then (i) λ M λ K i λ N over X. Proof Straightforward. Definition 11 Let N be a Γ-nearring and λ N a soft intersection Γ-nearring over X. Then, λ N is said to be a soft intersection-γ-ideal of N over X, if the following conditions hold: (i) λ N (x + y x) λ N (x) λ N (y), (ii) λ N (xαy) λ N (x), (iii) λ N (xα (y + z) xαy) λ N (z), for all x, y, z N and α Γ. If λ N is a soft intersection Γ-nearring over X and the conditions (i) and (ii) hold, then λ N is called a soft intersection right Γ-ideal of N over X and if conditions (i) and (iii) hold, then λ N is called a soft intersection left Γ-ideal of N over X. Example 5 Let N = {0, a, b, c} and Γ = {α,β} be non-empty sets. The binary operations defined as; + 0 a b c α 0 a b c β 0 a b c 0 0 a b c a a 0 c b a 0 a a a a 0 a 0 a b b c 0 a b 0 a b c b 0 b 0 c c c b a 0 c 0 0 c b c b Clearly, (N, +, Γ) is a Γ-nearring. Assume that N is the set of parameters and X = D 3 = { x, y : x 3 = y 2 = (xy) 2 = e, xy = yx 2 } = { e, x, x 2, y, yx, yx 2}, dihedral group, the universal set. We define a soft set λ N over X by λ N (0) = λ N (b) = D 3, λ N (c) = λ N (a) = {e, x, }. Then, clearly λ N is a soft intersection left Γ-ideal and right Γ-ideal of N over X. Therefore, λ N is a soft intersection Γ-ideal of N overs X.

9 S. Abdullah, M. Aslam, M. Naeem and K. Hila 623 Theorem 8 Let I be a non-empty subset of a Γ-nearring N. Then, S I is a soft intersection Γ-ideal of N over X if and only if I is a Γ-ideal of N. Proof The proof follows from 14. Theorem 9 Let N be a Γ-near-field and λ N a soft intersection Γ-ideal of N overs X. Then, λ N (0) λ N (1 N ) = λ N (x) for all 0 x N. Proof Suppose that λ N is a soft intersection Γ-ideal of N overs X, then λ N is a soft intersection Γ-nearring of N over X. Since λ N (0) λ N (x), so in particular λ N (0) λ N (1 N ). Now, let 0 x N. Then, λ N (x) = λ N (1 N αx) λ N (1 N ) = λ N ( xαx 1 ) λ N (x) Imply that λ N (x) = λ N (1 N ) for all 0 x N. For a nearring N, the zero-symmetric part of N denoted by N 0 is defined by N 0 = {N N N α0 = 0}. It is known that if N is a zero-symmetric Γ-nearring and I l N, then N ΓI N. Here, we have an analog for this case: Theorem 10 Let N = N 0 and λ N be a soft of N over X. Then, λ N (xα (y + z) xαy) λ N (z) implies that λ N (xαz) λ N (z) for all x, y, z N. Proof Let N be a zero-symmetric nearring, λ N be a soft soft over X and λ N (xα (y + z) xαy) λ N (z). Let y = 0 in λ N (xα (y + z) xαy) λ N (z) implies that λ N (xα (0 + z) xα0) λ N (z) implies that λ N (xαz) λ N (z). Theorem 11 If λ N and µ M are soft intersection Γ-ideals over X, then λ N µ M is also soft intersection Γ-ideal over X. Proof Let λ N and µ M are soft intersection Γ-ideals over X. Then, by Theorem 8, λ N µ M soft intersection Γ-nearrings over X. Let (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) N M and (γ 1, γ 2 ) Γ 1 Γ 2. Then, (λ N µ M ) ((x 1, y 1 ) + (x 2, y 2 ) (x 1, y 1 )) = (λ N µ M ) (x 1 + x 2 x 1, y 1 + y 2 y 1 ) = λ N (x 1 + x 2 x 1 ) µ M (y 1 + y 2 y 1 ) λ N (x 2 ) µ M (y 2 ) = (λ N µ M ) (x 2, y 2 ), (λ N µ M ) ((x 1, y 1 )(x 2, y 2 )) = (λ N µ M ) (x 1 x 2, y 1 y 2 ) = λ N (x 1 x 2 ) µ M (y 1 y 2 ) λ N (x 1 ) µ M (y 1 ) = (λ N µ M ) (x 1, y 1 ) and (λ N µ M ) ((x 1, y 1 ) (γ 1, γ 2 ) ((x 2, y 2 ) + (x 3, y 3 )) (x 1, y 1 ) (γ 1, γ 2 ) (x 2, y 2 )) = (λ N µ M ) (x 1 γ 1 (x 2 + x 3 ) x 1 γ 1 x 2, y 1 γ 2 (y 2 + y 3 ) y 1 γ 2 y 2 ) = λ N (x 1 γ 1 (x 2 + x 3 ) x 1 γ 1 x 2 ) µ M (y 1 γ 2 (y 2 + y 3 ) y 1 γ 2 y 2 ) λ N (x 3 ) µ M (y 3 ) = (λ N µ M ) (x 3, y 3 ) Hence, λ N µ M is a soft intersection Γ-ideal of N M over X.

10 624 Soft intersection Γ-nearrings and its applications Theorem 12 If λ N and µ M are soft intersection Γ-ideals over X 1 and X 2, then h N M = λ N λ M is also soft intersection Γ-nearring over X 1 X 2. Proof In Theorem 5, it is proved that if λ N and µ M are soft intersection Γ-nearrings over X 1 and X 2, then also h N M = λ N λ M. Let (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) N M and (γ 1, γ 2 ) Γ 1 Γ 2. Then, (λ N µ M ) ((x 1, y 1 ) + (x 2, y 2 ) (x 1, y 1 )) = (λ N µ M ) (x 1 + x 2 x 1, y 1 + y 2 y 1 ) = λ N (x 1 + x 2 x 1 ) µ M (y 1 + y 2 y 1 ) λ N (x 2 ) µ M (y 2 ) = (λ N µ M ) (x 2, y 2 ), (λ N µ M ) ((x 1, y 1 )(x 2, y 2 )) = (λ N µ M ) (x 1 x 2, y 1 y 2 ) = λ N (x 1 x 2 ) µ M (y 1 y 2 ) λ N (x 1 ) µ M (y 1 ) = (λ N µ M ) (x 1, y 1 ) and (λ N µ M ) ((x 1, y 1 ) (γ 1, γ 2 ) ((x 2, y 2 ) + (x 3, y 3 )) (x 1, y 1 ) (γ 1, γ 2 ) (x 2, y 2 )) = (λ N µ M ) (x 1 γ 1 (x 2 + x 3 ) x 1 γ 1 x 2, y 1 γ 2 (y 2 + y 3 ) y 1 γ 2 y 2 ) = λ N (x 1 γ 1 (x 2 + x 3 ) x 1 γ 1 x 2 ) µ M (y 1 γ 2 (y 2 + y 3 ) y 1 γ 2 y 2 ) λ N (x 3 ) µ M (y 3 ) = (λ N µ M ) (x 3, y 3 ) Hence, λ N µ M is a soft intersection Γ-ideal of N M over X 1 X 2. Theorem 13 If λ M and µ N are soft intersection Γ-nearrings over X, then λ M µ N is also soft intersection Γ-nearring over X. 4 Soft union-intersection product and soft characteristic function Definition 12 Let λ N and µ N be two soft sets over X. Then, λ N µ N, λ N Γ µ N are soft sets over X define as: { {λ N (y) µ N (z)}, if x = y + z, for some y, z N, (λ N µ N ) (x) = x=yz otherwise and and (λ N Γ µ N ) (x) = (λ N µ N ) (x) = { x=yαz for all x, y, z, n N and α Γ. {λ N (y) µ N (z)}, if x = yαz, for y, z N and α Γ, x=yα(z+n) yαz {λ N (y) µ N (n)}, if x = yα (z + n) yαz, otherwise otherwise,

11 S. Abdullah, M. Aslam, M. Naeem and K. Hila 625 Definition 13 Let A be a subset of Γ-nearring N. We denote by S A the soft characteristic function of A and define as: { X if x A, S A =: if x / A. Its clearly that soft characteristic function is a soft set over X, that is, S A : N P (X). Proposition 2 Let A and B be non-empty subsets of a Γ-nearring N. Then, we have (1) If A B, then S A S B, (2) If S A S B = S A B and S A S B = S A B, (3) If S A Γ S B = S AΓB and S A S B = S A B. Proof (1) and (2) are straightforward. (3) Let x N. Then, we have two cases i) If x = yαz, ii) If x yαz. i) If x AΓB, then x = yαz for y A, z B and α Γ. Thus, we have (S A Γ S B ) (x) = {S A (y) S B (z)} X X = X x=yαz (S A Γ S B ) (x) = X Because (S A Γ S B ) (x) X. and S AΓB (x) = X, because x AΓB. This aimplies S A Γ S B = S AΓB. If x / AΓB, then x yαz for y / A, z / B. If x = N αt for some N, t N, then we have (S A Γ S B ) (x) = {S A (N ) S B (t)} = x=n αt (S A Γ S B ) (x) = = (S A Γ S B ) (x) Because (S A Γ S B ) (x) X. i) x yαz, then clearly (S A Γ S B ) (x) = = (S A Γ S B ) (x). This completes the proof. Theorem 14 Let A be a non-empty subset of a Γ-nearring N. Then, S A is a soft subγ-nearring of N over X if and only if A is a subγ-nearring of N. Proof Assume that A is a subγ-nearring of a Γ-nearring N. Since { X If x A, S A (x) =; If x / A. So, S A is the soft set over X. Let x, y N. Then, we have two cases i) If x, y A; ii) If x, y / A. iii) If x A and y / A or x A and y / A. i) If x, y A, then x + y A. So, in this case ii) If x, y / A, then S A (x + y) = S A (x) S A (y). S A (x + y) = S A (x) S A (y). iii) If x A and y / A or x A and y / A, then S A (x + y) = S A (x) S A (y). Thus, in all cases we have S A (x + y) = S A (x) S A (y).

12 626 Soft intersection Γ-nearrings and its applications Now, let x N. Then, x A or x / A. For both cases we have S A (x) = S A ( x). Let x, y N and α Γ. Then, we have three cases: i) If x, y A, ii) If x, y / A, iii) If x A and y / A or x A and y / A. i) If x, y A, then xαy A for all α Γ. So, in this case S A (xαy) = S A (x) S A (y). ii) If x, y / A, then S A (xαy) = S A (x) S A (y). iii) If x A and y / A or x A and y / A, then S A (xαy) = S A (x) S A (y). Thus, in all cases we have S A (xαy) S A (x) S A (y). Conversely, assume that S A is the soft intersection Γ-nearring over X. Since { X If x A, S A (x) =; If x / A. So, A is a non-empty subset of N. Let x, y A. Then, S A (x + y) S A (x) S A (y) = X, whch implies that S A (x + y) = X, this imply x + y A. If x A. Then, X = S A (x) = S A ( x), this imply x A. Thus, A is a subgroup of N. Let x, y A and α Γ. Then, S A (xαy) S A (x) S A (y) = X, whch implies that S A (xαy) = X, this imply xαy A. Hence, A is a subγ-nearring of N. Theorem 15 Let λ N and µ N be soft intersection Γ-ideals over X. Then, λ N µ N is the smallest soft intersection Γ-ideal over X containing both λ N and µ N.

13 S. Abdullah, M. Aslam, M. Naeem and K. Hila 627 Proof Take any x, y N and α Γ. Then, (λ N µ N ) (x) (λ N µ N ) (y) = Now, let x, y N and α Γ. Then, = = [ x=p+q [ y=s+t x=p+q y=s+t x=p+q y=s+t {λ N (p) µ N (q)} ] {λ N (s) µ N (t)} {{λ N (p) µ N (q)} {λ N (s) µ N (t)}} {λ N (p) λ N (s) µ N (q) µ N (t)} As x y = p + q (s + t) = p + q s t = p + (q s q) + q + (s t s) and λ N (q s q) = λ N ( s) = λ N (s). x=p+q y=s+t x=p+q y=s+t x y=e+f { {λn (p) λ N (q s q)} {µ N (q) µ N (s t s)} { λn (p) λ N (q s q) µ N (q) µ N (s t s) ] } } {λ N (e) µ N (f)} = (λ N µ N ) (x y) (λ N µ N ) (x y) (λ N µ N ) (x) (λ N µ N ) (y). (λ N µ N ) (x) = {λ N (n 1 ) µ N (n 2 )} x=n 1 +n 2 {λ N (n 1 y) µ N (n 2 y)} x=n 1 +n 2 {λ N (m 1 ) µ N (m 1 )} xαy=m 1 +m 2 = (λ N µ N ) (xαy). Thus, (λ N µ N ) (xαy) (λ N µ N ) (x). Now (λ N µ N ) (x) = = = {λ N (n 1 ) µ N (n 2 )} x=n 1 +n 2 {λ N (y + n 1 y) µ N (y + n 2 y)} x=n 1 +n 2 {λ N (m 1 ) µ N (m 1 )}. y+x y=m 1 +m 2

14 628 Soft intersection Γ-nearrings and its applications Because for each x = n 1 + n 2 we have y + x y = y + n 1 + n 2 y = (y + n 1 y) + (y + n 2 y) and for each y + x y = m 1 + m 2 we have x = y + m 1 + m 2 + y = ( y + m 1 + y) + ( y + m 1 + y). = In order to show that y+x y=m 1 +m 2 {λ N (m 1 ) µ N (m 1 )} = (λ N µ N ) (y + x y). (λ N µ N ) (xα (m + n) xαm) (λ N µ N ) (n). We first show that, for each expression λ N (n 1 ) µ N (n 2 ) present in the following z=n 1 +n 2 {λ N (n 1 ) µ N (n 2 )}, the λ N (n 2 ) µ N (n 1 ) is also present in it. As z = n 1 + n 2 = n 2 n 2 + n 1 + n 2, so λ N (n 2 ) µ N ( n 2 + n 1 + n 2 ) is present. As µ N ( n 2 + n 1 + n 2 ) = µ N (n 1 ), so present. Now Now Thus, (λ N µ N ) (N ) = = {λ N (n 1 ) µ N (n 2 )} N =n 1 +n 2 {λ N (n 2 ) µ N (n 1 )} N =n 1 +n 2 xα (m + n) xαm = xα (m + (n 1 + n 2 )) xαm = xα ((m + n 1 ) + n 2 ) xα (m + n 1 ) + xα (m + n 1 ) xαm and λ N (n 2 ) λ N (xα ((m + n 1 ) + n 2 ) xα (m + n 1 )) µ N (n 2 ) µ N (xα (m + n 1 ) xαm) (λ N µ N ) (n) {λ N (xα ((m + n 1 ) + n 2 ) xα (n + n 1 )) µ N (xα (m + n 1 ) xαm)} n=n 1 +n 2 {λ N (m 1 ) µ N (m 1 )} = (λ N µ N ) (xα (m + n 1 ) xαm). xα(m+n) xαm=m 1 +m 2 Hence, λ N µ N is a soft intersection-γ-ideal over X. Now, (λ N µ N ) (n) = {λ N (n 1 ) µ N (n 2 )}. n=n 1 +n 2 Snce n = n + 0 and n = 0 + n, so (λ N µ N ) (n) λ N (n) and (λ N µ N ) (n) µ N (n).

15 S. Abdullah, M. Aslam, M. Naeem and K. Hila 629 If ν N is a soft intersection-γ-ideal over X such that ν N (n) λ N (n) and ν N (n) µ N (n) for all n N. Then, (λ N µ N ) (n) = {λ N (n 1 ) µ N (n 2 )} n=n 1 +n 2 {ν N (n 1 ) ν N (n 2 )} n=n 1 +n 2 = {ν N (n 1 ) ν N ( n 2 )} n=n 1 +n 2 {ν N (n 1 + n 2 )} = ν N (N ) n=n 1 +n 2 (λ N µ N ) (n) ν N (n). Ths completes the proof. 5 Soft Prime intersection-γ-ideals Definition 14 A soft intersection-γ-ideal λ N over X is a soft prime intersection-γ-ideals over X if for any soft intersection-γ-ideals µ N, δ N over X such that µ N Γ δ N λ N implies µ N λ N or δ N λ N. Equivalently, for all n 1, n 2 N and α Γ, the following holds λ N (n 1 ) λ N (n 2 ) λ N (n 1 αn 2 ). A soft intersection-γ-ideal λ N over X is a soft semiprime intersection-γ-ideals over X if for any soft intersection-γ-ideals µ N over X such that µ N Γ δ N λ N implies µ N λ N Theorem 16 Let P be a non-empty subset of a Γ-nearring N. Then, S P is a soft intersection prime Γ-ideal of N over X if and only if P is a prime Γ-ideal of N. Proof Assume that P is prime Γ-ideal N. Then, by Theorem 8, S P is the soft intersection Γ-ideal of N over X. Let n 1, n 2 N and α Γ. Then, we have either n 1 αn 2 P or n 1 αn 2 / P. If n 1 αn 2 P, then n 1 P or n 2 P. So, S P (n 1 αn 2 ) = X and S P (n 1 ) = X or S P (n 2 ) = X this imply S P (n 1 αn 2 ) = X = S P (n 2 ) S P (n 1 ). If n 1 αn 2 / P, then S P (n 1 αn 2 ) = S P (n 2 ) S P (n 1 ). Thus, for all n 1, n 2 N and α Γ, S P (n 2 ) S P (n 1 ) S P (n 1 αn 2 ). Conversley, suppose that S P is a soft intersection prime Γ-ideal of N over X. Then, by Theorem 8, is a Γ-ideal of N. Let n 1, n 2 N and α Γ such that n 1 αn 2 P. Then, S P (n 1 αn 2 ) = X, This imply S P (n 2 ) S P (n 1 ) S P (n 1 αn 2 ) = X, that is, S P (n 2 ) S P (n 1 ) = X, this imply n 1 P or n 2 P. Hence, P is prime Γ-ideal of N. 6 Applications of soft intersection Γ-nearring and soft intersection- Γ-ideals In this section, we give the applications of soft image, soft pre-image and upper-inclusion of sets to Γ-nearring theory with respect to soft intersection nearrings and soft intersection-ideals of a nearring.

16 630 Soft intersection Γ-nearrings and its applications Theorem 17 If λ N is a soft intersection-γ-ideal of Γ-nearring N over X, then N λ = {x N : λ N (x) = λ N (0)} is a Γ-ideal of N over X. Proof It is obvious that 0 N λ N. We need to prove that (i) x y N λ (ii) n+x n N λ, (iii) xαn N λ and nα (i + x) nαi N λ for all x, y N λ, N, i N and α Γ. If x, y N λ, then λ N (x) = λ N (y) = λ N (0). So, by Remark 1, it follows that f N (0) λ N (x y), f N (0) λ N (n + x n), f N (0) λ N (xαn) and f N (0) λ N (nα (i + x) nαi) for all x, y N λ, n, i N and α Γ. Since λ N is a soft intersection-γ-ideal of Γ-nearring N over X, so (i) λ N (x y) λ N (x) λ N (y) = λ N (0), (ii) λ N (n + x n) λ N (x) = λ N (0), (iii) λ N (xαn) λ N (x) = λ N (0) and (ii) λ N (nα (i + x) nαi) λ N (x) = λ N (0). This implies that (i) λ N (x y) = λ N (0), (ii) λ N (n + x n) = λ N (0), (iii) λ N (xαn) = λ N (0) and (ii) λ N (nα (i + x) nαi) = λ N (0) for all x, y N λ, n, i N and α Γ. Thus, N λ is a Γ-ideal of N over X. Theorem 18 If λ N is a soft intersection prime (semi-prime) Γ-ideal of Γ-nearring N over X, then N λ = {x N : λ N (x) = λ N (0)} is a prime (semi-prime) Γ-ideal of N over X. Theorem 19 Let λ N be a soft set over X and β be a subset of X such that β λ N (0). If λ N is a soft intersection-γ-ideal of Γ-nearring N over X, then λ β N = {x N : λ N (x) β} is a Γ-ideal of N over X. Proof Since λ N (0) β, so 0 λ β N and λ β N N. Take x, y λ β N, n, i N and α Γ, which implies that λ N (x) β and λ N (y) β. Now, we need to prove that x y λ β N (ii) n + x n λ β N, (iii) xαn λ β N and nα (i + x) nαi λ β N for all x, y λ β N, n, i N and α Γ. Since λ N is a soft intersection-γ-ideal of Γ-nearring N over X, so it follows that (i) λ N (x y) λ N (x) λ N (y) β β = β, (ii) λ N (n + x n) λ N (x) β, (iii) λ N (xαn) λ N (x) β and (iv) λ N (nα (i + x) nαi) λ N (x) β. Thus, this completes the proof. Theorem 20 Let λ N be a soft set over X and β be a subset of X such that β λ N (0). If λ N is a soft intersection-prime (semi-prime) Γ-ideal of Γ-nearring N over X, then λ β N = {x N : λ N (x) β} is a Γ-ideal of N over X. Proof The proof follows from 19. Theorem 21 Let λ N and λ M be soft sets over X and f be a Γ-nearring isomorphism from Γ-nearring N to Γ-nearring M. (i) If λ N is a soft intersection-γ-ideal of Γ-nearring N over X, then f (λ N ) is a soft intersection- Γ-ideal of Γ-nearring M over X (ii) If λ M is a soft intersection-γ-ideal of Γ-nearring M over X, then f 1 (λ M ) is a soft intersection-γ-ideal of Γ-nearring N over X.

17 S. Abdullah, M. Aslam, M. Naeem and K. Hila 631 Proof (i)let x 1, x 2 M. Since f is surjective, so there exist y 1, y 2 N such that f (y 1 ) = x 1, f (y 2 ) = x 2 and f (y 3 ) = x 3. We have f (λ N ) (x 1 x 2 ) = {λ N (y) : y N, f (y) = x 1 x 2 } = { λ N (y) : y N, y = f 1 (x 1 x 2 ) } = { λ N (y) : y N, y = f 1 (f (y 1 y 2 )) = y 1 y 2 } = {λ N (y 1 y 2 ) : y i N, f (y i ) = x i, i = 1, 2} {λ N (y 1 ) λ N (y 2 ) : y i N, f (y i ) = x i, i = 1, 2} = {λ N (y 1 ) : y 1 N, f (y 1 ) = x 1 } {λ N (y 2 ) : y 2 N, f (y 2 ) = x 2 } = f (λ N ) (x 1 ) f (λ N ) (x 2 ). Thus, f (λ N ) (x 1 x 2 ) f (λ N ) (x 1 ) f (λ N ) (x 2 ). Similarly, we can prove that f (λ N ) (x 1 α 1 x 2 ) f (λ N ) (x 1 ) f (λ N ) (x 2 ). Now, we prove that f (λ N ) (x 1 + x 2 x 1 ) = {λ N (y) : y N, f (y) = x 1 + x 2 x 1 } = { λ N (y) : y N, y = f 1 (x 1 + x 2 x 1 ) } = { λ N (y) : y N, y = f 1 (f (y 1 + y 2 y 1 )) = y 1 + y 2 y 1 } = {λ N (y 1 y 2 ) : y i N, f (y i ) = x i, i = 1, 2} {λ N (y 2 ) : y 2 N, f (y 2 ) = x 2, } = f (λ N ) (x 2 ). = f (λ N ) (x 2 ). Thus, f (λ N ) (x 1 + x 2 x 1 ) f (λ N ) (x 2 ). Now, let x 1, x 2 M, y 1, y 2 N, α Γ and α 1 Γ 1. Then, we have f (λ N ) (x 1 α 1 x 2 ) = {λ N (y) : y N, f (y) = x 1 α 1 x 2 } = { λ N (y) : y N, y = f 1 (x 1 α 1 x 2 ) } = { λ N (y) : y N, y = f 1 (f (y 1 αy 2 )) = y 1 αy 2 } = {λ N (y 1 αy 2 ) : y i N, f (y i ) = x i, i = 1, 2} {λ N (y 2 ) : y 2 N, f (y 2 ) = x 2, } = f (λ N ) (x 1 ) f (λ N ) (x 2 ). Now, let x 1, x 2, x 3 M, y 1, y 2, y 3 N, α Γ and α 1 Γ 1. Then, we have f (λ N ) (x 1 α 1 (x 2 + x 3 ) x 1 α 1 x 2 ) = {λ N (y) : y N, f (y) = x 1 α 1 (x 2 + x 3 ) x 1 α 1 x 2 } = { λ N (y) : y N, y = f 1 (x 1 α 1 (x 2 + x 3 ) x 1 α 1 x 2 ) } { λn (y) : y N, y = f = 1 } (f (y 1 α (y 2 + y 3 ) y 1 αy 2 )) = y 1 α (y 2 + y 3 ) y 1 αy 2 = {λ N (y 1 α (y 2 + y 3 ) y 1 αy 2 ) : y i N, f (y i ) = x i, i = 1, 2, 3} {λ N (y 3 ) : y 3 N, f (y 3 ) = x 3 } = f (λ N ) (x 3 ).

18 632 Soft intersection Γ-nearrings and its applications Hence, f (λ N ) is a a soft intersection-γ-ideal of Γ-nearring M over X. (ii) Let x 1, x 2, x 3 N and α 1 Γ and β 1 Γ 1. Then, ( f 1 (λ M ) ) (x 1 α 1 x 2 ) = λ M (f (x 1 α 1 x 2 )) = λ M (f (x 1 ) β 1 f (x 2 )) λ M (f (x 1 )) λ M (f (x 2 )) = ( f 1 (λ M ) ) (x 1 ) ( f 1 (λ M ) ) (x 2 ) Similarly, ( f 1 (λ M ) ) (x 1 x 2 ) ( f 1 (λ M ) ) (x 1 ) ( f 1 (λ M ) ) (x 2 ). Also, ( f 1 (λ M ) ) (x 1 + x 2 x 1 ) = λ M (f (x 1 + x 2 x 1 )) = λ M (f (x 1 ) + f (x 2 ) f (x 1 )) λ M (f (x 2 )) = ( f 1 (λ M ) ) (x 2 ) ( f 1 (λ M ) ) (x 1 + x 2 x 1 ) ( f 1 (λ M ) ) (x 2 ). Now, for x 1, x 2 N and α 1 Γ and β 1 Γ 1. ( f 1 (λ M ) ) (x 1 α 1 x 2 ) = λ M (f (x 1 α 1 x 2 )) = λ M (f (x 1 ) β 1 f (x 2 )) λ M (f (x 1 )) = ( f 1 (λ M ) ) (x 1 ) ( f 1 (λ M ) ) (x 1 α 1 x 2 ) ( f 1 (λ M ) ) (x 1 ). Finally, let x 1, x 2, x 3 N and α 1 Γ and β 1 Γ 1. Then, ( f 1 (λ M ) ) (x 1 α 1 (x 2 + x 3 ) x 1 α 1 x 2 ) = (λ M ) (f (x 1 α 1 (x 2 + x 3 ) x 1 α 1 x 2 )) = (λ M ) (f (x 1 ) β 1 f (x 2 ) + f (x 3 ) f (x 1 ) β 1 f (x 2 )) λ M (f (x 3 )) = ( f 1 (λ M ) ) (x 3 ). Hence, f 1 (λ M ) is a soft intersection-γ-ideal of Γ-nearring N over X. 7 Conclusion Fuzzy set theory, rough set theory and soft set theory are all mathematical tools for dealing with uncertainties. This paper is devoted to discussion of combination of soft set theory, set theory and Γ-nearring. By using soft sets and intersection operation of sets, we have defined a new concept, called soft intersection Γ-nearring. This new notion brings the soft set theory, set theory and Γ-nearring theory together and therefore is very functional for obtaining results by means of Γ-nearring structure. Based on the definition, we have introduced the notions of soft intersection subγ-nearrings, soft intersection-γ-ideals and soft intersection-prime (semi-prime) Γ-ideals of a Γ-nearring with illustrative examples. We have then investigated these notions with respect to soft image, soft pre-image and α-inclusion of soft sets. Finally, we give some applications of soft intersection Γ-nearrings to Γ-nearring theory. In further research, we wll focus on the following topics, 2) We will define soft intersectionbi-γ-ideals and soft intersection-quasi-γ-ideals of a Γ-nearring and its applications. 2) We will characterize Γ-nearrings by the properties of soft intersection Γ-ideals 3)We will define fuzzy soft intersection Γ-nearring, fuzzy soft intersection-γ-ideals of a Γ- nearring, 4) We extend this study, one can further study the other algebraic structures such as different algebras by means of soft intersections.

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21 S. Abdullah, M. Aslam, M. Naeem and K. Hila 635 [36] G. L. Booth Jacobson radicals of Γ-nearrings Proceedings of the Hobart Conference, Longman Sci. & Technical (1987) S. Abdullah Department of Mathematics Quaid-i-Azam University P.O. Box Islamabad, Pakistan M. Aslam Department of Mathematics Quaid-i-Azam University P.O. Box Islamabad, Pakistan M. Naeem Umm ul Qura University Makkah, Saudi Arab K. Hila Department of Mathematics & Computer Science Faculty of Natural Science University of Gjirokastra Albania kostaq

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