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Annals of Fuzzy Mathematics and Informatics Volume 4, No. 2, (October 2012), pp. 365–375 ISSN 2093–9310 http://www.afmi.or.kr

@FMI c Kyung Moon Sa Co. ° http://www.kyungmoon.com

On soft int-groups Kenan Kaygisiz Received 29 November 2011; Revised 20 January 2012; Accepted 14 February 2012

Abstract. In this paper, we present soft intersection groups (soft intgroups) on a soft set and obtain some other properties of soft int-groups. We also investigate some relations on α-inclusion, soft product and soft int-groups.

2010 AMS Classification: 03G25, 20N25, 08A72, 06D72 Keywords: Soft set, α-inclusion, Soft product, Soft int-group. Corresponding Author: Kenan Kaygisiz ([email protected])

1. Introduction

T

here are many problems in economy, engineering, environmental and social sciences that may not be successfully modeled by the classical mathematics because of various types of uncertainties. Zadeh [24] introduced the notion of a fuzzy set in 1965 to deal with such kinds of problems. In 1971, Azriel Rosenfeld [22] defined the fuzzy subgroup of a group. Rosenfeld’s paper made important contributions to the development of fuzzy abstract algebra. Since then, various researchers have studied on fuzzy group theory analogues of results derived from classical group theory. These include [1, 3, 6, 7, 8, 11, 12, 17, 21]. Mordeson et al. [20] combined all the above papers and many others in their book titled Fuzzy Group Theory. There is another theory, called soft sets, defined by Molodtsov [19] in 1999 in order to deal with uncertainties. Since then Maji et al.[18] studied the operations of soft sets, C ¸ a˘gman and Engino˘glu [10] modified definition and operations of soft sets and Ali et al. [5] presented some new algebraic operations for soft sets. Sezgin and Atag¨ un [23] analyzed operations of soft sets. Using these definitions, researches have been very active on the soft sets and many important results have been achieved in theoretical and practical aspects. An algebraic structure of soft sets was first studied by Akta¸s and C ¸ a˘gman [4]. They introduced the notion of the soft group and derived some basic properties.

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

Since then, many papers have been prepared on soft algebraic structures, such as [2, 13, 14, 15, 16]. C ¸ a˘gman et al. [9] studied on soft int-groups, which are different from the definition of soft groups in [4]. This new approach is based on the inclusion relation and intersection of sets. It brings the soft set theory, set theory and the group theory together. In this paper, we give some supplementary properties of soft sets and soft int-groups, analogues to classical group theory and fuzzy group theory. We, finally, present some important relations on α-inclusion, soft product and soft int-groups, to construct notions of soft group theory. 2. Preliminaries 2.1. Soft sets. In this section, we present basic definitions of soft set theory according to [10]. For more detailed explanations, we refer to the earlier studies [18, 19]. Throughout this paper, U refers to an initial universe, E is a set of parameters and P (U ) is the power set of U . ⊂ and ⊃ stands for proper subset and superset, respectively. Definition 2.1 ([19]). For any subset A of E, a soft set fA over U is a set defined by a function fA representing a mapping fA : E −→ P (U ) such that fA (x) = ∅ if x ∈ / A. A soft set over U can be represented by the set of ordered pairs fA = {(x, fA (x)) : x ∈ E, fA (x) ∈ P (U )} . Note that the set of all soft sets over U will be denoted by S(U ). From here on “soft set” will be used without over U . Definition 2.2 ([10]). Let fA be a soft set. If fA (x) = ∅ for all x ∈ E, then fA is called an empty soft set and denoted by ΦA . If fA (x) = U for all x ∈ A, then fA is called A-universal soft set and denoted by fAe. If fA (x) = U , for all x ∈ E, then fA is called a universal soft set and denoted by fEe . If fA ∈ S(U ), then the image(value class) of fA is defined by Im (fA ) = {fA (x) : x ∈ A} and if A = E, then Im(fE ) is called image of E under fA . Definition 2.3. Let fA be a soft set and A ⊆ E. Then, the set fA∗ defined by fA∗ = {x ∈ A : fA (x) 6= ∅} is called the support of fA . Definition 2.4. Let fA : E −→ P (U ) be a soft set and K ⊆ E. Then, the image of a set K under fA is defined by fA (K) = ∪ {fA (xi ) : xi ∈ K} . Definition 2.5 ([10]). Let fA , fB be two soft sets. Then, fA is a soft subset of fB , e B , if fA (x) ⊆ fB (x) for all x ∈ E. denoted by fA ⊆f e B , if fA (x) ⊆ fB (x) for fA is called a soft proper subset of fB , denoted by fA ⊂f all x ∈ E and fA (x) 6= fB (x) for at least one x ∈ E. 366

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fA and fB are called soft equal, denoted by fA = fB , if and only if fA (x) = fB (x) for all x ∈ E. e fB and interDefinition 2.6 ([10]). Let fA , fB be two soft sets. Then, union fA ∪ e section fA ∩fB of fA and fB are defined by fA∪e B (x) = fA (x) ∪ fB (x), fA∩e B (x) = fA (x) ∩ fB (x), respectively. 2.2. Definitions and basic properties of soft int-groups. In this section, we review soft int-groups and their basic properties according to paper by C ¸ a˘gman et al. [9]. Definition 2.7 ([9]). Let G be a group and fG be a soft set. Then, fG is called a soft intersection groupoid over U if fG (xy) ⊇ fG (x) ∩ fG (y) for all x, y ∈ G and is called a soft intersection group over U if it satisfies fG (x−1 ) = fG (x) for all x ∈ G as well. Throughout this paper, G denotes an arbitrary group with identity element e and the set of all soft int-groups with parameter set G over U will be denoted by SG (U ), unless otherwise stated. For short, instead of “fG is a soft int-group with the parameter set G over U ” we say “fG is a soft int-group”. Theorem 2.8 ([9]). Let fG be a soft int-group. Then (1) fG (e) ⊇ fG (x) for all x ∈ G, (2) fG (xy) ⊇ fG (y) for all y ∈ G if and only if fG (x) = fG (e). Theorem 2.9 ([9]). A soft set fG is a soft int-group if and only if fG (xy −1 ) ⊇ fG (x) ∩ fG (y) for all x, y ∈ G. Definition 2.10 ([9]). Let fG be a soft set. Then, e-set of fG , denoted by efG , is defined as efG = {x ∈ G : fG (x) = fG (e)} . If fG is a soft int-group, then the largest set in Im(fG ) is called the tip of fG , which is equal to fG (e). Theorem 2.11 ([9]). If fG is a soft int-group, then efG is a subgroup of G. 3. Some new results on soft sets and soft int-groups In this section, we first give some new results on soft sets and soft int-groups. Then, we define soft singleton and soft product, and give some additional properties of soft int-groups. Theorem 3.1. If fG is a soft int-group, then fG (xn ) ⊇ fG (x) for all x ∈ G where n ∈ N. Proof. Proof is direct from definition of soft int-group by induction. ¤ ¡ −1 ¢ Theorem 3.2. Let fG be a soft int-group and x, y ∈ G. If fG xy = fG (e), then fG (x) = fG (y). 367

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Proof. For any x, y ∈ G, ¡¡ ¢ ¢ ¡ ¢ fG (x) = fG xy −1 y ⊇ fG xy −1 ∩ fG (y) = fG (e) ∩ fG (y) = fG (y) and

¡ ¢ fG (y) = fG y −1 ¡ ¡ ¢¢ = fG x−1 xy −1 ¡ ¢ ¡ ¢ ⊇ fG x−1 ∩ fG xy −1 ¢ ¡ = fG x−1 ∩ fG (e) = fG (x) .

Hence fG (x) = fG (y).

¤

Theorem 3.3. If fG is a soft int-group and H ≤ G, then the restriction fG |H is a soft int-group with the parameter set H. Proof. Since H ≤ G, fG (xy −1 ) ⊇ fG (x) ∩ fG (y) for all x, y ∈ H. Let’s define fH (x) = fG (x) for all x ∈ H. Since H is a group, xy −1 ∈ H for all x, y ∈ H. Then for all x, y ∈ H fH (xy −1 ) = fG (xy −1 ) ⊇ fG (x) ∩ fG (y) = fH (x) ∩ fH (y), so fH is a soft int-group with the parameter set H.

¤

Theorem 3.4. Let Ai ≤ G for all i ∈ I and {fAi : i ∈ I} be a family of soft T int-groups. Then, f fAi is a soft int-group. i∈I

Proof. By C ¸ a˘gman et al. ([9, Theorem 7]), we have the proof for two soft int-groups. To prove the general form, let x, y ∈ G, then \ \© ¡ ¢ ¡ ¢ ª fAi xy −1 = fAi xy −1 : i ∈ I i∈I

⊇

\

{fAi (x) ∩ fAi (y) : i ∈ I} ´ ´ ³\ ³\ {fAi (y) : i ∈ I} {fAi (x) : i ∈ I} ∩ = Ã ! Ã ! \ \ = fAi (x) ∩ fAi (y) . i∈I

i∈I

So the proof is complete by Theorem 2.9.

¤

Lemma 3.5. Let fG be a soft int-group such that either fG (x) ⊆ fG (y) or fG (x) ⊇ fG (y) for any x, y ∈ G. If fG (x) 6= fG (y), then fG (xy) = fG (x) ∩ fG (y) for any x, y ∈ G. Proof. If fG (x) 6= fG (y), then either fG (x) ⊃ fG (y) or fG (x) ⊂ fG (y). Suppose (3.1)

fG (x) ⊂ fG (y)

then (3.2)

fG (x) = fG (xyy −1 ) ⊇ fG (xy) ∩ fG (y −1 ) = fG (xy) ∩ fG (y) 368

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

Thus, from (3.1) and (3.2) we have (3.3)

fG (x) ⊇ fG (xy) ∩ fG (y) ⊇ fG (xy) ⊇ fG (x) ∩ fG (y) = fG (x).

So, all expressions are equal in (3.3). Hence, fG (xy) = fG (x) ∩ fG (y). For the other case, proof is similar.

¤

Corollary 3.6. Let fG be a soft int-group as in Lemma 3.5. If fG (x) ⊂ fG (y) then fG (x) = fG (xy) = fG (yx) for all x, y ∈ G. Proof. Obvious from the Lemma 3.5 and (3.3).

¤

Remark 3.7. Corollary 3.6 is not true if we replace, in the hypothesis, the strict inclusion fA (x) ⊂ fA (y) with the inclusion fA (x) ⊆ fA (y). We show this fact by the following example. Example 3.8. Let G be the Dihedral group D3 , where D3 = {e, u, u2 , v, vu, vu2 }, and u3 = v 2 = e, uv = vu2 . Define a mapping fG : G −→ P (U ) such that U for x = e α for x = v fG (x) = β otherwise where φ ⊂ β ⊂ α ⊂ U. It is easy to verify that fG is a soft group. In the notation of Corollary 3.6, let x = u and y = vu, then although β = fG (u) ⊆ fG (vu) = β, fG ((u)(vu)) = fG (v) = α 6= β = fG (u). So fG (x) = fG (xy) is not true. Remark 3.9. The converse of Corollary 3.6 is not true. Let us show it by a counter example, using the Dihedral group given in Example Let¢x = u2 and y = vu2 . ¡ 3.8. 2 2 Although fG (x) = fG (u ) = β and fG (yx) = fG (vu )(u2 ) = fG (vu) = β, the inclusion β = fG (u2 ) = fG (x) ⊂ fG (y) = fG (vu2 ) = β is not true. Theorem 3.10. Let G be a cyclic group of prime order and A ⊆ G. Then, the soft set fA , defined by ½ α f or x = e fA (x) = β otherwise where α ⊃ β and α, β ∈ P (U ), is a soft int-group. Proof. For any x, y ∈ G, there are four conditions; (1) xy 6= e and neither x nor y equals to e. Then, fA (xy) = β ⊇ β ∩ β = fA (x) ∩ fA (y) and since x 6= e, then x−1 6= e, so fA (x) = fA (x−1 ) = β. (2) xy 6= e and only one of x or y equals to e. Firstly, let x = e. Then, fA (xy) = fA (ey) = β ⊇ α ∩ β = fA (x) ∩ fA (y). For the second condition of soft int-group, if x = e, then fA (x) = fA (e) = α = fA (e−1 ) = fA (x−1 ), and since y 6= e, then y −1 6= e so, fA (y) = β = fA (y −1 ). (3) xy = e and neither x nor y equals to e. Then, fA (xy) = α ⊇ β ∩ β = fA (x) ∩ fA (y) 369

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

and fA (x) = fA (x−1 ) = β, since x 6= e implies x−1 6= e. (4) The last condition is x = y = e, which satisfies all conditions as well.

¤

Definition 3.11 ([9]). Let fA be a soft set and α ∈ P (U ). Then, α-inclusion of the soft set fA , denoted by fAα , is defined as fAα = {x ∈ A : fA (x) ⊇ α} . 0

We define the set fAα = {x ∈ A : fA (x) ⊃ α}, which is called strong α-inclusion. Corollary 3.12. For any soft sets fA and fB , e fB , α ∈ P (U ) ⇒ fAα ⊆ fBα , (1) fA ⊆

(2) α ⊆ β, α, β ∈ P (U ) ⇒ fAβ ⊆ fAα , (3) fA = fB ⇔ fAα = fBα , for all α ∈ P (U ).

Theorem 3.13 ([9]). Let I be an index set and {fAi : i ∈ I} be a family of soft sets. Then, for any α ∈ P (U ), ´α S ¡ ¢ ³S (1) i∈I fAαi ⊆ e i∈I fAi , ´α T ¡ α ¢ ³T (2) fAi = e i∈I fAi . i∈I

Theorem 3.14. Let fA T be a soft set and S {αi : i ∈ I} be a non-empty subset of P (U ) αi and γ = i∈I αi . Then, the following assertions hold: for each i ∈ I. Let β = S (1) Ti∈I fAαi ⊆ fAβ , fAαi = fAγ . (2)

i∈I

i∈I

Proof. Let x ∈ A. Then, x

∈

[

fAαi ⇒ ∃i ∈ I such that x ∈ fAαi

i∈I

⇒

∃i ∈ I such that fA (x) ⊇ αi

⇒

∃i ∈ I such that fA (x) ⊇ αi ⊇

\

αi = β

i∈I

⇒

x ∈ fAβ .

So, the result follows. Second part is similar.

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Definition 3.15. Let fA be a soft set and α ∈ P (U ). Then, the soft set fAα , defined by, fAα (x) = α, for all x ∈ A, is called A − α soft set. If A is a singleton, say {w}, then fwα is called a soft singleton (or soft point). If α = U , then fAe is the characteristic function of A. Proposition 3.16. Let fAα be an A − α soft set. Then, e fBα = f(A∩B)α and fAα ∪ e fBα = f(A∪B)α , (1) fAα ∩ e fAe = fAα and fAα ∪ e fAe = fAe. (2) fAα ∩ 370

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

Lemma 3.17. Let fA be a soft set and f(f α )α be an (fAα ) − α soft set. Then, A [ [ f f fA = f(f α )α = f(f α )α . A A α∈P (U )

α∈Im(fA )

It is clear that, instead of A − α soft set of all α subset of U , it is enough to consider A − α soft set taken α from Im(fA ) . Proof. For any x ∈ A, [ [ f {α ∈ P (U ) : α ⊆ fA (x)} = fA (x) . f(f α )α (x) = A α∈P (U )

S So, fA = e α∈P (U ) f(f α )α . A Similarly, [ [ f f(f α )α (x) = {α ∈ Im(fA ) : α ⊆ fA (x)} = fA (x) A α∈Im(fA )

S and thus fA = e α∈Im(fA ) f(f α )α .

¤

A

Theorem 3.18. Let G be a group and α ∈ P (U ). Then, fG is a soft int-group if α and only if fG is a subgroup of G, whenever it is nonempty. Proof. The necessary condition is proven in ([9, Theorem 11]). To prove sufficient α α condition, suppose fG ≤ G for any nonempty fG . γ Let x, y ∈ G, fG (x) = β and fG (y) = δ, and let γ = β ∩ δ. Then, x, y ∈ fG and γ γ fG is a subgroup of G by hypothesis. So xy −1 ∈ fG . Hence, fG (xy −1 ) ⊇ γ = β ∩ δ = fG (x) ∩ fG (y). Thus, fG is a soft int-group.

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Definition 3.19. Let fG be a soft int-group. Then, the subgroups level subgroups of G for any α ∈ P (U ).

α fG

are called

Definition 3.20. Let G be a group and A, B ⊆ G. Then, soft product of soft sets fA and fB is defined as [ (fA ∗ fB )(x) = {fA (u) ∩ fB (v) : uv = x, u, v ∈ G} Inverse of fA is defined as

fA−1 (x) = fA (x−1 )

for all x ∈ G. The following theorem reduces the soft product to the product of singletons. Theorem 3.21. Let G be a group and fA , fB , fxα , fyβ be soft sets with the parameter set G. Then, for any x, y ∈ G and φ ⊂ α, β ⊆ U, (1) fxα ∗ fyβ = f(xy)(α∩β) , S S (2) fA ∗ fB = e (fxα ∗ fyβ ) = e {fxα ∗ fyβ : fxα ∈ fA , fyβ ∈ fB } . fxα ∈fA fyβ ∈fB

371

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

Proof. Let fxα , fyβ , fA , fB ∈ S(U ). (1) From the Definition 3.20, we have for any w ∈ G [ (fxα ∗ fyβ ) (w) = {fxα (u) ∩ fyβ (v) : uv = w, u, v ∈ G}. If u = x and v = y, then fxα (u) ∩ fyβ (v) 6= φ, otherwise fxα (u) ∩ fyβ (v) = φ. Hence, [ [ (fxα ∗ fyβ ) (w) = {φ, fxα (x) ∩ fyβ (y)} = {φ, α ∩ β} = α ∩ β for w = xy. (2) For any point w ∈ G, we may assume that, there exists u, v ∈ G such that uv = w and fA (u) 6= φ, fB (v) 6= φ without loss of generality. Then, S (fA ∗ fB ) (w) = {fA (u) ∩ fB (v) : uv = w and u, v ∈ G} S = {fxα (x) ∩ fyβ (y) : xy = w, fxα ∈ fA , fyβ ∈ fB } S = e (fxα ∗ fyβ ) (w) . ¤ fxα ∈fA fyβ ∈fB

Corollary 3.22. Let A ⊆ G, fA ∈ S(U ) and fxα , fyβ , fzγ be singletons in fA . Then, (1) (fxα ∗ fyβ ) ∗ fzγ = fxα ∗ (fyβ ∗ fzγ ) , (2) fxα ∗ fyβ = fyβ ∗ fxα , if G is commutative, (3) fxα ∗ fe(fA (e)) = fe(fA (e)) ∗ fxα = fxα , if fA ∈ SG (U ). Proof. Let fA ∈ S(U ). (1) For any fxα , fyβ , fzγ ∈ fA , (fxα ∗ fyβ ) ∗ fzγ

= = = =

f(xy)(α∩β) ∗ fzγ f((xy)z)((α∩β)∩γ) f(x(yz))(α∩(β∩γ)) fxα ∗ f(yz)(β∩γ)

= fxα ∗ (fyβ ∗ fzγ ) . (2) Clear from Theorem 3.21. (3) For any fxα ∈ fA , fxα ∗ fe(fA (e)) = f(xe)(α∩(fA (e))) = fxα = f(ex)((fA (e))∩α) = fe(fA (e)) ∗ fxα . ¤ Theorem 3.23. Let A, B, C ⊆ G and fA , fB , fC be soft sets. Then, (1) (fA ∗ fB ) ∗ fC = fA ∗ (fB ∗ fC ) , (2) fA ∗ fB = fB ∗ fA , if G is commutative, (3) If fA ∈ SG (U ), then the identity element of operation “∗” is fe(fA (e)) . Proof. Proofs are direct from definition of soft product, Theorem 3.21 and Conclusion 3.22. ¤ Theorem 3.24. Let A, B ⊆ G and fA , fB be soft sets. Then, [ [ (fA ∗ fB )(x) = (fA (v) ∩ fB (v −1 x)) = (fA (xv −1 ) ∩ fB (v)) v∈G

v∈G

for any x ∈ G. 372

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

Proof. For all v ∈ G, v(v −1 x) = x or (xv −1 )v = x includes all compositions of x in the definition of soft product, so equality holds. ¤ Corollary 3.25. Let A ⊆ G and fA , fuα be soft sets where α = fA (A). Then, for any x, u ∈ G, (fuα ∗ fA )(x) = fA (u−1 x) and (fA ∗ fuα )(x) = fA (xu−1 ). Proof. For any x, u ∈ G, we have (fuα ∗ fA )(x)

=

[

(fuα (v) ∩ fA (v −1 x))

v∈G

=

α ∩ fA (u−1 x)

=

fA (u−1 x)

by Theorem 3.24. Second part is similar.

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Theorem 3.26. Let Ai ⊆ G and fAi (i ∈ I) be soft int-groups. Then, Ã !−1 \ \ f f −1 fAi = (fAi ) . i∈I

i∈I

Proof. For all x ∈ G, Ã !−1 Ã ! \ \ ¡ −1 ¢ \ ¡ ¢ \ −1 f f fAi (x) = f Ai x = fAi x−1 = fAi (x) i∈I

i∈I

i∈I

i∈I

so, equality holds.

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Theorem 3.27. Let A, B ⊆ G and fA , fB be soft int-groups. Then, (fA ∗ fB )−1 = fB−1 ∗ fA−1 . Proof. For all x ∈ G, ¡ ¢ (fA ∗ fB )−1 (x) = (fA ∗ fB ) x−1 S = {fA (u) ∩ fB (v) : uv = x−1 , u, v ∈ G} ¡ ¢−1 ¡ ¢−1 −1 −1 S = {fB v −1 ∩ fA u−1 : v u = x, u−1 , v −1 ∈ G} ¡ ¢ ¡ ¢ S = {fB−1 v −1 ∩ fA−1 u−1 : v −1 u−1 = x, u−1 , v −1 ∈ G} ¡ ¢ = fB−1 ∗ fA−1 (x) .

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Theorem 3.28. Let fG be a soft set. Then, fG is a soft int-group if and only if fG satisfies the following conditions: e G, (1) (fG ∗ fG ) ⊆f −1 e −1 or f −1 ⊆f e G ). (2) fG = fG (or fG ⊆f G G Proof. Assume that fG ∈ SG (U ). Firstly, for all x ∈ G, [ (fG ∗ fG ) (x) = {fG (u) ∩ fG (v) : uv = x, u, v ∈ G} [ ⊆ {fG (uv) : uv = x, u, v ∈ G} =

fG (x) 373

Kenan Kaygisiz/Ann. Fuzzy Math. Inform. 4 (2012), No. 2, 365–375

e G. since fG ∈ SG (U ). So (fG ∗ fG ) ⊆f Second part is obvious by the definition of soft int-group since ¡ ¢ −1 fG (x) = fG x−1 = fG (x) for all x ∈ G. e G , then for all x ∈ G, (fG ∗ fG ) (x) ⊆ fG (x) . Conversely; suppose (fG ∗ fG ) ⊆f So, for all x ∈ G fG (x) ⊇ =

(fG ∗ fG ) (x) [ {fG (u) ∩ fG (v) : u, v ∈ G, uv = x} .

Hence, for any u, v ∈ G such that uv = x, we have fG (uv) ⊇ fG (u) ∩ fG (v) and by the second part of assumption, fG ∈ SG (U ). ¤ Theorem 3.29. Let A, B ⊆ G and fA , fB be soft int-groups. Then, fA ∗ fB is a soft int-group if and only if fA ∗ fB = fB ∗ fA . Proof. Assume that fA ∗ fB ∈ SG (U ). Then, −1

fA ∗ fB = fA−1 ∗ fB−1 = (fB ∗ fA )

= fB ∗ fA .

Conversely, suppose fA ∗ fB = fB ∗ fA . Then, (fA ∗ fB ) ∗ (fA ∗ fB ) = = = ⊆ and

−1

fA ∗ (fB ∗ fA ) ∗ fB fA ∗ (fA ∗ fB ) ∗ fB (fA ∗ fA ) ∗ (fB ∗ fB ) fA ∗ fB

−1

(fA ∗ fB ) = (fB ∗ fA ) = fA−1 ∗ fB−1 = fA ∗ fB . Consequently, by Theorem 3.28, fA ∗ fB is a soft int-group.

¤

4. Conclusions In this paper, we present soft int-groups on a soft set and give some of their supplementary properties. In addition, we give relations between α-inclusion, soft product and soft int-groups. This study affords us an opportunity to go further on soft group theory, that is, soft normal int-group, quotient group, isomorphism theorems etc. Acknowledgements. The author is highly grateful to Dr. Naim C ¸ a˘gman for his valuable comments and suggestions. References [1] S. Abou-Zaid, On fuzzy subgroups, Fuzzy Sets and Systems 55 (1993) 237–240. [2] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59 (20100 3458–3463. [3] M. Akg¨ ul, Some properties of fuzzy groups, J. Math. Anal. Appl. 133 (1988) 93–100. [4] H. Akta¸s and N. C ¸ a˘ gman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [5] M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) 1547–1553.

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[6] J. M. Anthony and H. Sherwood, Fuzzy subgroups redefined, J. Math. Anal. Appl. 69 (1979) 124–130. [7] M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39 (1991) 323–328. [8] K. R. Bhutani, Fuzzy sets, fuzzy relations and fuzzy groups: Some interrelations, Inform. Sci. 73 (1993) 107–115. [9] N. C ¸ a˘ gman, F. C ¸ ıtak and H. Akta¸s, Soft int-group and its applications to group theory, Neural Comput. Appl. DOI:10.1007/s00521-011-0752-x. [10] N. C ¸ a˘ gman and S. Engino˘ glu, Soft set theory and uni-int decision making, European J. Oper. Res. 207 (2010) 848–855. [11] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264–269. [12] V. N. Dixit, R. Kumar and N. Ajamal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990) 359–371. [13] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621–2628. [14] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413. [15] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (2008) 2466–2475. [16] Y. B. Jun, K. J. Lee and A. Khan, Soft ordered semigroups, MLQ Math. Log. Q. 56(1) (2010) 42–50. [17] J. G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. 80 (1994) 341–348. [18] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [19] D. A. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999) 19–31. [20] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, Springer, 2005. [21] N. P. Mujherjee and P. Bhattacharya, Fuzzy groups, some group theoretic analogs, Inform. Sci. 39 (1986) 247–268. [22] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [23] A. Sezgin and A. O. Atag¨ un , On operations of soft sets, Comput. Math. Appl. 61(5) (2011) 1457–1467. [24] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.

Kenan Kaygisiz ([email protected]) Gaziosmanpa¸sa University, Faculty of Arts and Sciences, Department of Mathematics, Tokat, Turkey.

375

@FMI c Kyung Moon Sa Co. ° http://www.kyungmoon.com

On soft int-groups Kenan Kaygisiz Received 29 November 2011; Revised 20 January 2012; Accepted 14 February 2012

Abstract. In this paper, we present soft intersection groups (soft intgroups) on a soft set and obtain some other properties of soft int-groups. We also investigate some relations on α-inclusion, soft product and soft int-groups.

2010 AMS Classification: 03G25, 20N25, 08A72, 06D72 Keywords: Soft set, α-inclusion, Soft product, Soft int-group. Corresponding Author: Kenan Kaygisiz ([email protected])

1. Introduction

T

here are many problems in economy, engineering, environmental and social sciences that may not be successfully modeled by the classical mathematics because of various types of uncertainties. Zadeh [24] introduced the notion of a fuzzy set in 1965 to deal with such kinds of problems. In 1971, Azriel Rosenfeld [22] defined the fuzzy subgroup of a group. Rosenfeld’s paper made important contributions to the development of fuzzy abstract algebra. Since then, various researchers have studied on fuzzy group theory analogues of results derived from classical group theory. These include [1, 3, 6, 7, 8, 11, 12, 17, 21]. Mordeson et al. [20] combined all the above papers and many others in their book titled Fuzzy Group Theory. There is another theory, called soft sets, defined by Molodtsov [19] in 1999 in order to deal with uncertainties. Since then Maji et al.[18] studied the operations of soft sets, C ¸ a˘gman and Engino˘glu [10] modified definition and operations of soft sets and Ali et al. [5] presented some new algebraic operations for soft sets. Sezgin and Atag¨ un [23] analyzed operations of soft sets. Using these definitions, researches have been very active on the soft sets and many important results have been achieved in theoretical and practical aspects. An algebraic structure of soft sets was first studied by Akta¸s and C ¸ a˘gman [4]. They introduced the notion of the soft group and derived some basic properties.

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Since then, many papers have been prepared on soft algebraic structures, such as [2, 13, 14, 15, 16]. C ¸ a˘gman et al. [9] studied on soft int-groups, which are different from the definition of soft groups in [4]. This new approach is based on the inclusion relation and intersection of sets. It brings the soft set theory, set theory and the group theory together. In this paper, we give some supplementary properties of soft sets and soft int-groups, analogues to classical group theory and fuzzy group theory. We, finally, present some important relations on α-inclusion, soft product and soft int-groups, to construct notions of soft group theory. 2. Preliminaries 2.1. Soft sets. In this section, we present basic definitions of soft set theory according to [10]. For more detailed explanations, we refer to the earlier studies [18, 19]. Throughout this paper, U refers to an initial universe, E is a set of parameters and P (U ) is the power set of U . ⊂ and ⊃ stands for proper subset and superset, respectively. Definition 2.1 ([19]). For any subset A of E, a soft set fA over U is a set defined by a function fA representing a mapping fA : E −→ P (U ) such that fA (x) = ∅ if x ∈ / A. A soft set over U can be represented by the set of ordered pairs fA = {(x, fA (x)) : x ∈ E, fA (x) ∈ P (U )} . Note that the set of all soft sets over U will be denoted by S(U ). From here on “soft set” will be used without over U . Definition 2.2 ([10]). Let fA be a soft set. If fA (x) = ∅ for all x ∈ E, then fA is called an empty soft set and denoted by ΦA . If fA (x) = U for all x ∈ A, then fA is called A-universal soft set and denoted by fAe. If fA (x) = U , for all x ∈ E, then fA is called a universal soft set and denoted by fEe . If fA ∈ S(U ), then the image(value class) of fA is defined by Im (fA ) = {fA (x) : x ∈ A} and if A = E, then Im(fE ) is called image of E under fA . Definition 2.3. Let fA be a soft set and A ⊆ E. Then, the set fA∗ defined by fA∗ = {x ∈ A : fA (x) 6= ∅} is called the support of fA . Definition 2.4. Let fA : E −→ P (U ) be a soft set and K ⊆ E. Then, the image of a set K under fA is defined by fA (K) = ∪ {fA (xi ) : xi ∈ K} . Definition 2.5 ([10]). Let fA , fB be two soft sets. Then, fA is a soft subset of fB , e B , if fA (x) ⊆ fB (x) for all x ∈ E. denoted by fA ⊆f e B , if fA (x) ⊆ fB (x) for fA is called a soft proper subset of fB , denoted by fA ⊂f all x ∈ E and fA (x) 6= fB (x) for at least one x ∈ E. 366

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fA and fB are called soft equal, denoted by fA = fB , if and only if fA (x) = fB (x) for all x ∈ E. e fB and interDefinition 2.6 ([10]). Let fA , fB be two soft sets. Then, union fA ∪ e section fA ∩fB of fA and fB are defined by fA∪e B (x) = fA (x) ∪ fB (x), fA∩e B (x) = fA (x) ∩ fB (x), respectively. 2.2. Definitions and basic properties of soft int-groups. In this section, we review soft int-groups and their basic properties according to paper by C ¸ a˘gman et al. [9]. Definition 2.7 ([9]). Let G be a group and fG be a soft set. Then, fG is called a soft intersection groupoid over U if fG (xy) ⊇ fG (x) ∩ fG (y) for all x, y ∈ G and is called a soft intersection group over U if it satisfies fG (x−1 ) = fG (x) for all x ∈ G as well. Throughout this paper, G denotes an arbitrary group with identity element e and the set of all soft int-groups with parameter set G over U will be denoted by SG (U ), unless otherwise stated. For short, instead of “fG is a soft int-group with the parameter set G over U ” we say “fG is a soft int-group”. Theorem 2.8 ([9]). Let fG be a soft int-group. Then (1) fG (e) ⊇ fG (x) for all x ∈ G, (2) fG (xy) ⊇ fG (y) for all y ∈ G if and only if fG (x) = fG (e). Theorem 2.9 ([9]). A soft set fG is a soft int-group if and only if fG (xy −1 ) ⊇ fG (x) ∩ fG (y) for all x, y ∈ G. Definition 2.10 ([9]). Let fG be a soft set. Then, e-set of fG , denoted by efG , is defined as efG = {x ∈ G : fG (x) = fG (e)} . If fG is a soft int-group, then the largest set in Im(fG ) is called the tip of fG , which is equal to fG (e). Theorem 2.11 ([9]). If fG is a soft int-group, then efG is a subgroup of G. 3. Some new results on soft sets and soft int-groups In this section, we first give some new results on soft sets and soft int-groups. Then, we define soft singleton and soft product, and give some additional properties of soft int-groups. Theorem 3.1. If fG is a soft int-group, then fG (xn ) ⊇ fG (x) for all x ∈ G where n ∈ N. Proof. Proof is direct from definition of soft int-group by induction. ¤ ¡ −1 ¢ Theorem 3.2. Let fG be a soft int-group and x, y ∈ G. If fG xy = fG (e), then fG (x) = fG (y). 367

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Proof. For any x, y ∈ G, ¡¡ ¢ ¢ ¡ ¢ fG (x) = fG xy −1 y ⊇ fG xy −1 ∩ fG (y) = fG (e) ∩ fG (y) = fG (y) and

¡ ¢ fG (y) = fG y −1 ¡ ¡ ¢¢ = fG x−1 xy −1 ¡ ¢ ¡ ¢ ⊇ fG x−1 ∩ fG xy −1 ¢ ¡ = fG x−1 ∩ fG (e) = fG (x) .

Hence fG (x) = fG (y).

¤

Theorem 3.3. If fG is a soft int-group and H ≤ G, then the restriction fG |H is a soft int-group with the parameter set H. Proof. Since H ≤ G, fG (xy −1 ) ⊇ fG (x) ∩ fG (y) for all x, y ∈ H. Let’s define fH (x) = fG (x) for all x ∈ H. Since H is a group, xy −1 ∈ H for all x, y ∈ H. Then for all x, y ∈ H fH (xy −1 ) = fG (xy −1 ) ⊇ fG (x) ∩ fG (y) = fH (x) ∩ fH (y), so fH is a soft int-group with the parameter set H.

¤

Theorem 3.4. Let Ai ≤ G for all i ∈ I and {fAi : i ∈ I} be a family of soft T int-groups. Then, f fAi is a soft int-group. i∈I

Proof. By C ¸ a˘gman et al. ([9, Theorem 7]), we have the proof for two soft int-groups. To prove the general form, let x, y ∈ G, then \ \© ¡ ¢ ¡ ¢ ª fAi xy −1 = fAi xy −1 : i ∈ I i∈I

⊇

\

{fAi (x) ∩ fAi (y) : i ∈ I} ´ ´ ³\ ³\ {fAi (y) : i ∈ I} {fAi (x) : i ∈ I} ∩ = Ã ! Ã ! \ \ = fAi (x) ∩ fAi (y) . i∈I

i∈I

So the proof is complete by Theorem 2.9.

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Lemma 3.5. Let fG be a soft int-group such that either fG (x) ⊆ fG (y) or fG (x) ⊇ fG (y) for any x, y ∈ G. If fG (x) 6= fG (y), then fG (xy) = fG (x) ∩ fG (y) for any x, y ∈ G. Proof. If fG (x) 6= fG (y), then either fG (x) ⊃ fG (y) or fG (x) ⊂ fG (y). Suppose (3.1)

fG (x) ⊂ fG (y)

then (3.2)

fG (x) = fG (xyy −1 ) ⊇ fG (xy) ∩ fG (y −1 ) = fG (xy) ∩ fG (y) 368

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Thus, from (3.1) and (3.2) we have (3.3)

fG (x) ⊇ fG (xy) ∩ fG (y) ⊇ fG (xy) ⊇ fG (x) ∩ fG (y) = fG (x).

So, all expressions are equal in (3.3). Hence, fG (xy) = fG (x) ∩ fG (y). For the other case, proof is similar.

¤

Corollary 3.6. Let fG be a soft int-group as in Lemma 3.5. If fG (x) ⊂ fG (y) then fG (x) = fG (xy) = fG (yx) for all x, y ∈ G. Proof. Obvious from the Lemma 3.5 and (3.3).

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Remark 3.7. Corollary 3.6 is not true if we replace, in the hypothesis, the strict inclusion fA (x) ⊂ fA (y) with the inclusion fA (x) ⊆ fA (y). We show this fact by the following example. Example 3.8. Let G be the Dihedral group D3 , where D3 = {e, u, u2 , v, vu, vu2 }, and u3 = v 2 = e, uv = vu2 . Define a mapping fG : G −→ P (U ) such that U for x = e α for x = v fG (x) = β otherwise where φ ⊂ β ⊂ α ⊂ U. It is easy to verify that fG is a soft group. In the notation of Corollary 3.6, let x = u and y = vu, then although β = fG (u) ⊆ fG (vu) = β, fG ((u)(vu)) = fG (v) = α 6= β = fG (u). So fG (x) = fG (xy) is not true. Remark 3.9. The converse of Corollary 3.6 is not true. Let us show it by a counter example, using the Dihedral group given in Example Let¢x = u2 and y = vu2 . ¡ 3.8. 2 2 Although fG (x) = fG (u ) = β and fG (yx) = fG (vu )(u2 ) = fG (vu) = β, the inclusion β = fG (u2 ) = fG (x) ⊂ fG (y) = fG (vu2 ) = β is not true. Theorem 3.10. Let G be a cyclic group of prime order and A ⊆ G. Then, the soft set fA , defined by ½ α f or x = e fA (x) = β otherwise where α ⊃ β and α, β ∈ P (U ), is a soft int-group. Proof. For any x, y ∈ G, there are four conditions; (1) xy 6= e and neither x nor y equals to e. Then, fA (xy) = β ⊇ β ∩ β = fA (x) ∩ fA (y) and since x 6= e, then x−1 6= e, so fA (x) = fA (x−1 ) = β. (2) xy 6= e and only one of x or y equals to e. Firstly, let x = e. Then, fA (xy) = fA (ey) = β ⊇ α ∩ β = fA (x) ∩ fA (y). For the second condition of soft int-group, if x = e, then fA (x) = fA (e) = α = fA (e−1 ) = fA (x−1 ), and since y 6= e, then y −1 6= e so, fA (y) = β = fA (y −1 ). (3) xy = e and neither x nor y equals to e. Then, fA (xy) = α ⊇ β ∩ β = fA (x) ∩ fA (y) 369

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and fA (x) = fA (x−1 ) = β, since x 6= e implies x−1 6= e. (4) The last condition is x = y = e, which satisfies all conditions as well.

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Definition 3.11 ([9]). Let fA be a soft set and α ∈ P (U ). Then, α-inclusion of the soft set fA , denoted by fAα , is defined as fAα = {x ∈ A : fA (x) ⊇ α} . 0

We define the set fAα = {x ∈ A : fA (x) ⊃ α}, which is called strong α-inclusion. Corollary 3.12. For any soft sets fA and fB , e fB , α ∈ P (U ) ⇒ fAα ⊆ fBα , (1) fA ⊆

(2) α ⊆ β, α, β ∈ P (U ) ⇒ fAβ ⊆ fAα , (3) fA = fB ⇔ fAα = fBα , for all α ∈ P (U ).

Theorem 3.13 ([9]). Let I be an index set and {fAi : i ∈ I} be a family of soft sets. Then, for any α ∈ P (U ), ´α S ¡ ¢ ³S (1) i∈I fAαi ⊆ e i∈I fAi , ´α T ¡ α ¢ ³T (2) fAi = e i∈I fAi . i∈I

Theorem 3.14. Let fA T be a soft set and S {αi : i ∈ I} be a non-empty subset of P (U ) αi and γ = i∈I αi . Then, the following assertions hold: for each i ∈ I. Let β = S (1) Ti∈I fAαi ⊆ fAβ , fAαi = fAγ . (2)

i∈I

i∈I

Proof. Let x ∈ A. Then, x

∈

[

fAαi ⇒ ∃i ∈ I such that x ∈ fAαi

i∈I

⇒

∃i ∈ I such that fA (x) ⊇ αi

⇒

∃i ∈ I such that fA (x) ⊇ αi ⊇

\

αi = β

i∈I

⇒

x ∈ fAβ .

So, the result follows. Second part is similar.

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Definition 3.15. Let fA be a soft set and α ∈ P (U ). Then, the soft set fAα , defined by, fAα (x) = α, for all x ∈ A, is called A − α soft set. If A is a singleton, say {w}, then fwα is called a soft singleton (or soft point). If α = U , then fAe is the characteristic function of A. Proposition 3.16. Let fAα be an A − α soft set. Then, e fBα = f(A∩B)α and fAα ∪ e fBα = f(A∪B)α , (1) fAα ∩ e fAe = fAα and fAα ∪ e fAe = fAe. (2) fAα ∩ 370

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Lemma 3.17. Let fA be a soft set and f(f α )α be an (fAα ) − α soft set. Then, A [ [ f f fA = f(f α )α = f(f α )α . A A α∈P (U )

α∈Im(fA )

It is clear that, instead of A − α soft set of all α subset of U , it is enough to consider A − α soft set taken α from Im(fA ) . Proof. For any x ∈ A, [ [ f {α ∈ P (U ) : α ⊆ fA (x)} = fA (x) . f(f α )α (x) = A α∈P (U )

S So, fA = e α∈P (U ) f(f α )α . A Similarly, [ [ f f(f α )α (x) = {α ∈ Im(fA ) : α ⊆ fA (x)} = fA (x) A α∈Im(fA )

S and thus fA = e α∈Im(fA ) f(f α )α .

¤

A

Theorem 3.18. Let G be a group and α ∈ P (U ). Then, fG is a soft int-group if α and only if fG is a subgroup of G, whenever it is nonempty. Proof. The necessary condition is proven in ([9, Theorem 11]). To prove sufficient α α condition, suppose fG ≤ G for any nonempty fG . γ Let x, y ∈ G, fG (x) = β and fG (y) = δ, and let γ = β ∩ δ. Then, x, y ∈ fG and γ γ fG is a subgroup of G by hypothesis. So xy −1 ∈ fG . Hence, fG (xy −1 ) ⊇ γ = β ∩ δ = fG (x) ∩ fG (y). Thus, fG is a soft int-group.

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Definition 3.19. Let fG be a soft int-group. Then, the subgroups level subgroups of G for any α ∈ P (U ).

α fG

are called

Definition 3.20. Let G be a group and A, B ⊆ G. Then, soft product of soft sets fA and fB is defined as [ (fA ∗ fB )(x) = {fA (u) ∩ fB (v) : uv = x, u, v ∈ G} Inverse of fA is defined as

fA−1 (x) = fA (x−1 )

for all x ∈ G. The following theorem reduces the soft product to the product of singletons. Theorem 3.21. Let G be a group and fA , fB , fxα , fyβ be soft sets with the parameter set G. Then, for any x, y ∈ G and φ ⊂ α, β ⊆ U, (1) fxα ∗ fyβ = f(xy)(α∩β) , S S (2) fA ∗ fB = e (fxα ∗ fyβ ) = e {fxα ∗ fyβ : fxα ∈ fA , fyβ ∈ fB } . fxα ∈fA fyβ ∈fB

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Proof. Let fxα , fyβ , fA , fB ∈ S(U ). (1) From the Definition 3.20, we have for any w ∈ G [ (fxα ∗ fyβ ) (w) = {fxα (u) ∩ fyβ (v) : uv = w, u, v ∈ G}. If u = x and v = y, then fxα (u) ∩ fyβ (v) 6= φ, otherwise fxα (u) ∩ fyβ (v) = φ. Hence, [ [ (fxα ∗ fyβ ) (w) = {φ, fxα (x) ∩ fyβ (y)} = {φ, α ∩ β} = α ∩ β for w = xy. (2) For any point w ∈ G, we may assume that, there exists u, v ∈ G such that uv = w and fA (u) 6= φ, fB (v) 6= φ without loss of generality. Then, S (fA ∗ fB ) (w) = {fA (u) ∩ fB (v) : uv = w and u, v ∈ G} S = {fxα (x) ∩ fyβ (y) : xy = w, fxα ∈ fA , fyβ ∈ fB } S = e (fxα ∗ fyβ ) (w) . ¤ fxα ∈fA fyβ ∈fB

Corollary 3.22. Let A ⊆ G, fA ∈ S(U ) and fxα , fyβ , fzγ be singletons in fA . Then, (1) (fxα ∗ fyβ ) ∗ fzγ = fxα ∗ (fyβ ∗ fzγ ) , (2) fxα ∗ fyβ = fyβ ∗ fxα , if G is commutative, (3) fxα ∗ fe(fA (e)) = fe(fA (e)) ∗ fxα = fxα , if fA ∈ SG (U ). Proof. Let fA ∈ S(U ). (1) For any fxα , fyβ , fzγ ∈ fA , (fxα ∗ fyβ ) ∗ fzγ

= = = =

f(xy)(α∩β) ∗ fzγ f((xy)z)((α∩β)∩γ) f(x(yz))(α∩(β∩γ)) fxα ∗ f(yz)(β∩γ)

= fxα ∗ (fyβ ∗ fzγ ) . (2) Clear from Theorem 3.21. (3) For any fxα ∈ fA , fxα ∗ fe(fA (e)) = f(xe)(α∩(fA (e))) = fxα = f(ex)((fA (e))∩α) = fe(fA (e)) ∗ fxα . ¤ Theorem 3.23. Let A, B, C ⊆ G and fA , fB , fC be soft sets. Then, (1) (fA ∗ fB ) ∗ fC = fA ∗ (fB ∗ fC ) , (2) fA ∗ fB = fB ∗ fA , if G is commutative, (3) If fA ∈ SG (U ), then the identity element of operation “∗” is fe(fA (e)) . Proof. Proofs are direct from definition of soft product, Theorem 3.21 and Conclusion 3.22. ¤ Theorem 3.24. Let A, B ⊆ G and fA , fB be soft sets. Then, [ [ (fA ∗ fB )(x) = (fA (v) ∩ fB (v −1 x)) = (fA (xv −1 ) ∩ fB (v)) v∈G

v∈G

for any x ∈ G. 372

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Proof. For all v ∈ G, v(v −1 x) = x or (xv −1 )v = x includes all compositions of x in the definition of soft product, so equality holds. ¤ Corollary 3.25. Let A ⊆ G and fA , fuα be soft sets where α = fA (A). Then, for any x, u ∈ G, (fuα ∗ fA )(x) = fA (u−1 x) and (fA ∗ fuα )(x) = fA (xu−1 ). Proof. For any x, u ∈ G, we have (fuα ∗ fA )(x)

=

[

(fuα (v) ∩ fA (v −1 x))

v∈G

=

α ∩ fA (u−1 x)

=

fA (u−1 x)

by Theorem 3.24. Second part is similar.

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Theorem 3.26. Let Ai ⊆ G and fAi (i ∈ I) be soft int-groups. Then, Ã !−1 \ \ f f −1 fAi = (fAi ) . i∈I

i∈I

Proof. For all x ∈ G, Ã !−1 Ã ! \ \ ¡ −1 ¢ \ ¡ ¢ \ −1 f f fAi (x) = f Ai x = fAi x−1 = fAi (x) i∈I

i∈I

i∈I

i∈I

so, equality holds.

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Theorem 3.27. Let A, B ⊆ G and fA , fB be soft int-groups. Then, (fA ∗ fB )−1 = fB−1 ∗ fA−1 . Proof. For all x ∈ G, ¡ ¢ (fA ∗ fB )−1 (x) = (fA ∗ fB ) x−1 S = {fA (u) ∩ fB (v) : uv = x−1 , u, v ∈ G} ¡ ¢−1 ¡ ¢−1 −1 −1 S = {fB v −1 ∩ fA u−1 : v u = x, u−1 , v −1 ∈ G} ¡ ¢ ¡ ¢ S = {fB−1 v −1 ∩ fA−1 u−1 : v −1 u−1 = x, u−1 , v −1 ∈ G} ¡ ¢ = fB−1 ∗ fA−1 (x) .

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Theorem 3.28. Let fG be a soft set. Then, fG is a soft int-group if and only if fG satisfies the following conditions: e G, (1) (fG ∗ fG ) ⊆f −1 e −1 or f −1 ⊆f e G ). (2) fG = fG (or fG ⊆f G G Proof. Assume that fG ∈ SG (U ). Firstly, for all x ∈ G, [ (fG ∗ fG ) (x) = {fG (u) ∩ fG (v) : uv = x, u, v ∈ G} [ ⊆ {fG (uv) : uv = x, u, v ∈ G} =

fG (x) 373

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e G. since fG ∈ SG (U ). So (fG ∗ fG ) ⊆f Second part is obvious by the definition of soft int-group since ¡ ¢ −1 fG (x) = fG x−1 = fG (x) for all x ∈ G. e G , then for all x ∈ G, (fG ∗ fG ) (x) ⊆ fG (x) . Conversely; suppose (fG ∗ fG ) ⊆f So, for all x ∈ G fG (x) ⊇ =

(fG ∗ fG ) (x) [ {fG (u) ∩ fG (v) : u, v ∈ G, uv = x} .

Hence, for any u, v ∈ G such that uv = x, we have fG (uv) ⊇ fG (u) ∩ fG (v) and by the second part of assumption, fG ∈ SG (U ). ¤ Theorem 3.29. Let A, B ⊆ G and fA , fB be soft int-groups. Then, fA ∗ fB is a soft int-group if and only if fA ∗ fB = fB ∗ fA . Proof. Assume that fA ∗ fB ∈ SG (U ). Then, −1

fA ∗ fB = fA−1 ∗ fB−1 = (fB ∗ fA )

= fB ∗ fA .

Conversely, suppose fA ∗ fB = fB ∗ fA . Then, (fA ∗ fB ) ∗ (fA ∗ fB ) = = = ⊆ and

−1

fA ∗ (fB ∗ fA ) ∗ fB fA ∗ (fA ∗ fB ) ∗ fB (fA ∗ fA ) ∗ (fB ∗ fB ) fA ∗ fB

−1

(fA ∗ fB ) = (fB ∗ fA ) = fA−1 ∗ fB−1 = fA ∗ fB . Consequently, by Theorem 3.28, fA ∗ fB is a soft int-group.

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4. Conclusions In this paper, we present soft int-groups on a soft set and give some of their supplementary properties. In addition, we give relations between α-inclusion, soft product and soft int-groups. This study affords us an opportunity to go further on soft group theory, that is, soft normal int-group, quotient group, isomorphism theorems etc. Acknowledgements. The author is highly grateful to Dr. Naim C ¸ a˘gman for his valuable comments and suggestions. References [1] S. Abou-Zaid, On fuzzy subgroups, Fuzzy Sets and Systems 55 (1993) 237–240. [2] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59 (20100 3458–3463. [3] M. Akg¨ ul, Some properties of fuzzy groups, J. Math. Anal. Appl. 133 (1988) 93–100. [4] H. Akta¸s and N. C ¸ a˘ gman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [5] M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) 1547–1553.

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Kenan Kaygisiz ([email protected]) Gaziosmanpa¸sa University, Faculty of Arts and Sciences, Department of Mathematics, Tokat, Turkey.

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