A NEW APPROACH TO GROUP THEORY VIA SOFT SETS AND L-FUZZY SOFT SETS

This paper aims to extend the notion of group to inside the algebraic structures of L-fuzzy soft sets. We firstly give some new notions such as product, extended product, restricted product of two L-fuzzy soft sets. By using these new notions we then introduce concept of L-fuzzy soft groups and study some of their properties. We also compare L-fuzzy soft groups to the related concept of soft groups. Furthermore, we show that L-fuzzy soft groups are more general concept than soft groups. We finally define L-fuzzy soft function and L-fuzzy soft group homomorphism, and then give theorem of homomorphic image and homomorphic pre-image under a L-fuzzy soft function. AMS Subject Classification: 08A72, 20N25


Introduction
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 uncertainties present in these problems.While probability theory, fuzzy set theory [20], rough set theory [18], and other mathematical tools are well-known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [16].In 1999, Molodtsov [16] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties.This so-called soft set theory is free from the difficulties affecting the existing methods.
At present, works on the soft set theory are making progress rapidly.Maji et al. [14] studied several operations on the theory of soft sets.Ali et al. [3] gave some new notions such as restricted intersection, restricted union, restricted difference and extended intersection of two soft sets.Some researches have studied algebraic properties of soft sets.Initially, Aktaş and C ¸agman [2] compared soft sets to the related concepts of fuzzy sets and rough sets.They also defined the notion of soft groups and derived their basic properties using Molodtsov's definition of the soft sets.Feng et al. [6] introduced the notions of soft semirings, soft ideals and idealistic soft semirings, and then investigated several related properties.Kazancı et al. [12] defined soft BCH-algebras and gave the theorems of homomorphic image and homomorphic pre-image of soft sets.Acar et al. [1] defined soft rings, and introduced basic notions of soft rings.C ¸elik et al. [5] defined some new binary relation on soft sets, and also they investigated some new properties of soft rings.Yamak et al. [19] introduced the notion of soft hypergroupoids.Some researches have studied algebraic properties of fuzzy soft sets.Firstly, Maji et al. [13] defined fuzzy soft set, and established some results on them.Jin-liang et al. [10] defined the operations on fuzzy soft groups, and proved some results on them.Aygünoglu and Aygün [4] gave the concept of fuzzy soft group, and defined fuzzy soft function and fuzzy soft homomorphism.Majumdar and Samanta [15] defined generalised fuzzy soft sets, and studied some of their properties.They also showed applications of generalised fuzzy soft sets.Jun et al. [11] derived the notion of fuzzy soft BCK/BCI algebras, and investigated its properties.Jiang et al. [9] proposed the notion of the interval-valued intuitionistic fuzzy soft set theory.Lastly, İnan and Öztürk [7] introduced the fuzzy soft ring, and (∈, ∈ ∨ q)-fuzzy soft subring.They also studied some of their basic properties.
In this paper, we introduce to group theory via soft sets and L-fuzzy soft sets.
The rest of this paper is organized as follows.In Section 2, we give the concepts of L-fuzzy subset, soft set, soft group and L-fuzzy soft set which will be used throughout the paper.We give some new notions such as product, extended product, restricted product of two L-fuzzy soft sets.In Section 3, we introduce the notion of L-fuzzy soft group and study its characteristic properties.We then give some related examples.We also investigated relations between soft groups and L-fuzzy soft groups.Furthermore, we show that Lfuzzy soft groups are more general concept than soft groups.Lastly, we define L-fuzzy soft function and L-fuzzy soft group homomorphism, and then give theorem of homomorphic image and homomorphic pre-image under a L-fuzzy soft function.

Preliminaries
In this section, we will give some known and useful definitions and notations regarding L-fuzzy subset, L-fuzzy subgrop, soft set, soft group, L-fuzzy soft set.
Throughout this paper, let (G, •) be a group, and L be a complete lattice.
Let G and H be two groups.A mapping f : G → H is called a homomorphism of groups if it satisfies for all a, b ∈ G.That is, the mapping f preserves the group operations.A group homomorphism f : G → H is called a monomorphism [resp.epimorphism, isomorphism] if it is an injective [resp.surjective, bijective] mapping.The sets Ker f = {g ∈ G | f (g) = e H } and Im f = {h ∈ H | f (g) = h for some g ∈ G} are called kernel and image of f , respectively.
A L-fuzzy subset µ of G is defined as a map from G to L. If L = [0, 1], then L-fuzzy subset is called fuzzy subset.The family of all L-fuzzy subsets of G is denoted by FL(G).The following are most popular operations on L-fuzzy subsets: ∀µ, ν ∈ FL(G), ω ∈ FL(H) and x ∈ G, y ∈ H; µ ≤ ν if and only if µ(x) ≤ ν(x) for all x ∈ G.For T ⊆ G, χ T ∈ FL(G) is called characteristic function of T , and defined by χ T (x) = 1 if x ∈ T and χ T (x) = 0 otherwise for all x ∈ G.

Soft Sets and Soft Groups
Molodtsov [16] defined the notion of a soft set in the following way: Let G be an inital universe set and E be a set of parameters.The power set of G is denoted by P(G) and A is a subset of E. A pair (F, A) is called a soft set over G, where F is a mapping given by F : A → P(G).In other words, a soft set over G is a parameterized family of subsets of the universe G.For x ∈ A, F (x) may be considered as the set of x-approximate elements of the soft set (F, A).Definition 6. [16] Let (F, A) and (K, B) be two soft sets over G.Then, (F, A) is said to be a soft subset of (K, B), [6]) Let (F, A) and (K, B) be two soft sets over G.Then, (1) The restricted intersection of soft sets (F, A) and (K, B) is defined as (2) The extended union of soft sets (F, A) and (K, B) is defined as the soft set (N, C) = (F, A) ∪ (K, B) over G, where C = A ∪ B, and If L is choosen closed unit interval [0, 1], then Definition 9 coincides with [2].
Proof.By Theorem 1, the proof can be achieved.
Corollary 12.By Theorem 11 (1), the subgroups of G can be embedded into L-fuzzy subgroups of G.By Theorem 11 (2), If L is chain, then L-fuzzy sublattice of G can be taken L-parameter soft group.
On the other words, soft groups are more general concept than subgroups and L-fuzzy subgroups.

L-Fuzzy Soft Sets and Operations
In this part, we define the concept of L-fuzzy soft sets, and give some notations regarding of them.
Definition 13.Let G be a common universe, E be a set of parameters and A ⊆ E. Then a pair (F, A) is called a L-fuzzy soft set over G, where F is a mapping given by F : A → FL(G).
Example 1.Let G = {h 1 , h 2 , h 3 , h 4 } be the set of four houses under consideration, E = {e 1 (costly), e 2 (beautiful), e 3 (green surroundings), e 4 (wooden), e 5 (luxurious be the set of parameters and A = {e 1 , e 2 , e 3 } ⊂ E. Let α i ∈ L, i ∈ {1, 2, ..., 9}, where α i ≤ α j for all i ≤ j.Then is the L-fuzzy soft set representing the 'attractiveness of the house' which Mr. X is going to buy. Definition 14.For two L-fuzzy soft sets (F, A) and (K, B) over a common universe G , we say that (F, A) is a L-fuzzy soft subset of (K, B) if    where C = A ∪ B = {1, 2, 3}, and for all n ∈ Z 4 , and for all n ∈ Z 4 (1,3), (2, 2), (2, 3)}, and for all n ∈ Z 4 (1,3), (2, 2), (2, 3)}, and for all n ∈ Z 4 Now, we can define a genaral binary operation on L-fuzzy soft sets in the following way: Suppose that ⊕ is a binary operation on P (E), and ⊗ is a binary operation on FL(G).Then for any two L-fuzzy soft set (F, A) and (K, B) over G, (F, A) ⊕ ⊗ (K, B) is defined as the L-fuzzy soft set (N, C) where C = A ⊕ B and Here we describe a general binary operation on L-fuzzy soft sets obtained as a special case of the some binary relations such as extended union, restricted union, extended intersection,restricted intersection, restricted product and extended product for the ⊕ ∈ {∪, ∩} and ⊗ ∈ {∨, ∧, •}.

A New Approach to Group Theory via Soft Sets and L-Fuzzy Soft Sets
The notion of fuzzy soft groups was introduced in ([4], [10]).In this section, we introduce the concept of L-fuzzy soft groups which extends the notion of group to include the algebraic structures of L-fuzzy soft sets, and give some new fundamental properties of L-fuzzy soft groups.
Definition 16.Let (F, A) be a L-fuzzy soft set over G. Then (F, A) is called a L-fuzzy soft group over G if F (a) is a L-fuzzy subgroup of G for all a ∈ A.
Example 3. Let µ ∈ FL S (G).Then the soft set (H, L) over G, defined by H(α) = χ µα for all α ∈ L, is a L-fuzzy soft group over G.
Clearly µ α ≺ G.By Theorem 4, χ µα is L-fuzzy subgroup over G. Hence (H, L) is a L-fuzzy soft group over G.
Example 4. Let L = {0, α, β, 1} be a lattice, where 0 Then it is easy to see that the L-fuzzy soft set (F, Example 5. Let G be a group.Let (F, N) be a L-fuzzy soft set over G defined by Proof.By Theorem 4, the proof can be achieved.
Theorem 18.Let U and U be a family of all soft groups and L-fuzzy soft groups over G, respectively.Then the mapping γ : U → U , defined by γ(F, L) = (F , L), is a one-to-one lattice homomorphism.
Proof.From the proof of Theorem 11 (2) and by Theorem 17, the proof can be achieved.
Corollary 19.By Theorem 18, a soft group can be embedded into L-fuzzy soft groups.
On the other words, L-fuzzy soft groups are more general concept than soft groups.
Theorem 20.Let (F, A) and (K, B) be two L-fuzzy soft groups over G.Then, (1) Proof. ( (2) It is similar to the proof of (1) Theorem 22.Let (F, A) and (K, A) be two L-fuzzy soft groups over G. Then (F, A) ⊙(K, A) is a L-fuzzy soft group over G if and only if Proof.By Theorem 5, the proof can be achieved.Proof.Let (F, A) be a L-fuzzy soft group over G. Then F (a) is a Lfuzzy subgroup of G for all a ∈ A. Also (F (a) • F (a))(x) ≤ F (a)(x) for all x ∈ G. Hence (F, A) ⊙ ∩ (F, A) ⊆(F, A).On the other hand, since Theorem 24.Let (F, A) be a L-fuzzy soft set over G. Then (F, A) is a L-fuzzy soft group over G if and only if (F, A) ⊙ ∩ (F, A) −1 ⊆(F, A).
Proof.It is similar to the proof of Theorem 23.
In this definition, If φ is a group homomorphism from G 1 to G 2 , then (φ, ψ) is said to be a L-fuzzy soft group homomorphism, and that (F, A) is L-fuzzy soft homomorphic to (K, B).The latter is denoted by (F, A) ∼ (K, B).If φ is an isomorphism from G 1 to G 2 and ψ is a bijective mapping from A onto B, then we say that (φ, ψ) is a L-fuzzy soft group isomorphism and that (F, A) is L-fuzzy soft isomorphic to (K, B).The latter is denoted by (F, A) ≃ (K, B).Theorem 28.Let (F, A) and (K, B) be two L-fuzzy soft groups over G 1 and G 2 , respectively.Let (φ, ψ) be a L-fuzzy soft group homomorphism from (F, A) to (K, B).Then, (1) If φ : G 1 → G 2 is a epimorphism of groups and ψ is a bijective mapping, then (φ(F ), B) is a L-fuzzy soft group over G 2 .
Proof.(1) Let y ∈ B. Since ψ is bijective, then there exist a unique x ∈ A such that ψ(x) = y.Since F (x) is a L-fuzzy subgroup of G 1 and φ epimorphism, then φ(F (x)) is a L-fuzzy subgroup of G 2 .Also, φ(F )(y) = φ(F (x)) is a L-fuzzy subgroup over G 2 .Hence, (φ(F ), B) is a L-fuzzy soft group over G 2 .
(2) Since ψ(x) ∈ B for all x ∈ A and (K, B) is a L-fuzzy soft group over G 2 , then K(ψ(x)) is a L-fuzzy subgroup of G 2 for all x ∈ A. Also, its homomorphic inverse image φ −1 (K(ψ(x))) is also a L-fuzzy subgroup of G 1 for all x ∈ A. Hence (φ −1 (K), A) is a L-fuzzy soft group over G 1 .

Conclusion
In this paper we studied the algebraic properties of L-fuzzy soft sets in group

( 1 )
for all a ∈ A. In this case, we write (F, A) ⊆(K, B).Definition 15.Let (F, A) and (K, B) be two L-fuzzy soft sets over G.Then, The extended union of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ∪(K, B) over G, where C = A ∪ B and

( 2 )( 3 )
The restricted union of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ∪ ℜ (K, B) over G, where C = A ∩ B, and N (c) = F (c) ∨ K(c) for all c ∈ C. The extenden intersection of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ∩ ε (K, B) over G, where C = A∪B, and

( 4 )( 5 )( 6 )( 9 )
The restricted intersection of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ∩(K, B) over G, where C = A ∩ B, and N (c) = F (c) ∧ K(c) for all c ∈ C. The ∧-intersection of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ∧(K, B) over G, where C = A × B, and N (a, b) = F (a) ∧ K(b) for all (a, b) ∈ A × B. The ∨-union of L-fuzzy soft sets (F, A) and (K, B) is defined as the Lfuzzy soft set (N, C) = (F, A) ∨(K, B) over G, where C = A × B, and N (a, b) = F (a) ∨ K(b) for all (a, b) ∈ A × B. (7) The cartesian product of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ×(K, B) over G, where C = A × B, and N (a, b) = F (a) × K(b) for all (a, b) ∈ A × B. (8) The product of L-fuzzy soft sets (F, A) and (K, B) is defined as the Lfuzzy soft set (N, C) = (F, A) ⊙(K, B) over G, where C = A × B, and N (a, b) = F (a) • K(b) for all (a, b) ∈ A × B. The extended product of L-fuzzy soft sets (F, A) and (K, B) is defined as the L-fuzzy soft set (N, C) = (F, A) ⊙ ∪ (K, B) over G, where C = A ∪ B, and

( 4 )( 6 )
fuzzy subgroup of G for all c ∈ A ∩ B. Hence, (F, A) ∪(K, B) is a L-fuzzy soft group over G.It is similar to the proof of (3) (5) Let (F, A) ∧(K, B) = (N, A × B).We know that F (a) and K(b) are Lfuzzy subgroups of G for all a ∈ A, b ∈ B. Since intersection of two L-fuzzy subgroups is a L-fuzzy subgroup, then N (a, b) = F (a) ∧ K(b) is also L-fuzzy subgroup of G. Hence (F, A) ∧(K, B) is a L-fuzzy soft group over G.It is similar to the proof of (3).Theorem 21.Let (F, A) and (K, B) be two L-fuzzy soft groups over G 1 and G 2 , respectively.Then (F, A) ×(K, B) is a L-fuzzy soft group over G 1 × G 2 .Proof.Let (F, A) ×(K, B) = (N, C), where C = A × B, and for all (a, b) ∈ A × B, we have N (a, b) = F (a) × K(b).Since (F, A) and (K, B) are L-fuzzy soft groups over G 1 and G 2 , then F (a) and K(b) are L-fuzzy subgroups of G 1 and G 2 , respectively, for all (a, b) ∈ A × B. Also, for all x ∈ G 1 and y ∈ G 2 , we write (F (a) × K(b))(x, y) = F (a)(x) ∧ K(b)(y).Hence, we obtain N (a, b) = F (a) × K(b) is a L-fuzzy subgroup of G 1 × G 2 for all (a, b) ∈ A × B. Consequently, (F, A) ×(K, B) is a L-fuzzy soft group over G 1 × G 2 .

Theorem 23 .
Let (F, A) be a L-fuzzy soft set over G. Then (F, A) is a Lfuzzy soft group over G if and only if (F, A) ⊙ ∩ (F, A) ⊆(F, A) and (F, A) ⊆(F, A) −1 .