Fuzzy soft sets as L -fuzzy sets

Abstract

In this paper, it is shown that an L-fuzzy soft set on X with a set E of parameters can be regarded as an L^X-fuzzy set on E, and it can also be regarded as an L^E-fuzzy set on X. Moreover an L^E-fuzzy set on X can also be regarded as an L-fuzzy sets on E × X. In particular, a soft set on X with a set E of parameters actually can be regarded as a 2^E-fuzzy set or a crisp subset of E × X. This shows that the concept of (fuzzy) soft set is redundant. Further it is shown that the notions of soft groups, soft rings and fuzzy soft semigroups are respectively special cases of the corresponding L-fuzzy groups, L-fuzzy rings and L-fuzzy semigroups. Therefore, theoretical workers should stop working in these restrictive and complicated programs, while applied workers should transfer the need or appropriateness of (fuzzy) soft set notions to lattice-valued fuzzy sets or fuzzy sets of type 2.

Keywords

soft set fuzzy set of type 2 complete lattice isomorphism

1 Introduction

In 1965, Zadeh [43] introduced the notion of fuzzy sets. In 1967, Goguen [17] generalized fuzzy sets to L-fuzzy sets. In 1975, Zadeh further introduced the notion of fuzzy sets of type n in [44]. In 1968, Chang [9] applied fuzzy set theory into topology and gave the definition of fuzzy topologies. In 1971, Rosenfeld applied fuzzy set theory into algebra and gave the definition of fuzzy group.

In order to generalize fuzzy sets, Molodtsov [27] initiated the theory of soft sets, whose aim is to provide a new mathematical tool for dealing with some uncertainties that traditional tools can not handle efficiently. In [23], Maji et al. combined fuzzy sets and soft sets, and introduced fuzzy soft sets. To continue the investigation on fuzzy soft sets, Ahmat and Kharal [3] presented some more properties of fuzzy soft sets.

Research works on (fuzzy) soft set theory and their applications are progressing rapidly in various fields, including topology [6 , 41], algebra [1 , 42]. In particular, soft sets and fuzzy soft sets have been extensively applied to decision making [2 , 47], and so on. Moreover soft sets and fuzzy soft set are also applied to rough sets and interval-valued fuzzy sets [29 , 49].

As an application of fuzzy soft sets, Tanay et al. [40] introduced the topological structure of fuzzy soft sets, which is called fuzzy soft topology. Later, Roy et al. [33] and Varol et al. [41] independently modified the definition of fuzzy soft sets and redefined fuzzy soft topology. Subsequently, Çetkin and Aygün [8] began to study other fuzzy soft topological structures in the context of fuzzy soft topological spaces.

As pointed out in [37, 38], soft topologies and fuzzy soft topologies are redundant and unnecessarily complicated in theoretical sense.

In 2007, Aktaş and Çaǧan introduced the notion of soft groups in [4]. In 2010, Acar, Koyuncu and Tanay introduced the notion of soft rings in [1]. In 2011, Yang introduced the notion of fuzzy soft semigroups and fuzzy soft ideals in [42]. Although (fuzzy) soft set approach has been applied to algebra and decision making and so on, there is one major question concerning the theoretical side of these programs of research:

Question: Are the ideas of (fuzzy) soft sets, (fuzzy) soft group and (fuzzy) soft rings mathematically redundant, and if so, at which level of discussion does the redundancy occur?

It is the purpose of this paper to resolve this question. It is pointed out that both soft sets and fuzzy soft sets are redundant. Further the notions of fuzzy soft semigroups, fuzzy soft groups and fuzzy soft rings are redundant.

2 Preliminaries

Let X be an initial universe set, E be a set of parameters and L be a complete lattice.

Definition 2.1. ([17, 43]) A fuzzy set A on X is a mapping A : X → [0, 1]. An L-fuzzy set A on X is a mapping A : X → L.

Definition 2.2. ([44]) A fuzzy set is of type 2, if its membership function ranges over fuzzy sets.

Remark 2.3. Let L^X denote the set of all fuzzy sets on X. The operations ≤, ∨ , ∧ in L are pointwisely extended to L^X as follows:

A ≤ B ⇔ A (x) ≤ B (x) for all x ∈ X.

(∨ _i∈ΩA_i) (x) = ∨ _i∈ΩA_i (x) for all x ∈ X.

(∧ _i∈ΩA_i) (x) = ∧ _i∈ΩA_i (x) for all x ∈ X.

It is easy to check that (L^X, ≤) is also a complete lattice.

In this paper, it is worth noting that we will not distinguish a crisp set from its characteristic function.

Definition 2.4. ([16]) An L-subset A of a semigroup S is called an L-subsemigroup of S if, for any a ∈ L, the a-cut A_[a] = {x ∈ S ∣ A (x) ≥ a} is a subsemigroup of S.

One can prove [16, 32] that A is an L-fuzzy subsemigroup if and only if, for any x ∈ S, A (xy) ≥ A (x) ∧ A (y).

Definition 2.5. ([28, 32]) Let G be a group and A ∈ L^G. Then A is said to be an L-fuzzy subgroup of G if A satisfies the following conditions:

(1) A (xy) ≥ A (x) ∧ A (y).

(2) A (x^-1) ≥ A (x).

Definition 2.6. ([21]) An L-fuzzy ring is a mapping μ : R → L, where (R, + , ·) is a ring, that satisfies:

(1) μ≢0.

(2) μ (x - y) ≥ μ (x) ∧ μ (y) for every x, y in R.

(3) μ (xy) ≥ μ (x) ∧ μ (y) for every x, y in R.

Theorem 2.7. ([36]) Let G be a group and A ∈ L^G, where L is a completely distributive lattice. Then the following conditions are equivalent.

A is an L-fuzzy subgroup of G.

∀a ∈ L, A_[a] is a subgroup of G, where A_[a] = {g ∈ G ∣ A (g) ≥ a}.

For all nonzero coprime element a in L, A_[a] is a subgroup of G.

Definition 2.8. ([27]) A pair (f, D) is called a soft set over X with a set E of parameters if D ⊆ E and f is a mapping given by f : A → 2^X, where 2^X is the powerset of X.

Definition 2.9. ([23]) A pair (f, D) is called a fuzzy soft set over X with a set E of parameters if D ⊆ E and f is a mapping given by f : D → [0, 1] ^X.

Researchers originally defined soft sets with respect to the subsets of the parameter set E. That is to say, for a soft set (f, D), the domain of the mapping f is D, which is a subset of E. This way causes some troubles sometimes. In order to study similarity measure of soft set, Majumdar and Samanta [25] extended a soft set (f, D) to a soft set type (f, E) and obtain the following definition.

Definition 2.10. ([25]) A soft set f_D on X with a set E of parameters is a mapping from E to 2^X, i.e., f_D : E → 2^X, where f_D (e)≠ ∅ if e ∈ D ⊆ E and f_D (e) =∅ if e ∉ D.

Analogous to the above ideas, the definition of “fuzzy soft set” was presented respectively in [33, 41] as follows.

Definition 2.11. ([33, 41]) A fuzzy soft set f_D on X with a set E of parameters is a mapping from E to I^X, i.e., f_D : E → I^X, where $f_{D} (e) \neq \underline{0}$ if e ∈ D ⊆ E and $f_{D} (e) = \underline{0}$ if e ∉ D.

Remark 2.12. Definitions 2.9 and 2.11 are different. However, every soft set (f, D) can be regarded as type f_D or f_E. Moreover, many researchers investigated fuzzy soft topologies by means of fuzzy soft sets in the sense of Definition 2.11. In the sequel, we will also adapt Definition 2.11.

In [33, 41], partial orders and algebraic operations of fuzzy soft sets are defined as follows.

Definition 2.13. Let f_A, g_B be two fuzzy soft sets on X with a set E of parameters. Then

f_A is called a fuzzy soft subset of g_B, if f_A (e) ≤ g_B (e) for all e ∈ E, written as f_A ⊑ g_B. f_A and g_B are said to be equal, denoted by f_A = g_B, if f_A ⊑ g_B and g_B ⊑ f_A.

The union of f_A and g_B, denoted by f_A ⊔ g_B, is the fuzzy soft set h_A∪B defined by h_A∪B (e) = f_A (e) ∨ g_B (e) for all e ∈ E. That is, h_A∪B = f_A ⊔ g_B.

The intersection of f_A and g_B, denoted by f_A ⊓ g_B, is the fuzzy soft set h_A∩B defined by h_A∩B (e) = f_A (e) ∧ g_B (e) for all e ∈ E. That is, h_A∩B = f_A ∧ g_B.

3 Soft sets and fuzzy soft sets as L-fuzzy sets

Definition 2.11 can be easily generalized to L-fuzzy setting as follows.

Definition 3.1. An L-fuzzy soft set on X with a set E of parameters is a map from E to L^X. The set of all L-fuzzy soft sets on X with a set of parameters is denoted by (L^X) ^E.

Remark 3.2. From Remark 2.3, we easily know that (L^X, ≤ , ∨ , ∧) is a complete lattice. Thus the operations ≤, ∨ , ∧ in L^X can be pointwisely extended to (L^X) ^E as follows.

Let f, g ∈ (L^X) ^E. f is said to be an L-fuzzy soft subset of g, denoted by f ⊑ g, if f (e) ≤ g (e) for all e ∈ E.

The union f of a family of L-fuzzy soft sets {f_j ∣ j ∈ J} is defined by $\forall e \in E, f (e) = \lor {f_{j} (e) ∣ j \in J} .$ f is written as f = ⊔ _j∈Jf_j or f = ⊔ {f_j ∣ j ∈ J}.

The intersection f of a family of L-fuzzy soft sets {f_j ∣ j ∈ J} is defined by $\forall e \in E, f (e) = \land {f_{j} (e) ∣ j \in J} .$ f is written as f = ⊓ _j∈Jf_j or f = ⊓ {f_j ∣ j ∈ J}.

It is easy to check that ((L^X) ^E, ⊑ , ⊔ , ⊓) is a complete lattice.

Remark 3.3. In [41], the authors gave the axioms of fuzzy soft topology in which the arbitrary union of fuzzy soft sets were needed. However, they did not gave the concrete form of this operation. It is easy to check that Definition 2.13 is the special case of Remark 3.2 when J = {1, 2} and L = [0, 1].

From Remark 3.2 we know that (L^X) ^E is exactly the set of all L^X-fuzzy sets on E, the operations ⊑, ⊔ and ⊓ are exactly the operations of L^X-fuzzy sets on E. Therefore an L-fuzzy soft set (in particular, fuzzy soft set and soft set) is exactly an L^X-fuzzy sets on E. Next we shall prove that an L^X-fuzzy sets on E can actually be regarded as an L-fuzzy set on E × X.

Theorem 3.4. Let X and E be two nonempty sets. Then ((L^X) ^E, ⊑ , ⊔ , ⊓), ((L^E) ^X, ⊑ , ⊔ , ⊓) and (L^X×E, ≤ , ∨ , ∧) are isomorphic as complete lattices.

Proof. (1) First we prove the following isomorphism. $({(L^{X})}^{E}, ⊑, ⊔, ⊓) ≅ ({(L^{E})}^{X}, ⊑, ⊔, ⊓) .$ Define two mappings φ : (L^X) ^E → (L^E) ^X and ψ : (L^E) ^X → (L^X) ^E satisfying the following conditions:

∀f ∈ (L^X) ^E, ∀ x ∈ X, ∀e ∈ E and F ∈ (L^E) ^X, $((φ (f)) (x)) (e) = (f (e)) (x)$ and $((ψ (F)) (e)) (x) = (F (x)) (e) .$ Then obviously we have φ (f) ∈ (L^E) ^X, (φ (f) ) (x) ∈ L^E, ((φ (f)) (x) ) (e) ∈ L; ψ (F) ∈ (L^X) ^E, (ψ (F) ) (e) ∈ L^X, ((ψ (F)) (e) ) (x) ∈ L.

Now we prove that φ : (L^X) ^E → (L^E) ^X is a complete lattice isomorphism and its inverse mapping is ψ : (L^E) ^X → (L^X) ^E.

In fact, ∀f ∈ (L^X) ^E, ∀ x ∈ X, ∀e ∈ E and ∀F ∈ (L^E) ^X, we have the following two equalities. $\begin{matrix} (ψ \circ φ (f) (e)) (x) & = & (ψ (φ (f)) (e)) (x) \\ = & (φ (f) (x)) (e) & = & (f (e)) (x) \end{matrix}$ and $\begin{matrix} (φ \circ ψ (F) (x)) (e) & = & (φ (ψ (F)) (x)) (e) \\ = & (ψ (F) (e)) (x) & = & (F (x)) (e) . \end{matrix}$ This implies ψ ∘ φ = 1_{L
^X} and φ ∘ ψ = 1_{L
^E}. Therefore ψ is a bijective mapping and its inverse mapping is φ.

Considering the preservation of arbitrary ⊔ and arbitrary ⊓, let {f_j ∣ j ∈ J} ⊆ (L^X) ^E. Denote f = ⊔ _j∈Jf_j and g = ⊓ _j∈Jf_j. Then we have $\begin{matrix} (φ (f) (x)) (e) & = & (φ (⊔_{j \in J} f_{j}) (x)) (e) \\ = & ((⊔_{j \in J} f_{j}) (e)) (x) \\ = & (\lor_{j \in J} f_{j} (e)) (x) \\ = & \lor_{j \in J} (f_{j} (e)) (x) \\ = & \lor_{j \in J} (φ (f_{j}) (x)) (e) \\ = & (⊔_{j \in J} φ (f_{j}) (x)) (e) \end{matrix}$ and $\begin{matrix} (φ (g) (x)) (e) & = & (φ (⊓_{j \in J} f_{j}) (x)) (e) \\ = & ((⊓_{j \in J} f_{j}) (e)) (x) \\ = & (\land_{j \in J} f_{j} (e)) (x) \\ = & \land_{j \in J} (f_{j} (e)) (x) \\ = & \land_{j \in J} (φ (f_{j}) (x)) (e) \\ = & (⊓_{j \in J} φ (f_{j}) (x)) (e) . \end{matrix}$ This proves φ preserves arbitrary ⊔ and arbitrary ⊓. Therefore φ : (L^X) ^E → (L^E) ^X is a complete lattice isomorphism and its inverse mapping is ψ : (L^E) ^X → (L^X) ^E.

(2) Next we prove the following isomorphism. $({(L^{E})}^{X}, ⊑, ⊔, ⊓) ≅ (L^{E \times X}, \leq, \lor, \land) .$ Define two mappings ι : (L^E) ^X → L^E×X and κ : L^E×X → (L^E) ^X satisfying the following conditions:

∀F ∈ (L^E) ^X, ∀ x ∈ X, ∀e ∈ E and ∀G ∈ L^E×X, $(ι (F)) (e, x) = (F (x)) (e)$ and $((κ (G)) (x)) (e) = G (e, x) .$ Analogous to the proof of (1) we can prove that ι : (L^E) ^X → L^E×X is a complete lattice isomorphism and its inverse mapping is κ : L^E×X → (L^E) ^X.□

Remark 3.5. By Remark 3.2 and Theorem 3.4 we know that an L-fuzzy soft set on X with a set E of parameters is exactly an L^X-fuzzy set on E. It can also be regarded as an L^E-fuzzy set on X and an L^E-fuzzy set on X can also be regarded as an L-fuzzy sets on E × X.

When L = [0, 1], we can obtain the following result.

Corollary 3.6. Let X and E be two nonempty sets. Then (([0, 1] ^X) ^E, ⊑ , ⊔ , ⊓) ≅ (([0, 1] ^E) ^X, ⊑ , ⊔ , ⊓) ≅ ([0, 1] ^X×E, ≤ , ∨ , ∧).

In particular, when L = {0, 1} = 2, we have the following corollary.

Corollary 3.7. Let X and E be two nonempty sets. Then ((2^X) ^E, ⊑ , ⊔ , ⊓) ≅ ((2^E) ^X, ⊑ , ⊔ , ⊓) ≅ (2^X×E, ⊆ , ⋃ , ⋂).

Remark 3.8. From Corollary 3.6 and Corollary 3.7 we know that a fuzzy soft set on X with a set E of parameters actually is a [0, 1] ^E-fuzzy set which is a fuzzy set of type 2 on X. It can be regarded as a fuzzy set on E × X. A soft set on X with a set E of parameters actually can be regarded as a 2^E-fuzzy set or a crisp subset of E × X.

Claim 1. Theorem 3.4 and Remark 3.8 imply that the concept of fuzzy soft set is redundant. (Fuzzy) soft set theory is a special subcase of fuzzy set theory. Analogously the notion of type 2 soft sets in [11] is also redundant.

4 Soft groups and fuzzy soft groups as L-fuzzy groups

In this section, we shall prove that soft groups and fuzzy soft groups can be regarded as special cases of L-fuzzy groups.

Definition 4.1. ([4]) Let (F, A) be a non-null soft set over a group G. Then (F, A) is called a soft group over G if F (x) is a subgroup of G for all x ∈ A.

Definition 4.2. ([1]) Let (F, A) be a non-null soft set over a ring R. Then (F, A) is called a soft ring over R if F (x) is a subring of R for all x ∈ A.

Definition 4.3. ([42]) Let (F, A) be a non-null fuzzy soft set over a semigroup S. Then (F, A) is called a fuzzy soft semigroup over S if F (x) is a fuzzy subsemigroup of S for all x ∈ A.

Theorem 4.4. Let (F, A) be a non-null soft set over a group G and φ is the ismorphism from (2^G) ^E to (2^E) ^G as in Theorem 3.4. Then (F, A) is a soft group over G if and only if φ (F) is a 2^E-fuzzy subgroup of G.

Proof. (Necessity). Let (F, A) is a soft group over G. Then for all x ∈ A, F (x) is a subgroup of G. For all x ∈ E - A, we define F (x) =∅ to be a subgroup of G. Now we prove that φ (F) is a 2^E-fuzzy subgroup of G. By Theorem 2.7, we only need to prove that for all nonzero coprime element a in lattice 2^E, the level cut set φ (F) _[a] is a subgroup of G. As we know, the set of all nonzero coprime elements in 2^E is exactly the set E. Thus we only need to prove that for all e ∈ E, the level cut set φ (F) _[{e}] is a subgroup of G. In fact, we have the following equality. $\begin{matrix} φ (F)_{[{e}]} & = & {g \in G ∣ {e} \subseteq φ (F) (g)} \\ = & {g \in G ∣ φ (F) (g) (e) = 1} \\ = & {g \in G ∣ F (e) (g) = 1} \\ = & F (e) . \end{matrix}$ This shows that φ (F) is a 2^E-fuzzy subgroup of G since F (e) is subgroup of G for each e ∈ E.

(Sufficency). Suppose that φ (F) is a 2^E-fuzzy subgroup of G. Then for all a ∈ 2^E, φ (F) _[a] is a subgroup of G. Especially φ (F) _[{e}] is a subgroup of G for all e ∈ E. By φ (F) _[{e}] = F (e) we know that F (e) is a subgroup of G for all e ∈ E. Hence (F, A) is a soft group of G.□

Analogous to Theorem 2.7 we have the following theorem.

Theorem 4.5. Let R be a ring and A ∈ L^G, where L is a completely distributive lattice. Then the following conditions are equivalent.

A is an L-fuzzy subring of G.

∀a ∈ L, A_[a] is a subring of G.

For all nonzero coprime element a in L, A_[a] is a subring of G.

By Theorem 4.5 and analogous to Theorem 4.4 we can obtain the following result.

Theorem 4.6. Let (F, A) be a non-null soft set over a ring R and φ is the ismorphism from (2^R) ^E to (2^E) ^R as in Theorem 3.4. Then (F, A) is a soft ring over R if and only if φ (F) is a 2^E-fuzzy subring of R.

Analogously we can obtain the following theorem.

Theorem 4.7. Let (F, A) be a non-null fuzzy soft set over a semigroup S and φ is the ismorphism from ([0, 1] ^S) ^E to ([0, 1] ^E) ^S as in Theorem 3.4. Then (F, A) is a fuzzy soft semigroup over S if and only if φ (F) is a [0, 1] ^E-fuzzy subsemigroup of S.

Claim 2. Theorem 4.4, Theorem 4.6 and Theorem 4.7 imply that the notions of soft groups, soft rings and fuzzy soft semigroups are redundant. They can be regarded as special cases of L-fuzzy (semi)groups and L-fuzzy rings.

5 Conclusions

As judged by the above criteria, we claim that a soft set f on X with a set E of parameters is exactly a 2^E-fuzzy sets on X and a fuzzy soft set on X with a set E of parameters is exactly a [0, 1] ^E-fuzzy sets on X, i.e., a fuzzy set of type 2. The notions of soft groups, soft rings and fuzzy soft semigroups are special cases of the corresponding L-fuzzy groups, L-fuzzy rings and L-fuzzy semigroups. Therefore, theoretical workers should stop working in these restrictive and complicated programs, while applied workers should transfer the need or appropriateness of (fuzzy) soft set notions to lattice-valued fuzzy sets or type 2 fuzzy sets.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation (11871097).

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